Answer:
a) 1.4% of the samples break during shipment
b) the probability is 4/7 ( 57.14%)
Step-by-step explanation:
a) defining the event B= the sample of laboratory glass breaks , then the probability is:
P(B)= probability that sample is shipped in small packaging * probability that the sample breaks given that was shipped in small packaging + probability that sample is shipped in large packaging * probability that the sample breaks given that was shipped in large packaging = 0.40* 0.02 + 0.60*0.01 = 0.014
b) we can use the theorem of Bayes for conditional probability. Then defining the event S= the sample is shipped in small packaging . Thus we have
P(S/B)= P(S∩B)/P(B) = 0.40* 0.02 / 0.014= 4/7 ( 57.14%)
where
P(S∩B)= probability that sample is shipped in small packaging and it breaks
P(S/B)= probability that sample was shipped in small packaging given that is broken
A textbook store sold a combined total of 219 history and chemistry textbooks in a week. The number of chemistry textbooks sold was 45 less than the number of history textbooks sold. How many textbooks of each type were sold?
Answer:
132 history textbooks, 87 chemistry textbooks
Step-by-step explanation:
[tex]C + H = 219\\C = H - 45\\[/tex]
[tex]1. H - 45 + H = 219\\2. 2H = 264\\3. H = 132[/tex]
H = 132,
C = 132 - 45 = 87
Answer: 87 Chemistry and 132 history textbooks.
Step-by-step explanation:
Let x represent the number of chemistry textbooks that was sold.
Let y represent the number of history textbooks that was sold.
The textbook store sold a combined total of 219 history and chemistry textbooks in a week. This means that
x + y = 219 - - - - - - - - - - - -1
The number of chemistry textbooks sold was 45 less than the number of history textbooks sold. This means that
x = y - 45
Substituting x = y - 45 into equation 1, it becomes
y - 45 + y = 219
2y = 219 + 45 = 264
y = 264/2 = 132
x = y - 45 = 132 - 45
x = 87
an=an−1−4 a1=15 to explicit formula
The explicit formula for the sequence given by an = an-1 - 4 with a1 = 15 is an = 19 - 4n, which is derived using the general formula for an arithmetic sequence.
Explanation:The question asks to derive the explicit formula for a sequence given by the recursive formula an = an-1 - 4 with the initial term a1 = 15. To find the explicit formula, we recognize this as an arithmetic sequence where the common difference (d) is -4 and the first term (a1) is 15.
The formula for the nth term of an arithmetic sequence is given by an = a1 + (n - 1)d. Substituting a1 = 15 and d = -4 into this formula gives us:
an = 15 + (n - 1)(-4)
Simplifying, we get an = 19 - 4n. This is the explicit formula for the given sequence, allowing us to find any term in the sequence without needing to calculate all the previous terms.
Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit. Let 1 = HIT and 0 = OUT. The output simulation was as follows.
1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1
Estimate the probability that he gets a hit. Round to three decimal places.
A. 0.286
B. 0.452
C. 0.476
D. 0.301
Answer:
Option C) 0.476
Step-by-step explanation:
We are given the following in the question:
1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1
where 1 means that Sam will get a hit and 0 mean Sam will be out.
Total number of outcomes = 42
Number of times Sam will get a hit = n(1) = 20
Number of times Sam will be out = n(0) = 22
We have to find the probability that Sam gets a hit.
Formula:
Thus, 0.476 is the probability that Sam will get a hit.
The probability that to gets a hit is 0.476.
What is mean by Probability?
The term probability refers to the likelihood of an event occurring.
Given that;
Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit.
Let Hit = 1 and Out = 0
Now,
Total number of outcomes = 42
And, Number of times Sammy get a hit = 20
Number of times Sammy will be out = 22
Since, The probability to get a hit is defined as;
Probability = Number of times he get hit / Total number of outcomes
= 20 / 42
= 0.476
Thus, The probability that to gets a hit is 0.476.
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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?
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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Suppose you have an experiment where you flip a coin three times. You then count the number of heads. a.)State the random variable. b.)Write the probability distribution for the number of heads.
Answer:
a. Number of heads
b.
x p(x)
0 1/8
1 3/8
2 3/8
3 1/8
Step-by-step explanation:
a)
A coin is flipped three times and the number of heads are counted.
We are interested in counting heads so, a random variable X is the number of heads appears on a coin.
b)
The sample space for flipping a coin three times is
S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
n(S)=8
The random variable X (number of heads) can take values 0,1,2 and 3 .
0 head={TTT}
P(0 heads)=P(X=0)=1/8
1 head={HTT,THT,TTH}
P(1 head)= P(X=1)=3/8
2 heads= {HHT,HTH,THH}
P(2 heads)=P(X=2)=3/8
3 heads={HHH}
P(3 heads)=1/8
The probability distribution for number of heads can be shown as
x p(x)
0 1/8
1 3/8
2 3/8
3 1/8
The random variable is the number of heads obtained when flipping a coin three times. The probability distribution for the number of heads can be found using the binomial probability formula.
Explanation:a) The random variable in this experiment is the number of heads obtained when flipping a coin three times. It can take on the values 0, 1, 2, or 3.
b) To write the probability distribution for the number of heads, we need to determine the probability of getting 0, 1, 2, or 3 heads. Since each coin flip is an independent event, we can use the binomial probability formula to calculate these probabilities.
For example, the probability of getting exactly 2 heads can be calculated as: P(X = 2) = (3 choose 2) * (0.5^2) * (0.5^1) = 3 * 0.25 * 0.5 = 0.375.
The probability distribution for the number of heads is:
X = 0, P(X = 0) = (3 choose 0) * (0.5^0) * (0.5^3) = 1 * 1 * 0.125 = 0.125
X = 1, P(X = 1) = (3 choose 1) * (0.5^1) * (0.5^2) = 3 * 0.5 * 0.25 = 0.375
X = 2, P(X = 2) = (3 choose 2) * (0.5^2) * (0.5^1) = 3 * 0.25 * 0.5 = 0.375
X = 3, P(X = 3) = (3 choose 3) * (0.5^3) * (0.5^0) = 1 * 0.125 * 1 = 0.125
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A manager checked production records and found that a worker produced 200 units while working 40 hours. In the previous week, the same worker produced 132 units while working 30 hours. a. Compute Current period productivity and Previous period productivity. (Round your answers to 2 decimal places.) Current period productivity Units / hr Previous period productivity Units / hr b. Did the worker's productivity increase, decrease, or remain the same
Answer:
a. Current: 5 units/hour. Previous: 4.4 units/hour
b. Increase
Step-by-step explanation:
a. Current period productivity is 200 / 40 = 5 units/hour
Previous period productivity is 132 / 30 = 4.4 units/hour
b. As this week's productivity = 5 units/hours which is larger than last week's productivity = 4.4 units/hour. The worker's productivity for this week has increased.
Write a formula that expresses Δ y in terms of Δ x . (Hint: enter "Delta" for Δ .) Suppose that y = 2.5 y=2.5 when x = 1.5 x=1.5. Write a formula that expresses y in terms of x x
Final answer:
To express Δy in terms of Δx, we can use the concept of slope. The formula that expresses Δy in terms of Δx is Δy = (2.5 - y1) / (1.5 - x1) * Δx.
Explanation:
To express Δy in terms of Δx, we can use the concept of slope. The formula for slope is:
Slope = (Δy) / (Δx)
To find the slope between two points, we can use the formula:
Slope = (y2 - y1) / (x2 - x1)
In this case, if y = 2.5 when x = 1.5, we can substitute these values into the formula and simplify:
Slope = (2.5 - y1) / (1.5 - x1)
Since we are only interested in expressing Δy in terms of Δx, we can solve for Δy:
Δy = Slope * Δx
Therefore, the formula that expresses Δy in terms of Δx is:
Δy = (2.5 - y1) / (1.5 - x1) * Δx
If we have y = mx + b and we only know one point (1.5, 2.5), we need a second point or more context for an exact equation. The change in y, Δy, with respect to a change in x, Δx, is found using: Δy = m * Δx.
To express Δy in terms of Δx, you can use the concept of derivatives and the definition of a linear function. Here's a step-by-step solution:
Given information: We know that "y = 2.5" when "x = 1.5."
Setting up the function: Let's assume that the relationship between y and x is linear. In a linear function, the rate of change of y with respect to x is constant. We can write the linear equation in the form: y = mx + b, where m is the slope and b is the y-intercept.
Finding the slope (m): Since linear functions have a constant slope, we need to calculate m. If we assume that y changes by some amount Δy when x changes by Δx, then the slope (m) can be represented as: m = Δy / Δx.
Using the derivative: For a linear equation, dy/dx = m. Therefore, Δy = m * Δx.
Given the specific solution: In the problem, we were given a point (x, y) = (1.5, 2.5). However, we need another point or more information to determine the exact form of the function y in terms of x. Without additional information, we cannot definitively determine the slope.
Assuming a direct variation: In simple cases, we might assume a direct variation (y = kx), but this requires more context. Based on the provided hint, if we use the ratio y/x = k, we can set up an initial formula to start with.
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters,are 0,17,58,77,95,106,118,127,63,,40 and 0. Use the Midpoint Rule with n = 5 to estimate the volume V of the liver.
Answer:
X XCX X X C C C C C CC C C C C C C C C C C C C C CC C C SC S DCSD VSDCS CS CSDV SD SDC D D D VD DFV DF DFV DF
Step-by-step explanation:
The estimated volume of the liver using the Midpoint Rule with n = 5 is approximately 1051.5 cubic centimeters.
To estimate the volume V of the liver using the Midpoint Rule with n = 5 , we need to first find the average area of adjacent cross-sections and then multiply it by the distance between these cross-sections.
Given the cross-sectional areas: 0, 17, 58, 77, 95, 106, 118, 127, 63, 40, and 0.
We will partition these areas into 5 equal intervals and use the midpoint of each interval to estimate the average area.
Interval 1: 0, 17
Interval 2: 17, 58
Interval 3: 58, 77
Interval 4: 77, 95
Interval 5: 95, 106
Interval 6: 106, 118
Interval 7: 118, 127
Interval 8: 127, 63
Interval 9: 63, 40
Interval 10: 40, 0
Now, we calculate the midpoints of each interval:
Midpoint 1: (0 + 17)/2 = 8.5
Midpoint 2: (17 + 58)/2 = 37.5
Midpoint 3: (58 + 77)/2 = 67.5
Midpoint 4: (77 + 95)/2 = 86
Midpoint 5: (95 + 106)/2 = 100.5
Midpoint 6: (106 + 118)/2 = 112
Midpoint 7: (118 + 127)/2 = 122.5
Midpoint 8: (127 + 63)/2 = 95
Midpoint 9: (63 + 40)/2 = 51.5
Midpoint 10: (40 + 0)/2 = 20
Next, we find the average area of these intervals:
[tex]\( A_1 = 8.5 \)[/tex], [tex]\( A_2 = 37.5 \)[/tex], [tex]\( A_3 = 67.5 \)[/tex], [tex]\( A_4 = 86 \)[/tex], [tex]\( A_5 = 100.5 \)[/tex], [tex]\( A_6 = 112 \)[/tex], [tex]\( A_7 = 122.5 \)[/tex], [tex]\( A_8 = 95 \)[/tex], [tex]\( A_9 = 51.5 \)[/tex], [tex]\( A_{10} = 20 \)[/tex]
Now, we use the Midpoint Rule formula to estimate the volume:
[tex]\[ V \approx \Delta x \sum_{i=1}^{n} A_i \][/tex]
Where [tex]\( \Delta x \)[/tex] is the distance between cross-sections, given as 1.5 cm, and n = 5 intervals.
V [tex]\approx[/tex] 1.5 * (8.5 + 37.5 + 67.5 + 86 + 100.5 + 112 + 122.5 + 95 + 51.5 + 20) \]
[tex]\[ V \approx 1.5 \times (701) \][/tex]
[tex]\[ V \approx 1051.5 \][/tex]
So, the estimated volume of the liver using the Midpoint Rule with n = 5 is approximately 1051.5 cubic centimeters.
The position of an object moving vertically along a line is given by the function s(t) = -16t^2 + 128t. Find the average velocity of the object over the following intervals.
a. [1, 4]
b. [1, 3]
c. [1, 2]
d. [1, 1 + h], where h > 0 is a real number
Answer:
a) 48
b) 64
c) 80
d) 96-16h
Step-by-step explanation:
a) s(1)=112 and s(4)=256
average velocity on [1,4] = (256-112)/(4-1) = 48
b) s(1)=112 and s(3)=240
average velocity on [1,3] = (240-112)/(3-1) = 64
c) s(1)=112 and s(2)=192
average velocity on [1,2] = (192-112)/(2-1) = 80
the next one's tricky to type. watch the parentheses carefully:
d) s(1)=112 and s(1+h)= -16(1+h)^2 + 128(1+h)
average velocity on [1,1+h] =
(s(1+h) - s(1))/((1+h)-1) = (-16(1+h)^2 +128(1+h) - (112))/h
= (-16(1+2h+h^2)+128+128h - 112)/h
= ( -16 -32h -16h^2 + 16 + 128h)/h
= ( 96h - 16 h^2)/h
= 96 - 16h
Select the null and the alternative hypotheses for the following tests:
a. Test if the mean weight of cereal in a cereal box differs from 18 ounces.
O H0: μ = 18; HA: μ ≠ 18
O H0: μ ≥ 18; HA: μ < 18
O H0: μ ≤ 18; HA: μ > 18
b. Test if the stock price increases on more than 60% of the trading days.
O H0: p ≤ 0.60; HA: p > 0.60
O H0: p ≥ 0.60; HA: p < 0.60
O H0: p = 0.60; HA: p ≠ 0.60
c. Test if Americans get an average of less than seven hours of sleep.
O H0: μ ≥ 7; HA: μ < 7
O H0: μ ≤ 7; HA: μ > 7
O H0: μ = 7; HA: μ ≠ 7
Answer:
Option A)
[tex]H_{0}: \mu = 18\text{ ounces}\\H_A: \mu \neq 18\text{ ounces}[/tex]
Option A)
[tex]H_{0}: p \leq 0.6\\H_A: p > 0.6[/tex]
Option A)
[tex]H_{0}: \mu \geq 7\text{ hours}\\H_A: \mu \leq 7\text{ hours}[/tex]
Step-by-step explanation:
We have to design null and alternate hypothesis for given test:
a) Test if the mean weight of cereal in a cereal box differs from 18 ounces.
Option A)
[tex]H_{0}: \mu = 18\text{ ounces}\\H_A: \mu \neq 18\text{ ounces}[/tex]
The null hypothesis means mean cereal box weight is 18 ounces and the alternate hypothesis state that it is different than 18 ounces.
b) Test if the stock price increases on more than 60% of the trading days.
Option A)
[tex]H_{0}: p \leq 0.6\\H_A: p > 0.6[/tex]
The null hypothesis states that the proportion is less than 0.6 that is stock price is less than or equal to 60% of the trading days and alternate hypothesis states that the proportion is greater than 0.6 that is stock price increases on more than 60% of the trading days.
c) Test if Americans get an average of less than seven hours of sleep.
Option A)
[tex]H_{0}: \mu \geq 7\text{ hours}\\H_A: \mu \leq 7\text{ hours}[/tex]
The null hypothesis states that the Americans get an average of greater than or equal to 7 hours of sleep where as the alternate hypothesis states that the Americans get a sleep less than 7 hours of sleep.
In Exercises 40-43, for what value(s) of k, if any, will the systems have (a) no solution, (b) a unique solution, and (c) infinitely many solutions?x - 2y +3z = 2x + y + z = k2x - y + 4z = k^2
Answer:
If k = −1 then the system has no solutions.
If k = 2 then the system has infinitely many solutions.
The system cannot have unique solution.
Step-by-step explanation:
We have the following system of equations
[tex]x - 2y +3z = 2\\x + y + z = k\\2x - y + 4z = k^2[/tex]
The augmented matrix is
[tex]\left[\begin{array}{cccc}1&-2&3&2\\1&1&1&k\\2&-1&4&k^2\end{array}\right][/tex]
The reduction of this matrix to row-echelon form is outlined below.
[tex]R_2\rightarrow R_2-R_1[/tex]
[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\2&-1&4&k^2\end{array}\right][/tex]
[tex]R_3\rightarrow R_3-2R_1[/tex]
[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&3&-2&k^2-4\end{array}\right][/tex]
[tex]R_3\rightarrow R_3-R_2[/tex]
[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&0&0&k^2-k-2\end{array}\right][/tex]
The last row determines, if there are solutions or not. To be consistent, we must have k such that
[tex]k^2-k-2=0[/tex]
[tex]\left(k+1\right)\left(k-2\right)=0\\k=-1,\:k=2[/tex]
Case k = −1:
[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-1-2\\0&0&0&(-1)^2-(-1)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-3\\0&0&0&-2\end{array}\right][/tex]
If k = −1 then the last equation becomes 0 = −2 which is impossible.Therefore, the system has no solutions.
Case k = 2:
[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&2-2\\0&0&0&(2)^2-(2)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&0\\0&0&0&0\end{array}\right][/tex]
This gives the infinite many solution.
We use matrix row reduction to determine the values of k that result in no solution, a unique solution, or infinitely many solutions in the system of equations.
Explanation:To determine the values of k for which the system of equations has no solution, a unique solution, or infinitely many solutions, we will use the concept of matrix row reduction. First, let's rewrite the system of equations in augmented matrix form:
[1 -2 3 2 | 0] [2 1 1 -1 | 0] [2 -1 4 -k^2 | 0]
Performing row reduction on this augmented matrix, we can find the values of k where each situation occurs. If there is a row of 0's followed by a non-zero constant (in the rightmost column), then the system has no solution. If the row reduction yields a matrix with a non-zero row followed by zeroes (except for the last row), then the system has infinitely many solutions. Otherwise, the system has a unique solution.
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The nutrition label on a bag of potato chips says that a one ounce (28 gram) serving of potato chips has 130 calories and contains ten grams of fat, with three grams of saturated fat. A random sample of 35 bags yielded a sample mean of 134 calories with a standard deviation of 17 calories.
a. Is there evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips?
b. State your null and alternative hypotheses, your computed p-value, and your decision based on the given random sample.
To determine if the nutrition label is accurate, a hypothesis test can be conducted using the provided sample data. Calculating the z-score and finding the p-value will determine if there is evidence that the label is inaccurate.
Explanation:In order to determine if there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips, we can conduct a hypothesis test using the given sample data. Let's state the null and alternative hypotheses:
Null Hypothesis (H0): The nutrition label provides an accurate measure of calories in the bags of potato chips.
Alternative Hypothesis (Ha): The nutrition label does not provide an accurate measure of calories in the bags of potato chips.
To test these hypotheses, we can calculate the z-score using the formula:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
In this case, the population mean is 130 calories (as stated on the nutrition label), the sample mean is 134 calories, the population standard deviation is 17 calories (as given), and the sample size is 35 bags (as given). Plugging in these values, we can calculate the z-score.
Once we have the z-score, we can find the p-value associated with it from a standard normal distribution table or using statistical software. If the p-value is less than the chosen significance level (usually 0.05), we reject the null hypothesis and conclude that there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips.
Without knowing the calculated p-value, we cannot make a decision based on the given random sample.
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Is there a real number whose square is −1? a. Is there a real number x such that ? b. Does there exist such that x2 = −1?
Answer:
a.[tex]x^2=-1[/tex]
b.a real number x
Step-by-step explanation:
We are given that statement.
We have to rewrite the given statement using variable or variables.
Statement:Is there a real number whose square is -1.
a.Let x bet the real number
The square of real number x written as [tex]x^2[/tex]
According to question
[tex]x^2=-1[/tex]
Therefore,
Is there a real number x such that [tex]x^2=-1[/tex]
b.Does there exist a real number x such that
[tex]x^2=-1[/tex]
There is no real number whose square is -1. However, in the domain of complex numbers, 'i' is defined as the square root of -1. Complex numbers include both real and imaginary parts.
Explanation:In the realm of real numbers, there isn't a real number whose square is -1. In the context of complex numbers, however, 'i' is defined to be the square root of -1. In other words, i2 = -1. It's important to note that complex numbers consist of a real part and an imaginary part (where 'i' is the basis for the imaginary part), and are beyond the usual scope of real numbers.
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At a certain college, 28% of the students major in engineering, 18% play club sports, and 8% both major in engineering and play club sports. A student is selected at random. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Given that the student is majoring in engineering, what is the probability that the student plays club sports
Answers: 0.286
Explanation:
Let E → major in Engineering
Let S → Play club sports
P (E) = 28% = 0.28
P (S) = 18% = 0.18
P (E ∩ S ) = 8% = 0.08
Probability of student plays club sports given majoring in engineering,
P ( S | E ) = P (E ∩ S ) ÷ P (E) = 0.08 ÷ 0.28 = 0.286
To find the probability that a student plays club sports given that they major in engineering, use conditional probability.
Explanation:To find the probability that a student plays club sports given that they major in engineering, we need to use conditional probability.
The formula for conditional probability is:
P(A|B) = P(A ∩ B) / P(B)
In this case, A represents playing club sports and B represents majoring in engineering. We are given that P(A ∩ B) = 8% and P(B) = 28%.
Plugging these values into the formula, we get:
P(A|B) = 8% / 28% = 0.2857
So the probability that a student plays club sports given that they major in engineering is approximately 0.2857, or 28.57%.
Which expressions represent a quadratic expression in factored form?
Answer:
3 and 4
Step-by-step explanation:
Well to start we have to know that they are asking us, a factorized form of a quadratic expression
a quadratic expression is of the form
ax ^ 2 + bx + c
Now the factored form is as follows
a ( x - x1 ) ( x - x2 )
Next, let's look at each of the options
In this case we lack a term with x since if we solve we have a linear equation
1. 5(x+9)
5x + 45
In this case if we pay attention they are being subtracted instead of multiplying, so we will not get a quadratic function
2. (x+4) - (x+6)
-2
In this case we have everything we need, now let's try to solve
3. (x-1) (x-1)
x^2 - x - x + 1
x^2 - 2x + 1 quadratic function
In this case we have everything we need, now let's try to solve
4. (x-3) (x+2)
x^2 -3x +2x -6
x^2 -x - 6 quadratic function
In this case we have a quadratic function but we do not have it in its factored form since we can observe the x ^ 2
5. x^2 + 8x
A factory makes rectangular sheets of cardboard, each with an area 2 1/2 square feet. Each sheet of cardboard can be cut into smaller pieces of cardboard measuring 1 1/6 square feet. How many smaller pieces of cardboard does each sheet of cardboard provide?
Answer:
Step-by-step explanation:
The area of each rectangular sheet of cardboard made by the factory is is 2 1/2 square feet. Converting
2 1/2 to improper fraction, it becomes 5/2 square feet.
Each sheet of cardboard can be cut into smaller pieces of cardboard measuring 1 1/6 square feet. Converting 1 1/6 to improper fraction, it becomes 7/6 square feet.
Therefore, the number of smaller pieces of cardboard that each sheet of cardboard provides is
5/2 ÷ 7/6 = 5/2 × 6/7 = 30/14
= 2.14 pieces
Madison is carrying a 11.3 liter jug of sports drink that weighs 7 kg.
What is the constant multiple of liters in a jug to the weight in kilograms?
Incorrect
Note: The constant multiple should be a reduced fraction,
not a mixed number.
Answer:
[tex]\large\boxed{\large\boxed{constat\text{ }multiple=\frac{113liter}{70kg}}}[/tex]
Explanation:
The constant multiple of liters in a jug to the weight in kilograms is the ratio or fraction that represents the number of liters of the sports drink in a jug to the weight.
[tex]constat\text{ }multiple=ratio=\frac{number\text{ }of\text{ }liters}{weight\text{ }in\text{ }kg}[/tex]
[tex]constat\text{ }multiple=\frac{11.3liter}{7kg}[/tex]
Convert the fraction into an equivalent fraction with integer numbers:
[tex]constat\text{ }multiple=\frac{11.3liter\times 10}{7kg\times 10}=\frac{113liter}{70kg}[/tex]
Since, the fraction cannot be reduced, that is the answer.
The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs seven times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.)
Answer:
[tex] x= (\frac{2V}{7})^{1/3}[/tex]
[tex] y= (\frac{2V}{7})^{1/3}[/tex]
[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]
Step-by-step explanation:
This is a minimization problem.
For this case we assume that we have a box and the volume is given by:
[tex] V = xyz[/tex] (1)
For this case we know that slate costs seven times as much (per unit area) as glass so then 7xy this value and if we find the cost function like this:
[tex] C(x,y,z) = 2yz+ 2xz + 7xy[/tex]
If we solve z from equation (1) we got:
[tex] z= \frac{V}{xy}[/tex] (2)
So then we can replace equation (2) into the cost equation and we got:
[tex] C(x,y,V/xy)= 2y (\frac{V}{xy}) +2x(\frac{V}{xy})+ 7xy[/tex]
And with this we have a function in terms of two variables x and y.
We can simplify the last equation and we got:
[tex] C(x,y,V/xy)= \frac{2V}{x} +\frac{2V}{y} + 7xy[/tex]
In order to solve the problem for the dimensions we can take the partial derivates respect to x and y and we got:
[tex] C_x = -\frac{2V}{x^2} +7y =0[/tex]
[tex] C_y = -\frac{2V}{y^2} +7x =0[/tex]
We can set the last two equations equal since are equal to 0 and we got:
[tex] -\frac{2V}{x^2} +7y =-\frac{2V}{y^2} +7x [/tex]
And the only possible solution for this case is [tex] x=y[/tex]
So then if we use x=y for the partial derivate of x we have:
[tex] C_x (x,y=x) = -\frac{2V}{x^2} +7x =0[/tex]
And solving for x we got:
[tex] \frac{2V}{x^2} =7x[/tex]
[tex] 7x^3 = 2V[/tex]
[tex] x= (\frac{2V}{7})^{1/3}[/tex]
And analogous we can do the same thing for the partial derivate of y and we got:
[tex] C_y (x=y,y) = -\frac{2V}{y^2} +7y =0[/tex]
And solving for x we got:
[tex] \frac{2V}{y^2} =7y[/tex]
[tex] 7y^3 = 2V[/tex]
[tex] y= (\frac{2V}{7})^{1/3}[/tex]
And for z we can replace and we got:
[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]
So then the dimensions in order to minimize the cost would be:
[tex] x= (\frac{2V}{7})^{1/3}[/tex]
[tex] y= (\frac{2V}{7})^{1/3}[/tex]
[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]
PLEASE HELP 50 COINS!!!!
Answer: 12.22
Step-by-step explanation:
Since it is a right angled triangle, we use the trigonometry method of solving triangles for this question.
The given angle is 42° and we recall our trigonometry functions of
Sin Φ = opposite/hypotenuse
Cos Φ= adjacent/hypotenuse
tan Φ = opposite/adjacent
Where
Φ =42°
Opposite of the angle = GH = 11
Adjacent of the angle = HI = ?.
Hence we use the tan Formula.
tan 42 = 11/HI
HI = 11/tan42
HI = 11/0.90
HI = 12.22
Find the domain of f and f −1 and its domain. f(x) = ln(ex − 3). (a) Find the domain of f. (Enter your answer using interval notation.) (−2,[infinity]) (b) Find f −1. f −1(x) = x+ln(3)
Answer:
a.Domain of f=(1.099,[tex]\infty)[/tex]
b.[tex]f^{-1}(x)=ln(e^x+3)[/tex]
Step-by-step explanation:
Let [tex]y=f(x)=ln(e^x-3)[/tex]
We know that domain of ln x is greater than zero
[tex]e^x-3>0[/tex]
Adding 3 on both sides of inequality
[tex]e^x-3+3>0+3[/tex]
[tex]e^x>3[/tex]
Taking on both sides of inequality
[tex]lne^x>ln 3[/tex]
[tex]x>ln 3[/tex]=1.099
By using [tex]lne^x=x[/tex]
Domain of f=(1.099,[tex]\infty)[/tex]
Let [tex]y=f^{-1}(x)=ln(e^x-3)[/tex]
[tex]e^y=e^x-3[/tex]
By using property [tex]lnx=y\implies x=e^y[/tex]
[tex]e^x=e^y+3[/tex]
Taking ln on both sides of equality '
[tex]lne^x=ln(e^y+3)[/tex]
[tex]x=ln(e^y+3)[/tex]
Replace x by y and y by x
[tex]y=ln(e^x+3)[/tex]
Substitute y=[tex]f^{-1}(x)[/tex]
[tex]f^{-1}(x)=ln(e^x+3)[/tex]
9 weeks 5 days - 1 week 6days =
Answer:
(9 weeks 5 days) - (1 week 6 days) = 55 days
Step-by-step explanation:
Answer:
Step-by-step explanation:
Two birds sit at the top of two different trees. The distance between the first bed and a birdwatcher on the ground is 34 feet the distance between the birdwatcher and the second bird is 47 feet What is the angle measure or angle of
depression between this bed and the birdwatcher? Round your answer to the nearest tenth
Answer:
Step-by-step explanation:
The given triangle is a right angle triangle.
The distance between the first bed and the bird watcher on the ground represents the opposite side of the right angle triangle.
The distance between the birdwatcher and the second bird is 47 feet. This represents the hypotenuse of the right angle triangle. To determine the angle of depression, x degrees, we would apply the Sine trigonometric ratio which is expressed as
Sin θ = opposite side/hypotenuse
Sin x = 34/47 = 0.723
x = Sin^-1(0.723)
x = 46.3 degrees to the nearest tenth.
Answer:
46.3 degrees or answer D
Step-by-step explanation:
lol
How do I solve this using the substitution method 3x+2y=9 x-5y=4
[tex]\bf \begin{cases} 3x+2y=9\\ x-5y=4 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{solving the 2nd equation for "y"}}{x-5y = 4\implies x-4-5y=0}\implies x-4=5y\implies \cfrac{x-4}{5}=y \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{substituting on the 1st equation}}{3x+2\left(\cfrac{x-4}{5} \right) = 9}\implies 3x+\cfrac{2(x-4)}{5}=9 \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{5}}{5\left( 3x+\cfrac{2(x-4)}{5} \right)=5(9)}\implies 15x+2(x-4)=45[/tex]
[tex]\bf 15x+2x-8=45\implies 17x-8=45\implies 17x=53\implies \boxed{x=\cfrac{53}{17}} \\\\\\ \stackrel{\textit{we know that}}{\cfrac{x-4}{5}=y}\implies \cfrac{\left(\frac{53}{17} -4 \right)}{5}=y\implies \cfrac{\left(\frac{53-68}{17} \right)}{5}=y\implies \cfrac{~~\frac{-15}{17}~~}{5}=y \\\\\\ \cfrac{~~\frac{-15}{17}~~}{\frac{5}{1}}=y\implies \cfrac{-15}{17}\cdot \cfrac{1}{5}=y\implies \boxed{-\cfrac{3}{17}=y} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \left( \frac{53}{17}~~,~~-\frac{3}{17} \right)~\hfill[/tex]
Answer: x = 57/17
y = - 3/17
Step-by-step explanation:
The given system of equations is expressed as
3x + 2y = 9 - - - - - - - - - - - - - -1
x - 5y = 4 - - - - - - - - - - - - - - -2
From equation 2, we would make x the subject of the formula by adding 5y to the left hand side and the right hand side of the equation. It becomes
x - 5y + 5y = 4 + 5y
x = 4 + 5y
Substituting x = 4 + 5y into equation 1, it becomes.
3(4 + 5y) + 2y = 9
12 + 15y + 2y = 9
15y + 2y = 9 - 12
-7y = - 3
y = - 3/17
Substituting y = - 3/17 into equation x = 4 + 5y, it becomes
x = 4 + 5 × - 3/17
x = 4 - 15/17
x = (68 - 15)/17
x = 53/17
From a sample with n = 32, the mean number of televisions per household is 4 with a standard deviation of 1 television. UsingChebychev's Theorem, determine at least how many of the households have between 2 and 6 televisions.At least ____ of the households have between 2 and 6 televisions.
Answer:
Atleast, 88.9% of the households have between 2 and 6 televisions.
Step-by-step explanation:
We are given the following in he question:
Sample size, n = 32
Mean, μ = 4
Standard Deviation, σ = 1
Chebychev's Theorem:
I states that atleast [tex]1 - \dfrac{1}{k^2}[/tex] percent of data lies within k standard deviations for a non normal data.For k = 2[tex]1-\dfrac{1}{2^2} = 0.75[/tex]
Atleast 75% of data lies within 2 standard deviation of mean.
For k = 3[tex]1-\dfrac{1}{3^2} = 0.889[/tex]
Atleast 88.9% of data lies within 3 standard deviation of mean.
[tex]2 = \mu - 2\sigma = 4 - 2(1)\\6 = \mu + 2\sigma = 4 +2(1)[/tex]
Thus, we have to find data within two standard deviations.
Atleast, 88.9% of the households have between 2 and 6 televisions.
Chebychev's Theorem allows us to determine the proportion of data within a certain number of standard deviations from the mean. In this case, using the formula z = (x - μ) / σ, we find that at least 75% of the households have between 2 and 6 televisions.
Explanation:Chebychev's Theorem allows us to determine the proportion of data within a certain number of standard deviations from the mean.
In this case, we want to find the proportion of households with between 2 and 6 televisions. To do this, we need to find out how many standard deviations away from the mean these values are.
The number of standard deviations away from the mean can be calculated using the formula z = (x - μ) / σ, where z is the number of standard deviations from the mean, x is the value we're interested in, μ is the mean, and σ is the standard deviation.
For x = 2, z = (2 - 4) / 1 = -2
For x = 6, z = (6 - 4) / 1 = 2
According to Chebychev's Theorem, no less than 1 - 1/k^2 of the data falls within k standard deviations from the mean. In this case, we're interested in the proportion of data between -2 and 2 standard deviations from the mean.
k = 2 (the distance between -2 and 2), so k^2 = 4.
Thus, the proportion of data within -2 and 2 standard deviations from the mean is equal to 1 - 1/4 = 3/4 = 0.75.
Therefore, at least 75% of the households have between 2 and 6 televisions.
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Ask Your Teacher Write out the form of the partial fraction decomposition of the function (See Example). Do not determine the numerical values of the coefficients. (If the partial fraction decomposition does not exist, enter DNE.) (a) x x2 + x − 20 (b) x2 x2 + x + 2
The partial fraction are:
a) [tex](x / (x^2 + x - 20))[/tex] = [tex]\dfrac{x}{x^2 + x - 20} = \dfrac{A}{(x - 4)} + \dfrac{B}{(x + 5)}[/tex]
b) [tex]\dfrac{x}{x^2 + x + 2} = \dfrac{A}{(x +1)} + \dfrac{B}{(x + 2)}[/tex]
(a) The partial fraction decomposition of the function [tex](x / (x^2 + x - 20))[/tex] can be written as:
[tex]\dfrac{x}{x^2 + x - 20} = \dfrac{A}{(x - 4)} + \dfrac{B}{(x + 5)}[/tex]
where A and B are constants.
(b) The partial fraction decomposition of the function [tex]\dfrac{x}{x^2 + x - 20}[/tex] can be written as:
[tex]\dfrac{x}{x^2 + x + 2} = \dfrac{A}{(x +1)} + \dfrac{B}{(x + 2)}[/tex]
where A and B are constants.
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The student is asked to perform partial fraction decomposition for two functions. In the first case, the given rational function decomposes to the form: A/(x - 4) + B/(x + 5). In the second case, the decomposition does not exist as the denominator can't be factored using real numbers.
Explanation:In mathematics, the concept under discussion is the partial fraction decomposition. This is a process used in algebra to break down complex fractions or rational expressions into simpler ones. Given (a) x/(x^2 + x - 20) and (b) x^2/(x^2 + x + 2), you are being asked to perform the decomposition.
For (a), the denominator, x^2 + x - 20, can be factored as (x - 4)(x + 5), so the partial fraction decomposition would have the form: x/(x^2 + x - 20) = A/(x - 4) + B/(x + 5).
For (b), since the denominator x^2 + x + 2 can't be factored using real numbers, the partial fraction decomposition doesn't exist. Here, the answer would be DNE (Does Not Exist).
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Mt. McKinley, in Alaska, is the highest mountain in North America at 20,320
feet. A climbing team made it 5/8 of the way to the summit before a storm forced them to turn back. What was their elevation when the storm hit?
Answer:
12700 feet
Step-by-step explanation:
Do 5/8 of 20,320
Do 20320 divided by 8 which is 2540
Do 2540 times 5 which is 12700
A solid is bounded below by the cone, z=x2+y2, and bounded above by the sphere of radius 2 centered at the origin. Find integrals that compute its volume using Cartesian and cylindrical coordinates. For your answers use θ= theta.
The cone [tex]z=\sqrt{x^2+y^2}[/tex] and the sphere [tex]z=\sqrt{4-x^2-y^2}[/tex] intersect in a circle of radius [tex]\sqrt 2[/tex] in the plane [tex]z=\sqrt2[/tex]:
[tex]\sqrt{x^2+y^2}=\sqrt{4-x^2-y^2}\implies 2x^2+2y^2=4\implies x^2+y^2=2[/tex]
[tex]\implies z=\sqrt{x^2+y^2}=\sqrt2[/tex]
In Cartesian coordinates, the volume is then given by the integral
[tex]\displaystyle\int_{-\sqrt2}^{\sqrt2}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{4-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
In cylindrical coordinates, the integral is
[tex]\displaystyle\int_0^{2\pi}\int_0^{\sqrt2}\int_r^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]
For the given problem, the volume of the structure can be calculated using both Cartesian and cylindrical coordinates. For Cartesian coordinates, the coordinates represents as [tex]z=x^2+y^2[/tex]and [tex]x^2+y^2+z^2=4.[/tex]For cylindrical coordinates, equations represent as [tex]z=r^2[/tex]and [tex]r^2+z^2=4[/tex]. Both integrations describe the volume of the structures.
Explanation:In the given problem, the volume contains two geometric shapes: the cone and the sphere. The volume of the shape can be calculated using both Cartesian and cylindrical coordinates.
Using Cartesian coordinates, first describe cone and sphere as [tex]z=x^2+y^2[/tex] and [tex]x^2+y^2+z^2=4[/tex] respectively. Define the volume by double integration:
∫∫ D (4 - z) dxdy
Where D is the region in the xy-plane bounded by the projection of the volume.
Using cylindrical coordinates, we represent the figures as [tex]z=r^2[/tex]and [tex]r^2+z^2=4.[/tex] The volume integral in cylindrical coordinates is then given by:
∫ (from 0 to 2pi) ∫ (from 0 to √2) ∫ (from [tex]r^2 \ to \ 2-r^2[/tex]) rdzdrdθ
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A small radio transmitter broadcasts in a 44 mile radius. If you drive along a straight line from a city 60 miles north of the transmitter to a second city 59 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
Answer: 87.03
Step-by-step explanation:
The outer parts (2) of the secant line containing α and β refers to the distance travelled where there is no signal. The middle part is where there is signal presence. To get the altitude of the triangle:
Hc= 2(A/c)
To find the area;
A= 1/2(59)(60)
A= 1770
Use Pythagoras theorem to get c:
C= =√(60)^2+(59)^2
=√7081
Hc=2(1770/√7081)
=3540√7081/7081
Solve for x using Pythagoras theorem:
x= (√44^2-Hc^2) + (√44^2-Hc^2)
where Hc= 3540√7081/7081
=87.03
By interpreting the problem geometrically and using Pythagoras' theorem, it can be concluded that for approximately 34 miles of the journey from the city 60 miles north of the transmitter to the city 59 miles east of the transmitter will be in range of the radio signal.
Explanation:This problem can be solved using geometry and the concept of a circle. If we imagine the area the radio transmitter can reach as a circle with the transmitter at the center, any point within a 44-mile radius from the transmitter can pick up its signal. Now, let's analyze the specific scenario proposed.
Firstly, the city 60 miles north is outside the signal range. However, as you drive towards the second city 59 miles east of the transmitter, you'll at some point enter the broadcast range. That's because, at the closest point, you're only about 15 miles away from the transmitter (60 miles - 44 miles), assuming you drive perpendicular to the diameter of the transmission circle.
You need to calculate the intersection of your driving path with the transmission circle. Using Pythagoras' theorem, it can be seen that for about 34 miles of your direct journey from the first city to the second city, you would be in range of the transmitter. The two cities form the hypotenuse of a right triangle, and that hypotenuse intersects the transmission circle creating a segment along which signal will be received.
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Which of the following values cannot be probabilities? 0.04, 5 divided by 3, 1, 0, 3 divided by 5, StartRoot 2 EndRoot, negative 0.59, 1.49 Select all the values that cannot be probabilities. A. 1.49 B. 1 C. three fifths D. StartRoot 2 EndRoot E. five thirds F. 0 G. negative 0.59 H. 0.04
Answer:
A. 1.49
D. √2
E. five thirds
G. - 0.59
Step-by-step explanation:
In order to be a probability, a value must be at least zero, or at most 1:
[tex]0 \leq P\leq 1[/tex]
Evaluating each of the given values:
A. 1.49
1.49 is at least zero but it is greater than one, therefore 1.49 cannot be a probability.
B. 1
1 represents a probability of 100%, therefore this value can be a probability
C. three fifths
[tex]0\leq \frac{3}{5} \leq 1[/tex]
Can be a probability
D. √2
[tex]\sqrt 2 =1.41 > 1[/tex]
Cannot be a probability
E. five thirds
[tex]\frac{5}{3}=1.67>1[/tex]
Cannot be a probability
F. 0
0 represents a probability of 0%, therefore this value can be a probability
G. - 0.59
Negative values cannot be probabilities.
H. 0.04
[tex]0\leq 0.04 \leq 1[/tex]
Can be a probability
Probabilities are values ranging from 0 to 1, inclusive. With this in mind, values 5/3, √2, -0.59, and 1.49 cannot be probabilities as they're either below 0 or above 1.
Explanation:In the field of mathematics, specifically in statistics, a probability represents the likelihood of an event occurring and is always a value between 0 and 1, inclusively. The value 0 means that an event will not happen, whilst 1 means the event is certain to happen. Therefore, any value less than 0 or greater than 1 cannot be a probability.
Given the values: 0.04, 5 divided by 3, 1, 0, 3 divided by 5, √2, negative 0.59, and 1.49, the values that cannot be probabilities are:
Value 5 divided by 3 (which equals approximately 1.67)Value √2 (which equals approximately 1.41)Negative 0.591.49These numbers do not lie within the range of 0 to 1, and hence, cannot represent probabilities.
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A local ice cream shop kept track of the number of cans of cold soda it sold each day, and the temperature that day, for two months during the summer. The data are displayed in the scatterplot below:A local ice cream shop kept track of the number of
The one outlier corresponds to a day on which the refrigerator for the soda was broken. Which of the following is true?
(a) A reasonable value of the correlation coefficient r for these data is 1.2.
(b) If the temperature were measured in degrees Celsius (C = 5/9*(F-32)), the value of r would change accordingly.
(c) If the outlier were removed, r would increase.
(d) If the outlier were removed, r would decrease.
(e) Both (b) and (c) are correct.
Final answer:
Option (a) is incorrect because the value of r cannot exceed the range of -1 to +1. Option (b) is incorrect as changing units does not alter the value of r. Option (c) is most likely correct because removing an outlier typically leads to an increased value of r.
Explanation:
The student's question pertains to the transformation of a scatterplot and the effects on the Pearson correlation coefficient, symbolized by the letter r, which measures the strength and direction of a linear relationship between two variables. The value of r ranges from -1.00 to +1.00, with positive values indicating a positive linear relationship and negative values indicating a negative linear relationship. The closer the value of r is to -1 or +1, the stronger the linear relationship is.
For option (a), it is not possible for r to have a value of 1.2 as it must be within the range of -1.00 to +1.00, making option (a) incorrect. Option (b) is also incorrect because changing the scale of the temperature from Fahrenheit to Celsius does not affect the value of r; the strength and direction of the correlation remain the same regardless of the units used. Regarding options (c) and (d), usually when an outlier that does not follow the overall pattern of the data is removed, the absolute value of r tends to increase, which means that if the outlier was negatively influencing the correlation, r would increase, indicating option (c) is correct. In the event the outlier has a positive influence on the correlation, r would decrease but this specific information is not provided.