3. The distances traveled by individuals to work is exponential, with a mean of 9 miles.

(a) What is the probability that they travel between 5 and 11 miles? (2)

(b) What is the 80^th percentile of this distribution? (2)

4. The annual number of crimes in a city has a normal distribution, with mean 200, and standard deviation 42.

(a) What is the probability of less than 150 crimes next year? (2)

(b) What is the 75^th percentile of this distribution? (2)

Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 15 cards in all. What is the probability of drawing at least 8 diamonds?

Exercise 5 (15) In the process of learning how to conduct a sign test, each student inevitably punches their computer screen several times. I have found that students who punch their screens during this exercise can be separated into two categories. The first category believes that this statistics course would be better if they did all of their statistics by hand. The second category is pleased that the course is taught using R. Below is the number of times a computer screen is punched by a student in each category. Assume all assumptions have been met.

1. By hand, test the hypothesis that there is no difference between the two categories in relation to how many times they punch their computer screen. Provide the scrap paper used to conduct this test.

2. By hand, calculate the 95% CI of the difference between the means.

(b) In a tutorial session of AMA1501, there are 11 accounting students, 6 marketing students and 8 financial service students. Among these 25 students, a group of 5 students is selected randomly for the first presentation.
i. How many different groups can be formed?
ii. What is the probability that this group consists of only accounting students?
iii. What is the probability that this group consists of exactly 2 accounting students and 3 marketing students?

The following statistics are computed by sampling from three normal populations whose variances are equal: (You may find it useful to reference the t table and the q table.)

xbar = 26.5, n₁ = 7; xbar−2= 32.2, n₂ = 8; xbar 3= 35.7, n₃ = 6; MSE = 33.2

a. Calculate 99% confidence intervals for μ₁ − μ₂, μ₁ − μ₃, and μ₂ − μ₃ to test for mean differences with Fisher’s LSD approach. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

b. Repeat the analysis with Tukey’s HSD approach. (If the exact value for n_T – c is not found in the table, then round down. Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

c. Which of these two approaches would you use to determine whether differences exist between the population means?

• Tukey's HSD Method since it protects against an inflated risk of Type I Error.

• Fisher's LSD Method since it protects against an inflated risk of Type I Error.

• Tukey's HSD Method since it ensures that the means are not correlated.

• Fisher's LSD Method since it ensures that the means are not correlated.

In Example 6.33 we found the distribution of the sum of two i.i.d. exponential variables with parameter λ. Call the sum X. Let Y be a third independent exponential variable with parameter λ. Use the convolution formula 6.8 to find the sum of three independent exponential random variables by finding the distribution of X + Y.

1. The reaction of an individual to a stimulus in a psychological experiment may take one of the two forms: A or B. An experimenter wishes to estimate the true proportion of people who will react in manner A. How many people must she include in the sample if she wants to estimate the true proportion with a 85% confidence interval of length 0. 06? Also, assume that a pilot study had revealed the proportion to be somewhere around 15%.

2. Suppose the sample mean lifetime of twenty-five batteries of a certain type (say Type A) was found to be 4. 1 hours with a corresponding standard deviation 0. 87 hours. Similarly, the sample mean lifetime of twenty-two batteries of another type (say Type B) was found to be 4. 9 hours with a corresponding standard deviation 0. 42 hours. At a significance level of alpha = 0. 02, is there enough evidence that Type A batteries is inferior to Type B?

A company wants to study the relationship between an employee's length of employment and their number of workdays absent. The company collected the following information on a random sample of seven employees.

1. What is the independent variable (X)?

2. What is the dependent variable (Y)?

3. What is the slope of the linear equation?

4. What is the Y intercept of the linear equation?

5.What is the least squares equation for the data?

6.What is the meaning of a negative slope?

7.What is the standard error of estimate?

How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 96% confidence the mean winning's for all the show's players.

13786 ,26730 ,25749 ,25857, 23425, 11656, 19845, 24269, 14632, 24309, 12774,17929,

16233, 21365, 25571

UCL =

LCL=

You are interested in finding a 90% confidence interval for the mean number of visits for physical therapy patients. The data below show the number of visits for 12 randomly selected physical therapy patients. Round answers to 3 decimal places where possible.

a. To compute the confidence interval use a ? z t  distribution.

b. With 90% confidence the population mean number of visits per physical therapy patient is between  and   visits.

c. If many groups of 12 randomly selected physical therapy patients are studied, then a different confidence interval would be produced from each group. About  percent of these confidence intervals will contain the true population mean number of visits per patient and about  percent will not contain the true population mean number of visits per patient.

Two plots at Rothamsted Experimental Station were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows.

Use a calculator to verify that, for this plot, the sample variance is s² ≈ 0.285.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

Use a calculator to verify that the sample variance for this plot is s² ≈ 0.449.

Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifcance. (a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²     H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

What assumptions are you making about the original distribution?

The populations follow independent normal distributions. We have random samples from each population. The populations follow independent chi-square distributions. We have random samples from each population.     The populations follow dependent normal distributions. We have random samples from each population. The populations follow independent normal distributions.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.200 0.100 < p-value < 0.200     0.050 < p-value < 0.100 0.020 < p-value < 0.050 0.002 < p-value < 0.020 p-value < 0.002

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.     At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots. Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.     Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots. Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.

Imagine our population of interest is all of the students who took a statsquiz. The average quiz score from all of the students in the class was 85 and the variance was 5. Imagine we could select several different samples from this population of 85 students. What would the standard deviation of the sampling distribution of the mean be for N=5?

A set of phones are normally distributed with a mean of 900 dollars and a standard deviation of 70 dollars. What proportion of phone prices are between 650 dollars and 925 dollars?

If the score variance were 40 and the error variance were 22, then the reliability would be?

Question 4 :In the planning of the monthly production for the next four months, in each month a company must operate either a normal shift or an extended shift (but not both) if it produces. It may choose not to produce in a month. A normal shift costs $100,000 per month and can produce up to 5,000 units per month. An extended shift costs $140,000 per month and can produce up to 7,500 units per month.

The cost of holding inventory is estimated to be $2 per unit per month (based on the average inventory held during each month) and the initial inventory is 3,000 units (i.e., inventory at the beginning of Month 1). The inventory at the end of month 4 should be at least 2,000 units. The demand for the company's product in each of the next four months is estimated to be as shown below:

Production constraints are such that if the company produces anything in a particular month it must produce at least 2,000 units. The company wants a production plan for the next four months to meet its demands. Formulate an integer programming model to solve the problem at minimum cost.

Decision variables :

Objective function :

Constraints :

Additional constraint 1 :The company can operate an extended shift in Month 1 only if it operates a normal shift in each of Month 2, Month 3 and Month 4.

Additional constraint 2 :The company must produce in either Month 1 or Month 2 (or both) if it does not produce in Month 3.

Additional constraint 3 :For each of Month 2, 3 and 4,the company cannot operate an extended shift in a month if it operates one in the previous month.

A consumer product testing organization uses a survey of readers to obtain customer satisfaction ratings for the nation's largest supermarkets. Each survey respondent is asked to rate a specified supermarket based on a variety of factors such as: quality of products, selection, value, checkout efficiency, service, and store layout. An overall satisfaction score summarizes the rating for each respondent with 100 meaning the respondent is completely satisfied in terms of all factors. Suppose sample data representative of independent samples of two supermarkets' customers are shown below.

(a)

Formulate the null and alternative hypotheses to test whether there is a difference between the population mean customer satisfaction scores for the two retailers. (Let μ₁ = the population mean satisfaction score for Supermarket 1's customers, and let μ₂ = the population mean satisfaction score for Supermarket 2's customers. Enter != for ≠ as needed.)

H₀:

H_a:

(b)

Assume that experience with the satisfaction rating scale indicates that a population standard deviation of 17 is a reasonable assumption for both retailers. Conduct the hypothesis test.

Calculate the test statistic. (Use

μ₁ − μ₂.

Round your answer to two decimal places.)

Report the p-value. (Round your answer to four decimal places.)

p-value =

At a 0.05 level of significance what is your conclusion?

Reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.    Do not reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Do not reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

(c)

Which retailer, if either, appears to have the greater customer satisfaction?

Supermarket 1

Supermarket 2    

neither

Provide a 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers. (Use

x₁ − x₂.

Round your answers to two decimal places.)

to

In a​ lottery, you bet on a six-digit number between 000000 and 111111. For a​ $1 bet, you win ​$700,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are μ=0.70 (that is, 70 cents) and σ=700.00. Joan figures that if she plays enough times every​ day, eventually she will strike it​ rich, by the law of large numbers. Over the course of several​ years, she plays 1 million times. Let x denote her average winnings.

a. Find the mean and standard deviation of the sampling distribution of x

b. About how likely is it that Joan​'s average winnings exceed​ $1, the amount she paid to play each​ time? Use the central limit theorem to find an approximate answer.