A library wants to determine the effectiveness of their summer literacy program among low-income children. Because surveying the large numbers of students in the program would require too many resources the library staff interviews 30 randomly chosen children among the low-income program attendees. The 30 sampled children are given a reading test before and after the program.A) The difference in the reading test scores (after – before) has mean 10 and standard deviation 4. Assuming the score differences are normally distributed, what percent of the children showed any improvement (difference > 0) in reading ability?B) What percent of children improved by more than 15 points?
Answers
Answer:
(A) P (D > 0) = 99.38%
(B) P (D > 15) = 10.56%
Step-by-step explanation:
The random variable D = difference, is defined as the difference between the reading test scores after and before the program.
The random variable D follows a normal distribution with mean, [tex]\mu_{D}=10[/tex] and standard deviation, [tex]\sigma_{D}=4[/tex].
(A)
Compute the probability that the children showed any improvement, i.e.
P (D > 0):
[tex]P(D>0)=P(\frac{D-\mu_{D}}{\sigma_{D}} >\frac{0-10}{4} )=P(Z>-2.5)=P(Z<2.5)[/tex]
Use the standard normal random variable to determine the probability.
[tex]P(D>0)=P(Z<2.5)=0.9938[/tex]
The percentage of children showed any improvement is:
0.9938 × 100 = 99.38%
Thus, 99.38% of children showed improvement.
(B)
Compute the probability that the children improved by more than 15 points, i.e. P (D > 15):
[tex]P(D>15)=P(\frac{D-\mu_{D}}{\sigma_{D}} >\frac{15-10}{4} )=P(Z>1.25)=1-P(Z<1.25)[/tex]
Use the standard normal random variable to determine the probability.
[tex]P(D>0)=1-P(Z<1.25)=1-0.8944=0.1056[/tex]
The percentage of children improved by more than 15 points is:
0.1056 × 100 = 10.56%
Thus, 10.56% of children showed improvement by more than 15 points.
50% of children showed some improvement, while 10.56% improved their reading scores by more than 15 points during the summer literacy program.
Explanation:The library staff is utilizing statistical analysis to assess the effectiveness of their summer literacy program. They have chosen a sample of 30 children out of the many who attended, and provided scores both before and after the program. A mean difference score of 10 and a standard deviation of 4 were determined. This question asks to find out the percentage of students who have improved based on these scores (positive score difference) and those who have improved by more than 15 points.
Firstly, we are assuming that the score differences follow a normal distribution. In a normal distribution, half of the results fall on either side of the mean. Since we are looking for an improvement, we only consider the side above the mean score difference, which is equivalent to 50% of all students.
Secondly, to find the percent of children who improved by more than 15 points, we need to calculate the z-score for the score difference of 15. Z-score is calculated as (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. So, Z = (15 - 10) / 4 = 1.25.
The z-score of 1.25 corresponds to an area of 0.8944 to the left under a standard table of normal distribution. To get the area to the right (which represents the students who improved by >15), we subtract this from 1. So, 1 - 0.8944 = 0.1056 or 10.56% students improved by more than 15 points.
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Related Questions
ln(x+1)3=3
ln
(
x
+
1
)
=
3
Question 7 options:
Answers
Answer:
x=e-1 0r 1.71828
Step-by-step explanation:
Step-by-step explanation:
[tex]In(x + 3)^{3} = 3 \\ \therefore \: 3In(x + 3) = 3 \:\\ \therefore \: In(x + 3) = \frac{3}{3} \: \\ \therefore \: In(x + 3) = 1\\ \therefore \: In(x + 3) = In \: 10\\..( \because \: In \: 10 = 1) \\ \therefore \:x + 3 = 10 \\ \therefore \:x = 10 - 3 \\ \: \: \: \: \: \huge \red{ \boxed{\therefore \:x = 7}}[/tex]
A manufacturer can produce digital recorders at a cost of 50 dollars each. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy q=120p recorders each month. a) Express the manufacturer's profit P as a function of q.
b)What is the average rate of profit obtained as the level of production increases from q=0 to q=15? (at dollars per unit)
c) At what rate is profit changing when q=15 recorders are produced? (at dollars per unit)
Answers
The manufacturer's profit is P(p) = 120p^2 - 6000p. The average profit for producing 0 to 15 recorders is 15 * 120. At 15 units, the rate of profit change is -3000.
Explanation:The function for the manufacturer's profit P is given by P(p) = pq - 50q where q=120p. Thus, P(p) = 120p * p - 50 * 120p, simplifying to P(p) = 120p^2 - 6000p.
To express the average rate of profit from q = 0 to q=15, we need to calculate P(15)-P(0) and divide it by 15 - 0. Thus, ((120 * 15^2 - 6000 * 15) - (120 * 0^2 - 6000 * 0)) / (15 - 0) = 15*120.
The rate at which profit is changing when q = 15 is the derivative of the profit function evaluated at q = 15. We obtain P'(q) = 240p - 6000, and by inserting 15 for p, we obtain -3000 (at dollars per unit).
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a) The manufacturer's profit can be expressed as P = (p - 50)q. b) The average rate of profit obtained as the level of production increases from q=0 to q=15 is p - 50 dollars per unit. c) When q=15 recorders are produced, the rate at which profit is changing is p - 50 dollars per unit.
Explanation:a) To express the manufacturer's profit P as a function of q, we can use the equation: P = (p - 50)q. This equation represents the total revenue minus the total cost, where the total revenue is the selling price p multiplied by the quantity q, and the total cost is the cost of production per unit (50 dollars) multiplied by the quantity q.
b) The average rate of profit obtained as the level of production increases from q=0 to q=15 can be found by calculating the change in profit divided by the change in quantity. In this case, the change in profit is (p - 50)(15) - (p - 50)(0) = 15(p - 50) dollars, and the change in quantity is 15 units. Therefore, the average rate of profit is (15(p - 50))/15 = p - 50 dollars per unit.
c) To find the rate at which profit is changing when q=15 recorders are produced, we can calculate the derivative of the profit function with respect to q. Taking the derivative of P = (p - 50)q with respect to q gives us dP/dq = p - 50 dollars per unit. So, when q=15, the rate at which profit is changing is p - 50 dollars per unit.
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Giorgio offers the person who purchases an 8-ounce bottle of Allure two free gifts, chosen from the following: an umbrella, a 1-ounce bottle of Midnight, a feminine shaving kit, a raincoat, or a pair of rain boots. If you purchased Allure, what is the probability you randomly selected an umbrella and a shaving kit in that order?a. 0.20 b. 0.00c. 0.05d. 1.00
Answers
Answer:
P(T) = 1/20 = 0.05
The probability of randomly selecting an umbrella and a shaving kit in that order is 0.05
Step-by-step explanation:
The probability of randomly selecting an umbrella and a shaving kit in that order.
P(T) = Probability of selecting umbrella first P(U) × probability of selecting shaving kit second P(S)
P(U) = 1/5 (1 umbrella out of five possible gifts)
P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)
P(T) = 1/5 × 1/4
P(T) = 1/20 = 0.05
The probability that a person will select an umbrella and a shaving kit in that order is 1/20.
The following can be deduced from the information given:
P(U) = 1/5 (1 umbrella out of five possible gifts)
P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)
Therefore, the probability that a person will select an umbrella and a shaving kit in that order will be:
P(T) = 1/5 × 1/4
P(T) = 1/20 = 0.05
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In a Caesar Cipher system, if encryption is done by the rule c = E(p+k) = p + 9, then the letter ‘q’ is encrypted as what other letter?
Answers
Answer:
i
Step-by-step explanation:
There has to be a mod 26 operation for c, since the alphabet only has 26 letters.
But otherwise
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
p is the initial position of the letter. So for example, a = 0, b = 1, c = 2, q = 16, z = 25...
9 is by how much the encrypted character will change relative to the original.
We want to cipher the letter q.
So p = 25.
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
[tex]c = 25 + 9 \mod{26}[/tex]
[tex]c = 34 \mod{26}[/tex]
The remainder of 34 divided by 26 is 8.
So
[tex]c = 8[/tex]
Which means that q is encrypted as 'i'.
In a Caesar Cipher system, when the encryption rule is shifting by 9 places, the letter 'q' is encrypted as 'c'.
Explanation:In a Caesar Cipher system, the letter 'q' is encrypted using the formula:
c = E(p + k)
Where:
c is the ciphertext (the encrypted letter).
p is the plaintext (the original letter).
k is the encryption key.
You mentioned that the encryption is done with the rule c = p + 9. In this case, k is 9.
To encrypt the letter 'q', you would substitute p with 'q' in the formula:
c = 'q' + 9
Adding 9 to 'q' in the alphabet results in:
'q' + 9 = 'z'
So, the letter 'q' is encrypted as the letter 'z' using this Caesar Cipher system.
Therefore, the encrypted letter for 'q' in a Caesar Cipher system with a shift of 9 is 'c'.
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In an area of the Midwest, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Which is the best predicted value for y given x = 7.3? Assume that the variables x and y have a significant correlation. 36.5 36.0 36.7 36.2
Answers
Answer:
The correct answer is 36.2
Step-by-step explanation:
Hello!
Given the data:
Rainfall(inch) x: 10.5, 8.8, 13.4, 12.5, 18.8, 10.3, 7.0, 15.6, 16.0
Yield (bushels per acre) y: 50.5, 46.2, 58.8, 59.0, 82.4, 49.2, 31.9, 76.0, 78.8
The response variable is
Y: Yield of wheat (bushels per acre)
Explanatory variable:
X: Rainfall in a certain period (inch)
I've calculated the equation of regression using a statistics software, the estimated equation is:
^Y= 4.27 + 4.38X
To calculate the value that will take the response variable for a given value of X:
^Y= 4.27 + 4.38*7.3= 36.24 bushels per acre.
I hope it helps!
Without the necessary data such as a regression equation or the correlation coefficient, we are unable to predict the yield of wheat when rainfall is 7.3 inches. A regression equation is a statistical tool that helps understand the relationship between predictor and response variables, in this case, rainfall and yield of wheat.
Explanation:In order to predict the value of yield of wheat (y) given the rainfall (x = 7.3), we would need to understand the relationship between x and y, typically via a regression equation. A regression equation is a statistical tool used to understand the relationship between predictor variables (in this case, rainfall) and response variables (in this case, yield of wheat).
However, from the details provided, it seems we do not have the necessary data to conduct a regression analysis or connect x and y values in a meaningful way. Therefore, we're unable to directly predict the specific yield of wheat when rainfall is 7.3 inches over this data gap.
An accurate value can only be predicted if we are provided with more data or the specific correlation coefficient between the two variables and the regression equation (y = ax + b).
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You go to Applebee’s and spend $98.42 on your meal. How much was the bill before 6% sales tax
Answers
(98.42•0.06)=5.9052
98.42-5.9052=92.5148
Answer: the bill before 6% sales tax is $92.85
Step-by-step explanation:
Let x represent the bill before the 6% sales tax.
It means that you paid 6% tax on x and the amount of tax paid would be
6/100 × x = 0.06 × x = 0.06x
Total amount that you paid for the meal including the 6% tax would be
x + 0.06x = 1.06x
If you spent $98.42 on the meal after the 6% tax, it means that
1.06x = 98.42
Dividing the left hand side and the right hand side of the equation by 1.06, it becomes
1.06x/1.06 = 98.42/1.06
x = $92.85
An urn contains 7 black and 9 green balls. Six balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 6 balls drawn from the urn are green? Round your answer to three decimal places.
Answers
Step-by-step explanation:
There are 16 balls total. Since the balls are replaced after selection, the probability that the ball is green is 9/16 each time. The probability that a green ball is selected 6 times is:
P = (9/16)^6
P = 0.032
A regular square pyramid has a height of 6 and a base with sides of length 12. What is the: Lateral area of the pyramid?
Answers
Answer: Lateral area of the pyramid is 144
Step-by-step explanation:
The formula for determining the lateral surface area of a regular pyramid is expressed as
Lateral area = (perimeter of base x slant height of pyramid) /2
From the information given,
Length of base = 12
Since the base is a square, the perimeter of the base is
4L = 4 × 12 = 48
Height of pyramid = 6
Therefore, lateral surface area of the pyramid is
(48 × 6)/2 = 288/2 = 144
Lateral area of square pyramid with height 6 and base side 12 is 144 square units.
To find the lateral area of a square pyramid, you need to calculate the area of each triangular face and then sum them up.
1. Find the slant height (l):
In a regular square pyramid, the slant height (l) can be found using the Pythagorean theorem. Each triangular face is an isosceles right triangle. So,
[tex]\[ l = \sqrt{(s/2)^2 + h^2} \][/tex]
Where:
[tex]- \( s \) is the length of one side of the square base (given as 12 in this case).- \( h \) is the height of the pyramid (given as 6).[/tex]
[tex]\[ l = \sqrt{(12/2)^2 + 6^2} \]\[ l = \sqrt{(6)^2 + 6^2} \]\[ l = \sqrt{36 + 36} \]\[ l = \sqrt{72} \]\[ l = 6\sqrt{2} \][/tex]
2. **Calculate the lateral area (A):**
Each triangular face has a base of length 12 and a height of 6 (same as the height of the pyramid). So, the area of each triangular face (A) can be calculated as:
Since there are four triangular faces in a square pyramid, the total lateral area (LA) is:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]\[ A = \frac{1}{2} \times 12 \times 6 \]\[ A = 36 \][/tex] [tex]\[ LA = 4 \times A \]\[ LA = 4 \times 36 \]\[ LA = 144 \][/tex]
So, the lateral area of the square pyramid is 144 square units.
Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (Enter your answer using interval notation.) (16 − t2)y' + 9ty = 5t2, y(−5) = 1
Answers
Answer: t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )
Step-by-step explanation:
The standard form of O.D.E is written as :
[tex]y^{1}[/tex] + [tex]p(t) = g(t)[/tex]
Equation given :
[tex](16-t^{2} )y^{1}[/tex] + [tex]9ty[/tex] = [tex]5t^{2}[/tex] , [tex]y(-5) = 1[/tex]
The first thing to do is to write the O.D.E in standard form , that is we will divide through by [tex]16 - t^{2}[/tex] , so we have
[tex]y^{1} + \frac{9ty}{16-t^{2}}=\frac{5t^{2}}{16-t^{2}}[/tex]
With this , we can see that [tex]p(t)[/tex] and [tex]g(t)[/tex] are both continuous in the same domain. Therefore , the intervals are :
[tex]I_{1}[/tex] = ( -∞ , -4 )
[tex]I_{2}[/tex] = ( - 4 , 4 )
[tex]I_{3}[/tex] = ( 4 , -∞ )
recall that y(−5) = 1 , then t = -5
This means that :
t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )
A university's housing and residence office wants to know how much students pay per month for rent in off-campus housing. The university does not have enough on-campus housing for students, and this information will be used in a brochure about student housing. They obtain a list of the 12,304 students who live in off-campus housing and have not yet graduated and mail a questionnaire to 200 students selected at random. Only 78 questionairres are returned 8.1 (a) What is the population in this study? Be careful: about what group do they want information? (b) What is the sample? Be careful: from what group do they actually obtain information? can redefine the population about which infor- The important message in this problem is that the sample mation is obtained
Answers
Answer:
a) For this case the population of study is " the 12,304 students who live in off-campus housing and have not yet graduated " since the objective of the study is know how much students pay per month for rent in off-campus housing. So thn th epopulation size is N = 12304.
b) They select 200 students who live in off-campus housing and have not yet graduated selected using random sampling from the total population of 12304. So the sample size is n=200
As we can see in the info they got response just for 78 questionairres, so then the response rate is (78/200)*100 =39% and the non response rate is 100-39= 61.
So at the end the final sample consist on 39% of the original sample with 78 responses available for the analysis.
For this case we can redefine the sample as "The k students selected at random who live in off-campus housing and have not yet graduated and answer the mail questionairries ". And as we can see we have a better definition fo the real sample for this situation.
Step-by-step explanation:
Previous concepts
The term population represent to the total set of observations in a sample space S defined, and with an specified characteristics.
The term sample represent a set of individuals or objects who are a subsample of the population who are "collected or selected from a statistical population by a defined procedure".
Part a
For this case the population of study is " the 12,304 students who live in off-campus housing and have not yet graduated " since the objective of the study is know how much students pay per month for rent in off-campus housing. So thn th epopulation size is N = 12304.
Part b
They select 200 students who live in off-campus housing and have not yet graduated selected using random sampling from the total population of 12304. So the sample size is n=200
As we can see in the info they got response just for 78 questionairres, so then the response rate is (78/200)*100 =39% and the non response rate is 100-39= 61.
So at the end the final sample consist on 39% of the original sample with 78 responses available for the analysis.
For this case we can redefine the sample as "The k students selected at random who live in off-campus housing and have not yet graduated and answer the mail questionairries ". And as we can see we have a better definition fo the real sample for this situation.
The population in this study is the 12,304 students who live in off-campus housing and have not yet graduated. The sample is the 200 students selected at random who received the questionnaire. The sample provides information about the population of students living in off-campus housing.
Explanation:(a) Population: The population in this study refers to all the 12,304 students who live in off-campus housing and have not yet graduated. This is the group about which the university's housing and residence office wants information.
(b) Sample: The sample in this study refers to the 200 students selected at random who received the questionnaire. This is the group from which the office obtains information. The sample can be seen as a representation of the population.
The important message in this problem is that the sample can help provide information about the population of students living in off-campus housing.
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Drilling down beneath a lake in Alaska yields chemical evidence of past changes in climate. Biological silicon, left by the skeletons of single-celled creatures called diatoms, measures the abundance of life in the lake. A rather complex variable based on the ratio of certain isotopes relative to ocean water gives an indirect measure of moisture, mostly from snow. As we drill down, we look farther into the past. Here are data from 2300 to 12,000 years ago:Isotope(%) Silicon(mg/g) Isotope(%) Silicon(mg/g) Isotope(%) Silicon(mg/g)?19.90 95 ?20.71 152 ?21.63 226?19.84 106 ?20.80 263 ?21.63 233?19.46 114 ?20.86 265 ?21.19 186?20.20 139 ?21.28 298 ?19.37 337
(b) Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)What is the correlation without this point? (Round your answer to two decimal places.)(c) Is the outlier also strongly influential for the regression line? Calculate the regression line with the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ? x ( )Calculate the regression line without the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ?( ) x
Answers
Answer:
The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.
What is the correlation with this point? (Round your answer to two decimal places.)
> cor(x,y)
[1] 0.34
What is the correlation without this point? (Round your answer to two decimal places.)
> cor(x1,y1)
[1] 0.79
Calculate the regression line with the outlier.
y = 35.2 x-523.9
Calculate the regression line without the outlier.
y = 77.9 x -1422.3
Step-by-step explanation:
We have the following data:
Isotope %: 19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37 (representing X)
Silicon : 85,152,226,106,263,233,114,265,186,139,298,337 (representing Y)
Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)
We can use the scatter plot in order to see any potential outlier. With the following R code:
> x<-c(19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37)
> y<-c(85,152,226,106,263,233,114,265,186,139,298,337)
> plot(x,y, main="Scatter plot Silicon vs Isotope")
And we can see the plot on the figure attached.
The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.
What is the correlation with this point? (Round your answer to two decimal places.)
> cor(x,y)
[1] 0.34
What is the correlation without this point? (Round your answer to two decimal places.)
> x1<-x[-12]
> x1
[1] 19.90 20.71 21.63 19.84 20.80 21.63 19.46 20.86 21.19 20.20 21.28
> y1<-y[-12]
> y1
[1] 85 152 226 106 263 233 114 265 186 139 298
> cor(x1,y1)
[1] 0.79
As we can see the correlation changes significantly without the outlier.
c) Is the outlier also strongly influential for the regression line? Calculate the regression line with the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ? x ( )
We can calculate the regression line with the following R code
> linearmod1<-lm(y~ x)
> linearmod1
Call:
lm(formula = y ~ x)
Coefficients:
(Intercept) x
-523.9 35.2
So our equation would be: y = 35.2 x-523.9
Calculate the regression line without the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ?( ) x
> linearmod2<-lm(y1~x1)
> linearmod2
Call:
lm(formula = y1 ~ x1)
Coefficients:
(Intercept) x1
-1422.26 77.85
The new equation would be y = 77.9 x -1422.3
So as we can see the outlier also changes significantly the estimation for the slope and the intercept of the linear model
To answer this question one would need to identify the outlier in the data then calculate both the correlation and regression lines with and without this outlier. Changes in these calculations can demonstrate the influence of the outlier.
Explanation:This question involves finding statistical outliers and calculating correlation and regression lines in a dataset. The outlier in a dataset is a data point that is remarkably distinct from the rest of the data. As your question doesn't provide a clear dataset, it's impossible to identify the outlier clearly. However, once identified, to determine if the point strongly influences the correlation you would find the correlation with the outlier and without the outlier and compare these two values.
If the
correlation
drastically changes when the outlier is removed, it can confirm that the point is strong influencing the correlation. The
regression line
could be calculate using the standard formulas for slope and intercept, both with and without the outlier in the data. If the regression line meaningfully shifts after removing the outlier, it indicates that outlier is impactful to the regression line as well. This analytical work typically requires skills in interpreting scatterplots and using statistical programs or calculators.
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A car rental company charges a one-time application fee of 40 dollars, 55 dollars per day, and 13 cents per mile for its cars. A) Write a formula for the cost, C, of renting a car as a function of the number of days, d, and the number of miles driven, m. B) If C = f(d, m), then f(5, 600) =
Answers
Answer:
A) C(d,m) = 40 + 55d + 0.13m
B) $448
Step-by-step explanation:
Let 'd' be the number of days and 'm' the number of miles driven.
A) The cost function that describes a fixed amount of $40, added to a variable amount of $55 per day (55d) and a variable amount of 13 cents per mile (0.13m) is:
[tex]C(d,m) = 40 +55d +0.13m[/tex]
B) If d = 5 and m =600, the total cost is:
[tex]C(5,600) = 40 +55*6 +0.13*600\\C(5,600)=\$448[/tex]
The cost is $448.
The formula for the cost, C of renting a car as a function of the number of days, d, and the number of miles driven, m is C f(d, m) = 40 + 55d + 0.13m
Given:
Application fee = $40
cost per day = $55
cost per mile = $0.13
let
Total cost = C
Number of days = d
Number of miles = m
Total cost, C = 40 + 55d + 0.13m
C f(d, m) = 40 + 55d + 0.13m
= 40 + 55(5) + 0.13(600)
= 40 + 275 + 78
= $393
Therefore, the cost of renting the car given the number of days is $393
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A bacteria population starts with 400 bacteria and grows at a rate of r(t) = (450.263)e1.12567t bacteria per hour. How many bacteria will there be after three hours?
Answers
Answer:
11,713 bacteria
Step-by-step explanation:
Integrating the growth rate function gives us the population of bacteria at any given moment 't', in hours:
[tex]r(t) = (450.263)e^{1.12567t} \\\int\ {r(t)} \, dt =P(t) = \frac{450.263}{1.12567}*e^{1.12567t} +C[/tex]
Since at t=0, P(t) = 400, the value of C is:
[tex]P(0) = \frac{450.263}{1.12567}*e^{1.12567*0} +C\\400 = 400*1+C\\C=0[/tex]
The number of bacteria after 3 hours is:
[tex]P(3) = 400*e^{1.12567*3}\\P(3) =11,713[/tex]
Suppose that out of 20% of all packages from Amazon are delivered by UPS, 12% of the packages that are delivered by UPS weighs 2 lbs or more. Also, 8% of the packages that are not delivered by UPS weighs less than 2 lbs.
a. What is the probability that a package is delivered by UPS if it weighs 2 lbs or more?
b. What is the probability that a package is not delivered by UPS if it weighs 2 lbs or more?
Answers
Answer:
(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more = 0.0316.
(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 0.9684 .
Step-by-step explanation:
We are given that 20% of all packages from Amazon are delivered by UPS, from which 12% of the packages that are delivered by UPS weighs 2 lbs or more and 8% of the packages that are not delivered by UPS weighs less than 2 lbs.
Firstly Let A = Package from Amazon is delivered by UPS.
B = Packages that are delivered by UPS weighs 2 lbs or more.
So, P(A) = 0.2 and P(A') = {Probability that package is not delivered by UPS}
P(A') = 1 - 0.2 = 0.8
P(B/A) = 0.12 {means Probability that package weight 2 lbs or more given it
is delivered by UPS}
P(B'/A') = 0.08 [means Probability that package weight less than 2 lbs given
it is not delivered by UPS}
Since, P(B/A) = [tex]\frac{P(A\bigcap B)}{P(A)}[/tex] , [tex]P(A\bigcap B)[/tex] = P(B/A) * P(A) = 0.12 * 0.2 = 0.024 .
Also P(B) { Probability that package weight 2 lbs or more} is given by;
Probability that package weight 2 lbs or more and it delivered by UPS.Probability that package weight 2 lbs or more and is not delivered by UPS.So, P(B) = [tex]P(B\bigcap A) + P(B\bigcap A')[/tex] = P(B/A) * P(A) + P(B/A') * P(A')
= 0.12 * 0.2 + 0.92 * 0.8 { Here P(B/A') = 1 - P(B'/A') = 1 - 0.08 = 0.92}
= 0.76
(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more is given by P(A/B);
P(A/B) = [tex]\frac{P(A\bigcap B)}{P(B)}[/tex] = [tex]\frac{0.024}{0.76}[/tex] = 0.0316
(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 1 - P(A/B) = 1 - 0.0316 = 0.9684 .
Similarly, approaching along the y-axis yields a limit equal to 0. Since these two limits are the same, we will examine another approach path. Approach (0, 0) along the curve y = x2. When x is positive, we have lim (x, y) → (0, 0) xy x2 + y2 =
Answers
Answer:
This approach to (0,0) also gives the value 0
Step-by-step explanation:
Probably, you are trying to decide whether this limit exists or not. If you approach through the parabola y=x², you get
[tex]\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{\sqrt{x^2+y^2}}=\lim_{(x,x^2)\rightarrow (0,0)}\frac{xx^2}{\sqrt{x^2+(x^2)^2}}=\lim_{x\rightarrow 0}\frac{x^3}{|x|\sqrt{1+x^2}}=0[/tex]
It does not matter if x>0 or x<0, the |x| on the denominator will cancel out with an x on the numerator, and you will get the term x²/(√(1+x²) which tends to 0.
If you want to prove that the limit doesn't exist, you have to approach through another curve and get a value different from zero.
However, in this case, the limit exists and its equal to zero. One way of doing this is to change to polar coordinates and doing a calculation similar to this one. Polar coordinates x=rcosФ, y=rsinФ work because the limit will only depend on r, no matter the approach curve.
A local fraternity is conducting a raffle where 55 tickets are to be sold—one per customer. There are three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what are the following probabilities? (Round your answers to five decimal places.) (a) What is the probability that the four organizers win all of the prizes? (b) What is the probability that the four organizers win exactly two of the prizes? (c) What is the probability that the four organizers win exactly one of the prizes? (d) What is the probability that the four organizers win none of the prizes?
Answers
Answer:
(a) 0.0152%
(b) 1.1663%
(c) 19.4297%
(d) 79.3787%
Step-by-step explanation:
Tickets bought by organizers = 4
Number of tickets = 55
Prizes = 3
(a) The probability that the four organizers win all of the prizes is:
[tex]P = \frac{4}{55}*\frac{3}{54}*\frac{2}{53}\\P=0.0152\%[/tex]
(b) The probability that the four organizers win exactly two of the prizes is:
[tex]P = \frac{4}{55}*\frac{3}{54}*\frac{51}{53}+\frac{4}{55}*\frac{51}{54}*\frac{3}{53}+\frac{51}{55}*\frac{4}{54}*\frac{3}{53}\\P=1.1663\%[/tex]
(c) The probability that the four organizers win exactly one of the prizes is:
[tex]P = \frac{4}{55}*\frac{51}{54}*\frac{50}{53}+\frac{51}{55}*\frac{4}{54}*\frac{50}{53}+\frac{51}{55}*\frac{50}{54}*\frac{4}{53}\\P=19.4397\%[/tex]
(d) The probability that the four organizers win none of the prizes is:
[tex]P = \frac{51}{55}*\frac{50}{54}*\frac{49}{53}}\\P=79.3787\%[/tex]
A small grocery store had 10 cartons of milk, 2 of which were sour. If you are going to buy the 6th carton of milk sold that day at random, find the probability of selecting a carton of sour milk.
Answers
Answer:
The probability that the sixth customer buys sour milk is [tex]\frac{1}{5}[/tex].
Step-by-step explanation:
The grocery store has a total of 10 cartons of milk.
The number of cartons of milk that are sour is, 2.
If none of the sour cartons of milk were bought by the first 5 buyers, then the probability of this event is:P (Both the sour cartons are available to be sold to the sixth customer)
= [tex]P(2\ sour\ cartons)=\frac{2}{5}[/tex]
If only one sour carton of milk is sold to the first 5 buyers then the probability is:P (Only one sour cartons is available to be sold to the sixth customer)
= [tex]P(1\ sour\ cartons)=\frac{1}{5}[/tex]
If both the sour carton of milk is sold to the first 5 buyers then the probability is:P (None of the sour cartons is available to be sold to the sixth customer)
= [tex]P(0\ sour\ cartons)=\frac{0}{5}[/tex]
Compute the probability that the sixth customer buys sour milk:
= P (Both sour milk is available for the 6th customer) +
P (Only one sour milk is available for the 6th customer) +
P (None of the sour milk is available for the 6th customer)
[tex]=\frac{{8\choose 5}{2\choose 0}}{{10\choose 5}} \times\frac{2}{5} +\frac{{8\choose 4}{2\choose 1}}{{10\choose 5}} \times\frac{1}{5} +\frac{{8\choose 3}{2\choose 2}}{{10\choose 5}} \times\frac{0}{5} \\=\frac{56\times2}{252\times5} +\frac{140\times1}{252\times5} +0\\=\frac{1}{5}[/tex]
Thus, the probability that the sixth customer buys sour milk is [tex]\frac{1}{5}[/tex].
Trying to find volume of a right circular cone.
Answers
Volume of the cone is 117.5 π ft³
Step-by-step explanation:
Lateral area of the cone = πrs
s is the slant height = 15 ft
From the above formula, we can find the radius as, 5 ft.
Volume of the cone = π r² h/3
s = √ (5²+ h²)
Squaring on both sides, we will get,
s² = 15² = (5² + h²)
15² - 5² = h²
225 - 25 = 200 = h²
h = √200 = 14.1 ft
Volume = π × 5² × 14.1 / 3 = 117.5 π ft³
Determine if b is a linear combination of a1 a2, and a3. a1 = [ 1 -2 0 ], a2 = [ 0 1 3 ], a3 = [ 6 -6 18 ], b = [ 2 -2 6 ] Choose the correct answer below. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column. Vector b is not a linear combination of a1, a2, and a3. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
Answers
Answer: Vector b is not a linear combination
Step-by-step explanation:
First of all we put the vectors in terms of different variables, such as:
a1(1,-2,0)=(a,-2a,0);
a2(0,1,3)=(0,b,3b);
a3(6,-6,18)=(6c,-6c,18c);
To know that a vector is a linear combination we need to express it like a sum of other different vectors.
(2,-2,6)=(a,-2a,0)+(0,b,3b)+(6c,-6c,18c)
(2,-2,6)=(a+0+6c,-2a+b-6c,0+3b+18c)
We express this sum like a system of equations.
a+6c=2
-2a+b-6c=-2
3b+18c=6
We solve this system of equations and we can note that the system don't have a solution, so the vector b is not a linear combination of a1, a2, and a3.
Upon forming a system of linear equations and solving, a solution would imply that vector b is indeed a linear combination of vectors a1, a2, and a3. The observed placement of pivots in the corresponding echelon matrix backs this conclusion.
Explanation:In this question, you are asked to determine if vector b is a linear combination of vectors a1, a2, and a3. A vector is a linear combination of others if it can be written as a weighed sum of those vectors. To solve this problem, we need to form a system of linear equations based on the vectors and solve this system. If all of the coefficients can be expressed as real numbers, it means that the vector b is a linear combination of a1, a2, and a3.
In this case, our system of equations looks like this:
x∗a1 + y∗a2 + z∗a3 = b
In matrix form it can be written as:
|1 0 6|x| = |2|, |-2 1 -6|y| = |-2|, |0 3 18|z| = |6|.
Solve this system through methods like Gauss-Jordan elimination or row reduction. The pivots in the corresponding echelon matrix should be in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
This suggests that vector b can indeed be a linear combination of vectors a1, a2, and a3.
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The amount of juice in a grapefruit is directly proportional to the cube of its diameter. Let j represent the amount of juice (in fluid ounces) in a grapefruit, let d represent the diameter (in inches) of the grapefruit, and let k be the constant of proportionality. Write an equation that relates j to d.
Answers
Answer:
The equation relating j to d can be written as:
If J = the amount of juice in fluid ounces and D = its diameter, then:
J=kD³ where k = the constant of proportionality.
The equation that represents the relationship between the amount of juice in a grapefruit (j) and its diameter (d) with constant of proportionality (k), given that they are directly proportional, is j = kd^3.
Explanation:The question given is a classic example of direct variation, more specifically, cubic direct variation since the amount of juice in a grapefruit varies with the cube of its diameter. When two quantities are directly proportional or vary directly, they form a relationship that can be described by an equation of the form y = kx^n, where k is the constant of proportionality, x is one quantity and y is the other. Here, the amount of juice in a grapefruit (j) is directly proportional to the cube of the diameter (d^3), with k being the constant of proportionality. Therefore, the equation that represents this relationship is j = kd^3.
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Determine whether the results below appear to have statistical significance, and also determine whether the results have practical significance.
In a study of a weight loss program, 5 subjects lost an average of 50 lbs. It is found that there is about 28% chance of getting such results with a diet that has no effect.
Part A. Does the weight loss program have statistical significance?
a. No, the program is not statistically significant because the results are likely to occur by chance.
b. Yes, the program is statistically significant because the results are unlikely to occur by chance.
c. No, the program is not statistically significant because the results are unlikely to occur by chance.
d. Yes, the program is statistically significant because the results are likely to occur by chance.
PART B. Does the weight loss program have practical significance?
a. Yes, the program is practically significant because the results are too unlikely to occur by chance.
b. No, the program is not practically significant because the amount of weight lost is trivial.
c. No, the program is not practically significant because the results are likely to occur even if the weight loss program has no effect.
d. Yes, the program is practically significant because the amount of lost weight is large enough to be considered practically significant.
Answers
Answer:
A: d. Yes, the program is statistically significant because the results are likely to occur by chance.
B: d. Yes, the program is practically significant because the amount of lost weight is large enough to be considered practically significant.
Step-by-step explanation:
A likely event is the one which is likely to occur or show the same results.
An unlikely event is one in which there's a chance of not getting the desired result or may not show the same results.
In the given scenario the program is statistically significant because even if the 28 % results occur likely by chance the rest 72 % will show some desired results.
In part B the program is statistically practical as the remaining 72 % represent a significant stats.
Ralph wants to estimate the percentage of coworkers that use the company's healthcare. He asks a randomly selected group of 200 coworkers whether or not they use the company's healthcare. What is the parameter?
a.the percentage of surveyed coworkers that use the company's healthcare
b.the 200 coworkers surveyed
c.specific "yes" or "no" responses to the survey
d.all coworkers that use the company's healthcare
e.the percentage of all coworkers that use the company's healthcare
Answers
Answer:
e.
Step-by-step explanation:
The parameter is used to measure the population or in other words the measurement taken from a population is known as parameter. Here, the population will be all the co-workers that use company's healthcare. The percentage calculated for all co-workers that use company's healthcare will be the measurement calculated from population. Thus, the parameter will be the percentage of all the co-workers that use the company's healthcare.
Final answer:
The parameter Ralph is trying to estimate is 'e. the percentage of all coworkers that use the company's healthcare,' which reflects the actual percentage for the entire population of coworkers, not just the surveyed sample.
Explanation:
The question posed by the student is concerned with identifying the parameter involved in a statistical study. In the scenario provided, Ralph wants to estimate the percentage of coworkers that use the company's healthcare by surveying a random sample of 200 coworkers. The parameter, in this case, is the actual percentage of all coworkers at the company who use the company's healthcare, because it describes the entire population that is being studied, not just the sample that was surveyed. Therefore, the correct answer is e. the percentage of all coworkers that use the company's healthcare.
A parameter is a summary measure that describes an entire population, whereas a statistic is a summary measure that describes a sample taken from the population. An example that illustrates this distinction is an insurance company determining the proportion of all medical doctors who have been involved in malpractice lawsuits by surveying a random sample of 500 doctors. The true proportion of all doctors who have faced lawsuits is a parameter, while the proportion calculated from the sample of 500 doctors is a statistic.
A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Make a system of equations. After how many minutes will they have a same amount of water in the tank?
Answers
Let [tex]t[/tex] be the number of minutes.
The first tank starts with 1500 liters, and loses 4 liters per minute, so after [tex]t[/tex] minutes there will be
[tex]1500-4t[/tex]
liters of water.
The second tank is filled at 6 liters per second, i.e.
[tex]6\times 60=360[/tex] liters per minute.
So, there will be
[tex]300+360t[/tex]
liters of water in the second tank after [tex]t[/tex] minutes.
The two quantities will be equal when
[tex]1500-4t=300+360t \iff 1200=364t \iff t=\dfrac{1200}{364}\approx 3.3[/tex]
so, approximately, after 3.3 minutes.
Answer: it will take about 3.3 minutes for both tanks to have the same amount of water.
Step-by-step explanation:
Let x represent the number of minutes it will take both tanks to have same amount of water.
A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. This means that in x minutes, the volume of water in the tank would be
1500 - 4x
At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Converting 6 liters per second to minutes, it becomes 60 × 6 = 360 liters per minute. This means that in x minutes, the volume of water in the tank would be
300 + 360x
For both tanks to have same amount of water, then
1500 - 4x = 300 + 360x
360x + 4x = 1500 - 300
364x = 1200
x = 1200/364 = 3.3 minutes
We have N cars on a circular one-way road; they have the same make, same model, same year and the same fuel economy. The total amount of gas in all cars is sufficient to make the full circle.
Prove by induction that it is always possible to find a car that can make the full circle, taking gas from other cars as it passes them.
Answers
Answer: Satisfied for n=1, n=k and n=k+1
Step-by-step explanation:
The induction procedure involves two steps
First is
Basic Step
Here we consider that for the value n=1, there is one car and it will always make the full circle.
Induction Step
Since basic step is satisfied for n=1
Now we do it for n=k+1
Now according to the statement a car makes full circle by taking gas from other cars as it passes them. This means there are cars that are there to provide fuel to the car. So we have a car that can be eliminated i.e. it gives it fuels to other car to make full circle so it is always there.
Now ,go through the statement again that the original car gets past the other car and take the gas from it to eliminate it. So now cars remain k instead of k+1 as it's fuel has been taken. Now the car that has taken the fuel can make the full circle. The gas is enough to make a circle now.
So by induction we can find a car that satisfies k+1 induction so for k number of cars, we can also find a car that makes a full circle.
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
Determine the sample space for the experiment.
Answers
Answer:
The sample space for the experiment is {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
Step-by-step explanation:
Consider the provided information.
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
The sample space for the experiment is:
{(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}
Hence, the possible outcomes are {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
Using your knowledge of exponential and logarithmic functions and properties, what is the intensity of a fire alarm that has a sound level of 120 decibels?
A.
1.0x10^-12 watts/m^2
B.
1.0x10^0 watts/m^2
C.
12 watts/m^2
D.
1.10x10^2 watts/m^2
Answers
Option B:
[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
Solution:
Given sound level = 120 decibel
To find the intensity of a fire alarm:
[tex]$\beta=10\log\left(\frac{I}{I_0} \right)[/tex]
where [tex]I_0=1\times10^{-12}\ \text {watts}/ \text m^2}[/tex]
Step 1: First divide the decibel level by 10.
120 ÷ 10 = 12
Step 2: Use that value in the exponent of the ratio with base 10.
[tex]10^{12}[/tex]
Step 3: Use that power of twelve to find the intensity in Watts per square meter.
[tex]$10^{12}=\left(\frac{I}{I_0} \right)[/tex]
[tex]$10^{12}=\left(\frac{I}{1\times10^{-12}\ \text {watts}/ \text m^2} \right)[/tex]
Now, do the cross multiplication,
[tex]I=10^{12}\times1\times\ 10^{-12} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1\times\ 10^{12-12} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
Option B is the correct answer.
Hence [tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex].
Final answer:
To find the intensity of a sound at 120 dB, we use the formula SIL = 10 log(I / I0). With I0 as 10⁻¹² W/m², we find that I = 1.0 x 10⁰ W/m², corresponding to choice B.
Explanation:
To determine the intensity of a fire alarm that has a sound level of 120 decibels (dB), we use the relationship between sound intensity level and intensity in watts per meter squared (W/m²). The formula to convert decibel level to intensity is:
SIL = 10 log(I / I0)
Where SIL is the sound intensity level in decibels, I is the intensity of the sound, and I0 is the reference intensity, usually taken as 10⁻¹² W/m², the threshold of human hearing. To find the unknown intensity I, we can rearrange the formula:
I = I0 × 10(SIL/10)
For a sound level of 120 dB, the calculation would be:
I = 10⁻¹² W/m² × 10¹²⁰/¹⁰
I = 10⁻¹² W/m² × 10¹²
I = 1.0 × 10⁰ W/m²
Therefore, the correct answer is B. 1.0 x 10⁰ watts/m².
class is made up of 40% women and has 20 women in it. What is the total number of students in the class?
Answers
The total number of students in the class is 50, which is calculated based on the given that 20 women make up 40% of the class.
Explanation:The question deals with the concept of percentages. In this case, the number of women in the class represents 40% of the total number of students. We have been told that there are 20 women in the class.
Here's how we can solve it step by step:
Given that 20 women represent 40% of all students in the class.So, if we want to find 100% (the total number of students), we'll divide 20 by 40 to find the value that represents 1%, which equals 0.5.Then we multiply 0.5 by 100 to get the total number of students which equals 50.Therefore, the total number of students in the class is 50.
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An electronics firm sells four models of stereo receivers, three CD decks, and six speaker brands. When the three types of components are sold together, they form a "system." How many different systems can the electronics firm offer?
A. 169
B. 72
C. 13
D. 36
Answers
Answer:
B. 72
Step-by-step explanation:
The total number of different systems that can be bundled together is the product of the possible number of ways to select 1 out of 4 stereo receivers, by 1 out of 3 CD decks and by 1 out of 6 speakers. Assuming that the order at which products are picked does not matter, the number of different systems is:
[tex]n=\frac{4!}{(4-1)!1!}*\frac{3!}{(3-1)!1!}*\frac{6!}{(6-1)!1!}\\n=4*3*6\\n=72\ systems[/tex]
The electronics firm can offer 72 different systems.
The electronics firm can offer 72 different systems by combining 4 models of stereo receivers, 3 CD decks, and 6 speaker brands together.
To find the number of different systems the electronics firm can offer, we use the combinations formula. Since there are 4 models of stereo receivers, 3 CD decks, and 6 speaker brands, the total number of different systems can be calculated as:
4 models * 3 CD decks * 6 speaker brands = 72 different systems
Therefore, the answer to the question is 72 different systems (option B).
Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a king for the second card drawn, if the first card, drawn without replacement, was a king? Express your answer as a fraction or a decimal number rounded to four decimal places.
Answers
Answer:
the probability of choosing a king for the second card drawn is 3/51 , if the first card, drawn without replacement, was a king
Step-by-step explanation:
defining the variable F= choosing a king in the first drawn , then the probability is
P(F)= 4/52
then using the theorem of Bayes for conditional probability and denoting the event S= choosing a king in the second drawn , then
P(S/F)= P(S∩F)/P(F)
where
P(S∩F) = probability of choosing a king in the first drawn and second drawn = 4/52* 3/51
P(S∩F) =probability of choosing a king in the second drawn given that a king was chosen in the first drawn
then
P(S/F)= P(S∩F)/P(F) = 4/52* 3/51 / 4/52 = 3/51
The probability of choosing a king for the second card drawn is 1/17 or approximately 0.0588.
Explanation:To find the probability of choosing a king for the second card drawn, given that the first card was a king, we need to consider the total number of cards and the number of kings remaining after the first card was drawn.
After the first king is drawn, there are 51 cards left in the deck, with 3 remaining kings. Therefore, the probability of choosing a king for the second card drawn is 3/51, which can be simplified to 1/17 or approximately 0.0588.
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1. A class of 16 students contains five Math majors, eight Engineering majors and three Physics majors. A group of four students from the class is to be selected to form a team for an academic competition.How many teams can be formed that have a representative from each major?
Answers
Answer:
495
Step-by-step explanation:
Using combination without repetition, the formular is given by n!/(r!(n-1)!
= (12 x11 ×10 ×9) /4 !
=11880/24
=495
From the group of 16 students with 5 math majors, 3 physics major and 8 engineering majors a team of 4 students where there has been a student from each majors total no. of teams that can be formed is 111.
What is combination ?Combination is selecting some things in which order does not matter.
According to the given question
A class of 16 students contains 5 Math majors, 8 Engineering majors and 3 Physics majors. A group of four students from the class is to be selected to form a team for an academic competition.
Here three cases will be formed as group is formed by 4 students and the no. of majors is 3.
First case when 2 math majors, 1 physics major and 1 engineering major is chosen from 16 students.
= ⁵C₂ + ³C₁ + ⁸C₁
= 8 + 5 + 3
= 16.
Second case when 1 math major, 2 physics major and 1 engineering major is chosen from 16 students.
= ⁵C₁ + ³C₂ + ⁸C₁
= 8 + 20 + 3
= 31.
In the third case when 1 math major, 1 physics major and 2 engineering major is chosen from 16 students.
= ⁵C₁ + ³C₁ + ⁸C₂
= 56 + 5 + 3
= 64.
∴total no of teams that can be formed is
= 16 + 31 + 64
= 111.
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