A consumer products company found that 44​% of successful products also received favorable results from test market​ research, whereas 13​% had unfavorable results but nevertheless were successful. That​ is, P(successful product and favorable test ​market) = 0.44 and​ P(successful product and unfavorable test ​market) = 0.13. They also found that 32​% of unsuccessful products had unfavorable research​ results, whereas 11​% of them had favorable research​ results, that is​ P(unsuccessful product and unfavorable test ​market) = 0.32 and​ P(unsuccessful product and favorable test ​market) = 0.11.
Find the probabilities of successful and unsuccessful products given known test market​ results, that​ is, P(successful product given favorable test​ market), P(successful product given unfavorable test​ market), P(unsuccessful product given favorable test​ market), and​ P(unsuccessful product given unfavorable test​ market).

Answer:

Correct answer is option C i.e 28 minutes

Step-by-step explanation:

Number of students arriving at adviser's office per hour = x = 28

Number of students get approved = [tex]\frac{1}{2min}[/tex]  = 30/hour

    ∴ y = 30

Number of students on average on waiting =Lq

Lq = [tex]\frac{x^{2} }{y(y-x)}[/tex]

=  [tex]\frac{28^{2} }{30(30-28)}[/tex]

= 13.07

Average time student has to spend in

Waiting = Wq = [tex]\frac{x}{y(y-x)}[/tex]

= [tex]\frac{28}{30(30-28)}[/tex]

= 0.466 hours

= 28 minutes

Answer:

a) And if we replace we got: [tex]\bar X= 78[/tex]

[tex] s = 15.391[/tex]

b) [tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]    

[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]    

So on this case the 99% confidence interval would be given by (62.182;93.818)    

Step-by-step explanation:

Dataset given: 109 67 58 76 65 80 96 86 71 72

Part a

For this case we can calculate the sample mean with the following formula:

[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

And if we replace we got: [tex]\bar X= 78[/tex]

And the deviation is given by:

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 15.391[/tex]

Part b

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=10-1=9[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,9)".And we see that [tex]t_{\alpha/2}=3.25[/tex]

Now we have everything in order to replace into formula (1):

[tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]    

[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]    

So on this case the 99% confidence interval would be given by (62.182;93.818)    

To Plotting lines , use the PFloor line tool and click your mouse for the first and second end-points, ensuring that the line intersects both the D1 and S1 lines and has a height greater than 50.

To plot the line described in the question, follow these steps:

Select the line tool on the tool palette labeled PFloor.

Click your mouse for the first end-point on the vertical axis.

Move your mouse to the right and click again for the second end-point.

The new line should intersect both the D1 and S1 lines and have a height greater than 50 on the vertical axis.

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Answer:

Option D

75.39

Step-by-step explanation:

When provided with the co-ordinates (x, y) and(a, b) then the distance between them is given by  [tex]\sqrt {(x-a)^{2}+(y-b)^{2}}[/tex]

Since  u = (25, $350) and v = (53, $420) then the Euclidean distance will be

[tex]\sqrt {(53-25)^{2}+(350-420)^{2}}=75.3923073\approx 75.39[/tex]

The dissimilarity between the two observations using Euclidean distance is approximately 75.39(Option d).

To calculate the dissimilarity between two observations using Euclidean distance, we need to find the distance between the corresponding elements in the two observations and then calculate the square root of the sum of their squared differences.

In this case, we have:

u = (25, $350) and v = (53, $420)

The distance between the ages is 53 - 25 = 28.

The distance between the amounts spent is $420 - $350 = $70.

Now we can use the formula for Euclidean distance:

Distance = sqrt((28)^2 + (70)^2) = sqrt(784 + 4900) = sqrt(5684) ≈ 75.39

Therefore, the dissimilarity between the two observations using Euclidean distance is approximately 75.39.

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Answer:

Number of births to women below 22 years of age: 830

% of births occurred to women of age between 34 and 36: 6.35%

Step-by-step explanation:

There are two parts to this question:

To calculate the number of births, we look at the histogram below.

We see that each bar has a number on top, suggesting that particular for the age limit. Number of births to women below age of 22, we start adding all numbers below the mark of 22 on x-axis:

⇒ 39+134+29+363 = 830

To calculate the percentage, we divide the number of births in that particular interval by total number of births and then multiply by 100.

To calculate the total number of births, we add all the numbers on the top of the bars:

⇒ 39+ 134+294+363+391+425+460+474+432+343+250+163+98+49+17+4+1 Total births = 3937

[tex]\frac{250}{3937} \times 100\\ 0.0635 \times 100\\= 6.35 \%[/tex]

Answer:

Null hypothesis:[tex]p=0.114[/tex]  

Alternative hypothesis:[tex]p \neq 0.114[/tex]  

[tex]z=\frac{0.152 -0.114}{\sqrt{\frac{0.114(1-0.114)}{850}}}=3.49[/tex]  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z>3.49)=0.00049[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.

[tex]0.152 - 1.96 \sqrt{\frac{0.152(1-0.152)}{850}}=0.128[/tex]

[tex]0.152 - 1.96 \sqrt{\frac{0.152(1-0.152)}{850}}=0.176[/tex]

And the 95% confidence interval would be given (0.128;0.176).

And support the conclusion obtained on the hypothesis test since the value of 0.114 is not in the confidence interval, so we have enough evidence to reject the null hypothesis.

Step-by-step explanation:

Data given and notation

n=850 represent the random sample taken

X=129 represent the number of housing units that are vacant.

[tex]\hat p=\frac{129}{850}=0.152[/tex] estimated proportion of housing units that are vacant.

[tex]p_o=0.114[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the proportion is equal is 0.114 or not:  

Null hypothesis:[tex]p=0.114[/tex]  

Alternative hypothesis:[tex]p \neq 0.114[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.152 -0.114}{\sqrt{\frac{0.114(1-0.114)}{850}}}=3.49[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level assumed is [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z>3.49)=0.00049[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.

Confidence interval

The confidence interval would be given by this formula

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.96[/tex]

And replacing into the confidence interval formula we got:

[tex]0.152 - 1.96 \sqrt{\frac{0.152(1-0.152)}{850}}=0.128[/tex]

[tex]0.152 - 1.96 \sqrt{\frac{0.152(1-0.152)}{850}}=0.176[/tex]

And the 95% confidence interval would be given (0.128;0.176).

And support the conclusion obtained on the hypothesis test since the value of 0.114 is not in the confidence interval, so we have enough evidence to reject the null hypothesis.

We first set the null and alternative hypothesis and then conduct a z-test for proportions to calculate the z-score and subsequently the p-value. We use the p-value to decide whether to reject the null hypothesis. Finally, we construct a 95% confidence interval for the true proportion of vacant houses and check if this supports our test conclusion.

Firstly, define the proportion of vacant housing units in the country as p0 and in the randomly selected county as p. The null hypothesis (H0) states that the county isn't different, so H0: p = p0 = 0.114. The alternative hypothesis (Ha) would be that the county is different, so Ha: p ≠ 0.114.

Let's conduct a Test of Hypothesis using a z-test for proportions. The z-score is calculated as (p - p0) / sqrt((p0 * (1 - p0)) / n), where n represents the sample size. Substituting in your values, the z score will be calculated. This z-score can be used to find the p-value from a standard normal (Z) distribution table.

If the p-value is less than 0.05 (which is α, significance level), we reject the null hypothesis in favor of alternative hypothesis, else we do not reject the null hypothesis. Thus, our conclusion is formulated based on this p-value.

To construct a 95% confidence interval for the true proportion of vacant houses, we use the formula: p ± Z * sqrt((p * (1 - p)) / n). Here, Z will be the Z score corresponding to the desired confidence level, 95% (which is 1.96 for two-tailed test).

If the national proportion (0.114) doesn't lie within this interval, it supports our test conclusion of rejecting the null hypothesis and vice versa.

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We are required to calculate the percentage increase in tax and determine who is right.

The percentage increase in tax is 25% and Linda is very correct

percentage increase = difference in tax /

percentage increase = difference in tax / old tax × 100

old tax = 2%

New tax = 2.5%

Difference = New tax - old tax

= 2.5% - 2%

= 0.5%

percentage increase = difference in tax /

percentage increase = difference in tax / old tax × 100

= 0.5% / 2% × 100

= 0.25 × 100

= 25%

Therefore, the percentage increase in tax is 25% and Linda is very correct

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Answer:

linear: 5s

quadratic: 50s

log-linear: 0.75 s

cubic: 500s

Step-by-step explanation:

Let [tex]t_1,t_2[/tex] be the running time associated with the input of sizes [tex] s_1,s_2[/tex]

If the running time is linear

[tex]t_2 = t_1\frac{s_2}{s_1} = 0.5*\frac{1000}{100} = 0.5*10 = 5s[/tex]

If the running time is quadratic

[tex]t_2 = t_1\left(\frac{s_2}{s_1}\right)^2 = 0.5*\left(\frac{1000}{100}\right)^2 = 0.5*10^2 = 50s[/tex]

If the running time is log-linear

[tex]t_2 = t_1\frac{log(s_2)}{log(s_1)} = 0.5*\frac{log(1000)}{log(100)} = 0.5*1.5 = 0.75s[/tex]

If the running time is cubic:

[tex]t_2 = t_1\left(\frac{s_2}{s_1}\right)^3 = 0.5*\left(\frac{1000}{100}\right)^3 = 0.5*10^3 = 500s[/tex]

Answer:

Step-by-step explanation:

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: The distribution of severity of psoriasis cases at the end and prior are same.

Alternative hypothesis: The distribution of severity of psoriasis cases at the end and prior are different.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square goodness of fit test of the null hypothesis.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 4 - 1

D.F = 3

(Ei) = n * pi

Category            observed Num      expected num      [(Or,c -Er,c)²/Er,c]

Remission             380                         20                           6480

Mild

symptoms               520                         770                       81.16883117

Moderate

symptoms                 95                         160                         24.40625

Severe

symptom                  5                             50                          40.5

Sum                          1000                       1000                       6628.075081

Χ2 = Σ [ (Oi - Ei)2 / Ei ]

Χ2 = 6628.08

Χ2Critical = 7.81

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, Ei is the expected frequency count for level i, Oi is the observed frequency count for level i, and Χ2 is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 6628.08.

We use the Chi-Square Distribution Calculator to find P(Χ2 > 19.58) =less than 0.000001

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we cannot accept the null hypothesis.

We reject H0, because 6628.08 is greater than 7.81. We have statistically significant evidence at alpha equals to 0.05 level to show that distribution of severity of psoriasis cases at the end of the clinical trial for the sample is different from the distribution of the severity of psoriasis cases prior to the administration of the drug suggesting the drug is effective.

The chi-square test is a statistical method that determines if there's a significant difference between observed and expected frequencies in different categories, such as symptom status in this clinical trial. Without post-treatment numbers, we can't run the exact test. However, if the test statistic exceeded the critical value, we could conclude that the drug significantly affected symptom statuses.

This question pertains to the use of a chi-squared test, which is a statistical method used to determine if there's a significant difference between observed frequencies and expected frequencies in one or more categories. For this case, the categories are the symptom statuses (remission, mild, moderate, and severe).

To conduct a chi-square test, you first need to know the observed frequencies (the initial percentages given in the question) and the expected frequencies (the percentages after treatment). As the question doesn't provide the numbers after treatment, I can't perform the exact chi-square test.

If the post-treatment numbers were provided, you would compare them to the pre-treatment numbers using the chi-squared formula, which involves summing the squared difference between observed and expected frequencies, divided by expected frequency, for all categories. The result is a chi-square test statistic, which you would then compare to a critical value associated with a chosen significance level (commonly 0.05) to determine if the treatment has a statistically significant effect.

To interpret a chi-square test statistic, if the calculated test statistic is larger than the critical value, it suggests that the drug made a significant difference in the distribution of symptom statuses. If not, we can't conclude the drug was effective.

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Answer:

ML = {bba, ab, a, bbaabb, ababb, aabb}

Step-by-step explanation:

By application of Union of a set.

M = {bba,ab, a}

L = {Λ,abb}

ML = {bba, ab, a, bbaabb, ababb, aabb}

Answer:

The variance of given sample is 33.64 square pounds.

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 16

Standard deviation, s = 5.8 pounds

We have to find the variance of the given sample.

Variance is the square of the standard deviation.

[tex]\text{Variance} = (\text{Standard Deviation})^2\\= (5.8\text{ pounds})^2\\=33.64\text{ pound}^2[/tex]

Thus, the variance of given sample is 33.64 square pounds.

Answer:

300 million people

Step-by-step explanation:

If the participation rate is 60% and 200 million people 16 years or older are not in the labor force, it means that 200 million corresponds to 40% of people 16 years or older. Since 60% of people 16 years or older are in the labor force, the total number of people in the labor force is given by:

[tex]n=\frac{200}{0.4}-200\\ n= 300\ million\ people[/tex]

300 million people are in the labor force in this economy.

Answer:

(44/4)*30 = $330

Step-by-step explanation:

divide by four and multiply by 30

Answer:

0.2637

Step-by-step explanation:

We see from the question that the 5-card hand contains all 4 suits as shown below;

Number of cards = 52

Number of suits = 4

For the favorable cases therefore, we will choose two cards from the suit in which two cards are drawn. Then we will proceed to choose one card from each of the other suits.

4 suits will divide into 52 cards to give = (52 / 4) = 13 cards

Hence, the required probability;

[tex]= {\frac{4 *13c_2*13c_1*13c_1*13c_1}{52c_5}}\\= {\frac{2197}{8330}}\\= 0.2637[/tex]

Answer:

12x + 2

Step-by-step explanation:

Let the number be represented by x.

Then five times the number = 5*x

Seven times the number = 7*x

Sum of 5 times the number minus -2 = [tex]\[5*x - (-2)\][/tex] = [tex]\[5x +2\][/tex]

Adding seven times the number to this expression yields, [tex]\[5x+2+7x\][/tex]

[tex]\[= (5+7)x+2\][/tex]

[tex]\[= 12x+2\][/tex]

So the simplified expression corresponds to 12x + 2.

Answer:

The question is incomplete, below is the complete question,"There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.

a) What is the probability that the individual needn't stop at either light?

b) What is the probability that the individual must stop at exactly one of the two lights? c) What is the probability that the individual must stop just at the first light?"

Answer:

A. 0.63

B. 0.24

C. 0.07

Step-by-step explanation:

Data given,

P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.

From the question, we can conclude that the event are dependent, hence

a. P(needn't stop at either light) = 1 - P(Need to stop at either light)

P(EUF)' =1-P(EUF)

P(EUF)' =1- (P(E)+P(F) -P(E ∩ F))

P(EUF)' =1-(0.2+0.3-0.13)

P(EUF)' =1-0.37

P(EUF)' =0.63

b. P(must stop at exactly one of the two lights) = P(must stop at either light) - P(must stop at both lights)

P(must stop at exactly one of the two lights)  = P(E u F) - P(En F)

but P(E u F)=0.37,

P(En F)=0.13,

P(must stop at exactly one of the two lights) = 0.37 - 0.13 = 0.24

c. P(must stop at just the first light) = P(must stop at either light) - P(must stop at the second light)

P(must stop at just the first light) = P(E u F)-P(F)

P(must stop at just the first light) = 0.37 - 0.3 = 0.07

The question deals with the topic of Probability in Mathematics. It presents the probabilities of two events, denoted as E and F, which are stopping at the first and second traffic lights, respectively. The question also provides the concurrent occurrence of both events.

The mathematics topic this question deals with is Probability. In the scenario given, E represents the event that Darlene must stop at the first traffic light and F represents the event that she needs to stop at the second traffic light. The probabilities of these events are given as P(E)=0.2 and P(F)=0.3, respectively. Additionally, we're given that the probability of both events happening (denoted P(E ∩ F)) is 0.13.

In order to analyze the situation, we can leverage the rule of joint probability, which states that the probability of two independent events both happening is the product of their individual probabilities. However, in this case the events E and F are not independent (since the probability of the intersection P(E ∩ F) is not equal to the product of probabilities P(E)*P(F)) so we know that the occurrence of E does influence the occurrence of F, and vice versa.

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Answer:

(a) There are 70 different ways set up 4 computers out of 8.

(b) The probability that exactly three of the selected computers are desktops is 0.305.

(c) The probability that at least three of the selected computers are desktops is 0.401.

Step-by-step explanation:

Of the 9 new computers 4 are laptops and 5 are desktop.

Let X = a laptop is selected and Y = a desktop is selected.

The probability of selecting a laptop is = [tex]P(Laptop) = p_{X} = \frac{4}{9}[/tex]

The probability of selecting a desktop is = [tex]P(Desktop) = p_{Y} = \frac{5}{9}[/tex]

Then both X and Y follows Binomial distribution.

[tex]X\sim Bin(9, \frac{4}{9})\\ Y\sim Bin(9, \frac{5}{9})[/tex]

The probability function of a binomial distribution is:

[tex]P(U=k)={n\choose k}\times(p)^{k}\times (1-p)^{n-k}[/tex]

(a)

Combination is used to determine the number of ways to select k objects from n distinct objects without replacement.

It is denotes as: [tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]

In this case 4 computers are to selected of 8 to be set up. Since there cannot be replacement, i.e. we cannot set up one computer twice or thrice, use combinations to determine the number of ways to set up 4 computers of 8.

The number of ways to set up 4 computers of 8 is:

[tex]{8\choose 4}=\frac{8!}{4!(8-4)!}\\=\frac{8!}{4!\times 4!} \\=70[/tex]

Thus, there are 70 different ways set up 4 computers out of 8.

(b)

It is provided that 4 computers are randomly selected.

Compute the probability that exactly 3 of the 4 computers selected are desktops as follows:

[tex]P(Y=3)={4\choose 3}\times(\frac{5}{9})^{3}\times (1-\frac{5}{9})^{4-3}\\=4\times\frac{125}{729}\times\frac{4}{9}\\ =0.304832\\\approx0.305[/tex]

Thus, the probability that exactly three of the selected computers are desktops is 0.305.

(c)

Compute the probability that of the 4 computers selected at least 3 are desktops as follows:

[tex]P(Y\geq 3)=1-P(Y<3)\\=1-[P(Y=0)+P(Y=1)+P(Y=2)]\\=1-[({4\choose 0}\times(\frac{5}{9} )^{0}\times (1-\frac{5}{9} )^{4-0}+({4\choose 1}\times(\frac{5}{9} )^{1}\times (1-\frac{5}{9} )^{4-1}+({4\choose 2}\times(\frac{5}{9} )^{2}\times (1-\frac{5}{9} )^{4-2}]\\=1-0.59918\\=0.40082\\\approx0.401[/tex]

Thus, the probability that at least three of the selected computers are desktops is 0.401.

To sketch the curve of intersection, we substitute the equation of the parabolic cylinder into the equation of the ellipsoid. We use the discriminant to determine the nature of the curve and find its parametric equations.

To sketch the curve of intersection of the parabolic cylinder and the top half of the ellipsoid, we can substitute the equation of the parabolic cylinder into the equation of the ellipsoid and then solve for the remaining variable. By doing this, we obtain a quadratic equation.

We can then use the discriminant to determine the nature of the solutions, which will help us identify if the curve is a parabola or an ellipse. Based on the discriminant, we can find the parametric equations for the curve and determine its shape.

For example, if the quadratic equation has two distinct real solutions, then the curve is an ellipse, but if it has one repeated real solution, the curve is a parabola.

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Answer:

Step-by-step explanation:

Check attachment for solution

Answer:

The trip consists of two parts. The rowing part is the hypotenuse of right angled triangle

whose sides are the distance from P to the island, which is 5, and the distance between P and

the landing point of the rowboat on the shore, which is x

so this part of trip is sqrt(25+ x^2)

The 2nd part is the walking part, which is (8-x)

Distance = rate times time (D = rt), so to get the time you have t = D/r. We must divide each

of the trip by the appropriate rate to get the time.

a) T(x) = sqrt(25+x^2)/3 + (8-x)/4

To find minimum time required take derivative of the T(x) function and find it's zeros

T'(x) = x/(3(sqrt(25+x^2)) - 1/4 = 0

x/(3(sqrt(25+x^2)) = 1/4

4x = 3sqrt(25+x^2

16x^2 = 9(25+x^2) = 225 + 9x^2

7x^2 = 225

x^2 = 225/7

x = sqrt(225/7) = 5.669467 miles

T(x) = 3.602386382 hours

Step-by-step explanation:

The point where the boat should be landed can be found by expressing

the distance travelled on the boat and walking as a function of time.

The point where the boat should be landed is the point 3.4 miles from the

point P towards the town.

Reasons:

x represent the distance from point P to the boat landing point.

Therefore, distance of rowing the boat = √((12 - x)² + 3²)

The total time, t, is therefore;

[tex]t = \dfrac{12-x}{4} +\dfrac{\sqrt{x^2 + 3^2} }{3}[/tex]

When the time is minimum, we get;

[tex]\dfrac{dt}{dx} = \dfrac{d}{dx} \left( \dfrac{12-x}{4} +\dfrac{\sqrt{x^2 + 3^2} }{3} \right) = \dfrac{12\cdot \left(-3 + 4 \cdot\dfrac{2 \cdot x }{2\cdot \sqrt{x^2 + 9} } \right)}{144}[/tex]

[tex]\dfrac{12\cdot \left(-3 + 4 \cdot\dfrac{2 \cdot x }{2\cdot \sqrt{x^2 + 9} } \right)}{144} = -\dfrac{1}{4} +\dfrac{x }{3 \cdot \sqrt{x^2 +9} }[/tex]

[tex]\dfrac{x }{3 \cdot \sqrt{x^2 +9} } = \dfrac{1}{4}[/tex]

4·x = 3·√(x² + 9)

16·x² = 9·(x² + 9)

7·x² = 81

[tex]x = \dfrac{9 \cdot \sqrt{7} }{7}[/tex]

x ≈ 3.4 miles.

The point where the boat should be landed is the point approximately 3.4

miles from the point P in the direction of the town.

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Answer:

You did the same on both exams.

Step-by-step explanation:

To compare both the scores, we need to compute the z scores of both the exams and then compare the values. The formula for z-score is:

Z = (X - μ)/σ

Where X = score obtained

           μ = mean score

           σ = standard deviation

For Exam 1:

Z = (95 - 79)/8

  = 16/8

Z = 2

For Exam 2:

Z = (90 - 60)/15

  = 30/15

Z = 2

The z-scores for both the tests are same hence the third option is correct i.e. you did the same on both exams.

Your performance on exam 1 and exam 2 can be compared using Z-scores, which measure how many standard deviations a score is from the mean. You scored 2 standard deviations above the mean on both exams, so you did the same on both exams.

In this question, your performance on exams is being compared relative to the mean of the class scores and their standard deviation. This is a concept in statistics known as Z-scores. The Z-score tells us how many standard deviations an observation (your score) is from the mean. The formula for Z-score is (observation - mean) / standard deviation.

For exam 1 your Z - score is (95-79) / 8 which equals 2. This means you scored 2 standard deviations above the mean on exam 1. For exam 2, your Z-score is (90-60) / 15 which equals 2. Again, this means you scored 2 standard deviations above the mean on exam 2. Because your Z-score for both exams is the same, you did the same on both exams.

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Answer:

[tex]z=\frac{\bar d -0}{\frac{\sigma_d}{\sqrt{n}}}=\frac{-0.625 -0}{\frac{6.818}{\sqrt{8}}}=-0.259[/tex]

[tex]p_v =2*P(z<-0.259) =0.796[/tex]

So the p value is higher than the significance level given [tex]\alpha=0.01[/tex], then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is equal to 0. So we can conclude that we don't have significant differences between the two populations.

Step-by-step explanation:

Previous concepts  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Solution to the problem

Let's put some notation :

x=values popoulation 2 , y = values population 1

x: 64 69 83 60 66 84 60 81

y: 58 76 77 70 62 76 67 76

The system of hypothesis for this case are:

Null hypothesis: [tex]\mu_y- \mu_x = 0[/tex]

Alternative hypothesis: [tex]\mu_y -\mu_x \neq 0[/tex]

The first step is calculate the difference [tex]d_i=y_i-x_i[/tex] and we obtain this:

d: -6,7,-6,10,-4,-8, 7, -5

The second step is calculate the mean difference  

[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}= \frac{-5}{8}=-0.625[/tex]

The third step would be calculate the standard deviation for the differences, and we got:

[tex]\sigma_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n} =6.818[/tex]

The 4 step is calculate the statistic given by :

[tex]z=\frac{\bar d -0}{\frac{\sigma_d}{\sqrt{n}}}=\frac{-0.625 -0}{\frac{6.818}{\sqrt{8}}}=-0.259[/tex]

Now we can calculate the p value, since we have a two tailed test the p value is given by:

[tex]p_v =2*P(z<-0.259) =0.796[/tex]

So the p value is higher than the significance level given [tex]\alpha=0.01[/tex], then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is equal to 0. So we can conclude that we don't have significant differences between the two populations.

Answer:

x = 368     production of country A (millions of (MT)

y = 215      production of country B (millions of (MT)

z =  93       production of country C (millions of (MT)

Step-by-step explanation:

Let call production as follows

Country A   production  x   millions on MT

Country B   production  y   millions on MT

Country C   production  z   millions on MT

Then according to problem statement

x  +  y  +  z   =  676        (1)

x = 60 +  y  + z

y = 122 + z

That system  ( 3 equations and three unknown varables ) could be solved by any of the available procedures.

By subtitution we get

x  =  60 + 122 + z + z     ⇒  x  =  182 + 2*z

And      

182 +  2*z  + 122 + z + z = 676

Solving for z

304  +  4*z  =  676     ⇒   4*z  =  676 - 304   ⇒   4*z  =  372

z  =  372/4       ⇒     z  =  93  millions of  (MT)

And

y  =  122 + z       ⇒  y  =  122 + 93       ⇒  y = 215  millions of (MT)

x  =  182  + 2*z    ⇒  x  =  182  + 2 ( 93)   ⇒ x  =  182 + 186

x = 368  millions of (MT)

We can cheked in equation 1

x = 368

y = 215

z =  93

Give a total of 676 millions of (MT)

Answer:

a) [tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]

b) The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean

Step-by-step explanation:

Part a

The best point of estimate for the population mean is the sample mean given by:

[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

Since is an unbiased estimator [tex] E(\bar X) = \mu[/tex]

Data given: 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1

So for this case the sample mean would be:

[tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]

Part b

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

The margin of error is given by this formula:

[tex] ME=t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]    (2)

And for this case we know that ME =0.89 with a confidence of 95%

So then the limits for our confidence level are:

[tex] Lower= \bar X -ME= 1.75- 0.89=0.86[/tex]

[tex] Upperr= \bar X +ME= 1.75+0.89=2.64[/tex]

So then the best answer for this case would be:

The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean

Answer:

[tex] -3*(3y+2) + 9y = -6[/tex]

[tex] -9y -6 + 9y = -6[/tex]

[tex]-6=-6[/tex]

So then as we can see we can have infinite solutions.

[tex]S= [(x, \frac{x-2}{3}) , x \in R][/tex]

Step-by-step explanation:

Assuming the following system of equations:

[tex] 2x-6y =4[/tex]   (1)

[tex] -3x+9y =-6[/tex]   (2)

For this case we can use the substitution method in order to find the possible solutions for the system.

If we solve for x from equation (1) we got:

[tex] 2x = 6y +4[/tex]

[tex] x = 3y +2 [/tex]   (3)

Now we can replace equation (3) into equation (2) and we got:

[tex] -3*(3y+2) + 9y = -6[/tex]

[tex] -9y -6 + 9y = -6[/tex]

[tex]-6=-6[/tex]

So then as we can see we can have infinite solutions.

And the possible solutions are for a fixed value of x, we can solve y from equation (3) and we got:

[tex] y = \frac{x-2}{3}[/tex]

So the solution would be: [tex]S= [(x, \frac{x-2}{3}) , x \in R][/tex]

Answer:

a) 0.501

b) 0.166

c) 0.3

Step-by-step explanation:

We have the following information:

1967 qualified applicants

985 accepted

327 waitlisted

655 turned away for lack of space

a. The relative frequency of accepted qualified students (to three places after the decimal) is:________

This is the number of accepted qualified students divided by the number of qualified students.

So

985/1967 = 0.501

b. The proportion of waitlisted students (to three places after the decimal) is:________

This is the number of waitlisted students divided by the number of qualified students.

So

327/1967 = 0.166

c. The percentage of students turned away from lack of space (to one place after the decimal) is:______

This is the number of students turned away by lack of space divided by the number of qualified students.

So

655/1967 = 0.3

The relative frequency of accepted students is approximately 0.501, the proportion of waitlisted students is 0.166, and the percentage of students turned away for lack of space is roughly 33.3%.

To find the relative frequency, proportion, and percentage from the given data for the school district's magnet schools program, we will use simple mathematical computations. We have a total of 1967 qualified applicants, where 985 were accepted, 327 were waitlisted, and 655 were turned away due to lack of space.

Answer:

Step-by-step explanation:

Given that A is a square matrix and A is idempotent

[tex]A^2 = A[/tex]

Consider I-A

i) [tex](I-A)^2 = (I-A).(I-A)\\= I^2 -2A.I+A^2\\= I-2A+A\\=I-A[/tex]

It follows that I-A is also idempotent

ii) Consider the matrix 2A-I

[tex](2A-I).(2A-I)=\\4A^2-4AI+I^2\\= 4A-4A+I\\=I[/tex]

So it follows that 2A-I matrix is its own inverse.

Answer:

0.0726 or 7.26%

Step-by-step explanation:

When choosing the first vial, there is a 16 in 58 chance that the vial does not have a hairline crack. When choosing the second vial, since on good vial was already picked, there is a 15 in 57 chance that the vial does not have a hairline crack. The probability that none of the 2 vials have hairline cracks is:

[tex]P = \frac{16}{58}*\frac{15}{57}\\P=0.0726[/tex]

There is a 0.0726 or 7.26% chance that none of the 2 vials have a hairline crack.

The answer are (a) 169 (b) 341 (c) 125 (d) 87

A Venn diagram is an illustration that uses circles to show the commonalities and differences between things or groups of things.

Given that, A department store surveyed 428 shoppers, and the following information was obtained: 216 shoppers made a purchase, and 294 were satisfied with the service they received. If 47 of those who made a purchase were not satisfied with the service,

Refer to the Venn diagram attached.

The total number of shoppers surveyed is, N = 428.

Number of shoppers who made a purchase, n (P) = 216

Number of shoppers who were satisfied with the service they received,

n (S) = 294

Number of shoppers who made a purchase but were not satisfied with the service, n(S' ∩ P)  = 47

(a) The number of shoppers who made a purchase and were satisfied with the service = n(S ∩ P)

n(S ∩ P) = n(P)-n(S'∩P)

= 216 - 47 = 169

(b) The numbers of shoppers who made a purchase or were satisfied with the service = n (P ∪ S)

n (P ∪ S) = n(P)+n(S)-n(S∩P)

= 216+294-169

= 341              

(c) The numbers of shoppers who were satisfied with the service but did not make a purchase = n(S∩P')

= n(S)-n(S∩P)

= 241-169

= 125

(d) The number of shoppers who were not satisfied and did not make a purchase  = n(S'∩P')

= N-n (S ∪ P)

= 428-341

= 87

Hence, the answer are (a) 169 (b) 341 (c) 125 (d) 87

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a. 169 shoppers made a purchase and were satisfied with the service.

b. 341 shoppers made a purchase or were satisfied with the service.

c. 125 shoppers were satisfied with the service but did not make a purchase.

d. 381 shoppers were not satisfied and did not make a purchase.

Let's break down the information given:

Total shoppers surveyed = 428

Shoppers who made a purchase = 216

Shoppers satisfied with the service = 294

Shoppers who made a purchase and were not satisfied = 47

We are asked to find:

a. Shoppers who made a purchase and were satisfied with the service.

To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers who made a purchase:

216 − 47 = 169

b. Shoppers who made a purchase or were satisfied with the service.

To find this, we add the shoppers who made a purchase and the shoppers who were satisfied, but we need to be careful not to count the overlap twice (those who made a purchase and were satisfied):

216+294−169=341

c. Shoppers who were satisfied with the service but did not make a purchase.

To find this, we subtract the shoppers who made a purchase and were satisfied from the total shoppers who were satisfied:

294−169=125

d. Shoppers who were not satisfied and did not make a purchase.

To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers surveyed:

428−47=381

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Answer: 29%

Step-by-step explanation:

I considered the question to be: for what proportion of cars was the number of occupants more than 1 standard deviation and greater than the mean.

The first step is to calculate the weighted mean of the number of occupants in the cars. The next step is to determine the standard deviation using the formula √[ Σ ( xi - μ )² / N ]. Subsequently, identify the number of occupants that are more than one standard deviation greater than the mean.

In this question, we're asked to calculate the proportion of cars that had more than one standard deviation above the mean number of occupants. First, we need to calculate the weighted mean (average) of the number of occupants in the cars. Based on the data, the mean can be calculated as:

Mean = (1*71 + 2*16 + 3*8 + 4*3 + 5*2) / 100 = 1.66

Next, we find the standard deviation. The standard deviation tells us how spread out the numbers are from the mean. Calculating standard deviation is a bit involved. The formula is √[ Σ ( xi - μ )² / N ]. Once you have these, look at the number of occupants that are more than one standard deviation greater than this mean.

Note: This is a statistical analysis question that requires knowledge of the concepts of mean and standard deviation.

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