D.        Suppose a nationwide study reports that the average price for a gallon of self-serve regular unleaded gasoline is $3.76. You believe that the figure is higher in your area of the country. You decide to test this claim for your part of the United States by randomly calling gasoline stations. Your random survey of 25 stations produces the following prices as shown in the table. Assume gasoline prices for a region are normally distributed. Do the data you obtained provide enough evidence to reject the null hypothesis? Use a 1% level of significance. Interpret your result.

D1.      The appropriate test should be:

            a.         a right-hand-side tailed t-test for the mean unleaded gasoline price in your area.

            b.         a right-hand-side tailed Z-test for the proportion of gasoline price in your area that                        is higher than the national average.

            c.         a left-hand-side tailed t-test for the mean unleaded gasoline price in your area.

            d.         a Chi-square test for the mean unleaded gasoline price in your area.

D2.      The appropriate observed value or test statistic is

            a.         1.9578

            b.         2.575

            c.         2.101

            d.         1.645

D3.      The appropriate critical value is, under a 1% level of significance:

            a.         2.575

            b.         1.645

            c.         2.492

            d.         2.101

D4.      Is it true that the true average unleaded gasoline price in your area is higher than the         national average of $3.76, based on the results of your test?

            a.         Yes

            b.         No, sample evidence does not support this statement.

            c.         Can’t decide.

College Graduation Rates. Data from the College Results Online website compared the 2011 graduation rate and median SAT score for 92 similar-sized public universities and colleges in the United States. The scatterplot below shows the relationship between these two variables along with the least squares fit. Round all calculated results to 4 decimal places. 1. The relationship between median SAT score and graduation rate is , , and . 2. The explanatory variable is and the response variable is . The summary statistics for graduation rate and median SAT score are listed below. The correlation between graduation rate and median SAT score is 0.673. Median SAT score: mean = 1038.1, standard deviation = 77.5 Graduation rate: mean = 49.4, standard deviation = 15 3. The equation of the regression line is y = + x 4. Complete the following sentence to interpret the slope of the regression line: An increase of in Median SAT score corresponds to a/an of in Graduation Rate. 5. The recorded median SAT score for Northern Michigan University is 1030. Use the regression equation to estimate the graduation rate for Northern Michigan University. 6. The recorded graduation rate for Northern Michigan University is 46.4. Complete the following sentence. The residual for Northern Michigan University is . This means the graduation rate at Northern Michigan University is A. the same as B. lower than C. higher than the rate predicted by the regression model. 7. Stanford University (an elite private university in California not included in this data set) has a median SAT score of 1455. Would it be appropriate to use this linear model to predict the graduation rate for Stanford? A. Yes, because 1455 is a reasonable median SAT score for an elite university. B. No, because 103.705% is too large to be a reasonable graduation rate, even for an elite university. C. No, because 1455 is beyond the range of the data used to build the regression model.

In general, high school and college students are the most pathologically sleep-deprived segment of the population. Their alertness during the day is on par with that of untreated narcoleptics and those with untreated sleep apnea. Not surprisingly, teens are also 71 percent more likely to drive drowsy and/or fall asleep at the wheel compared to other age groups. (Males under the age of twenty-six are particularly at risk.) The accompanying data set represents the number of hours 25 college students at a small college in the northeastern United States slept and is from a random sample. Enter this data into C1 of Minitab Express. 6 7 6 7 6 7 7 7 8 6 6 6 8 8 8 5 4 6 7 8 5 8 7 6 7 For the analyses that follow, we shall use 90%, 95%, and 99% as the confidence levels for the confidence interval. 5% as the level of significance (?) for the hypothesis test. 7 hours sleep as the null hypothesis (according to The Sleep Foundation). Use Minitab Express to: (i) create a boxplot – GRAPHS, Boxplot - and (ii) normal probability plot – GRAPHS, Probability Plot, and (iii) calculate descriptive statistics - STATISTICS, Describe, Descriptive Statistics. Under the “Descriptive Statistics” dialog window, click on the Statistics tab and check only Mean, SE of mean, Standard deviation, and N. Include these with the submission of your project

Use Minitab Express to: (i) create a boxplot – GRAPHS, Boxplot - and (ii) normal probability plot – GRAPHS, Probability Plot, and (iii) calculate descriptive statistics - STATISTICS, Describe, Descriptive Statistics. Under the “Descriptive Statistics” dialog window, click on the Statistics tab and check only Mean, SE of mean, Standard deviation, and N. Include these with the submission of your project.

Case Study 1 Results Count! What if all organizations focused their performance approaches to results? Would that work in all cases? It does work at Mabel’s Labels, an Ontario company that designs, manufactures and sells waterproof labels for identifying personal belongings. In its brief 13 years in business, it has expanded into Walmart in both Canada and the United States. Recognizing that people have many priorities in their lives—personal and work—the company has ensured that its business success is measured by results. This means that individual performance is also measured by results. In this way, Mabel’s people can work at almost any location. To quote one of them, “Work is what you do, not where you go.” Every week, staff determine where they will work. In making the decision, consideration is given to whether the team’s needs are being met, and whether the person’s own goals and deliverables can be met. One individual indicates that their best design work is done when alone; however, this same person expressed the value of office collaboration in coming up with design ideas. Does such a focus create a culture of results only? Not according to the employees. People get together outside of work for socializing as well as volunteering for such things as food drives and helping at a local charity for seniors. The company’s approach to performance and employee engagement earned it an award in Canada’s Top 100 Small & Medium Employers for 2015. Sources: Adapted from “2015 Canada’s Top Small & Medium Employers,” The Globe and Mail, March 2015;

Question 1: Would a results focus work in a financial services institution? Why or why not?

Question 2: If Mabel’s Labels decided to use a different approach to performance, what approach would you suggest? Why?

You have now been asked to study the yearly mean sales of cameras of two competing models at stores throughout the United States. You will also study the proportions of cameras sold that include certain lenses at a large store that sells both lenses. The specific questions you will be asked to answer are stated below. In addition, appropriate sample data for the studies you will be accomplishing are given below. Answer the following questions concerning the situations posed.

1. You first wish to know whether there is a difference in the yearly mean sales of two different camera bodies, the Nikon D5 and a competing Canon model. The population standard deviation in the sales of the Nikon model is still thought to equal 50 units. The population standard deviation in the sales of the competing Canon model is thought to equal 40 units. Random samples of 35 stores that sell the Nikon D5 and of 40 stores that sell the competing Canon model are selected. The sample mean number of Nikon D5 bodies that is sold is found to be 177.5. The sample mean number of Canon bodies that is sold is found to equal 154.4. At each of the 10% and 5% levels of significance, is the mean number of Nikon D5 bodies different from the mean number of Canon bodies sold for these two populations of stores? If, at either level of significance, you observe that there is a difference in the yearly mean sales between the populations of stores, is that difference different from 5 camera bodies? If the software makes it possible, find also both 90% and 95% confidence intervals for the difference in the yearly mean sales of the two camera bodies. Explain their meanings. Then, use these intervals to answer all previous questions posed in the problem that were addressed using hypothesis testing techniques. Use the 90% confidence interval to test at the 10% level of significance and the 95% confidence interval to test at the 5% level of significance.

R-Code

3. A rival music streaming company wishes to make inference for the proportion of individuals in the United States who subscribe to Spotify. They plan to take a survey. Let S1, . . . , Sn be the yet-to-be observed survey responses from n individuals, where the event Si = 1 corresponds to the ith individual subscribing to Spotify and the event Si = 0 corresponds to the ith individual does not subscribe to Spotify (i = 1, . . . , n). Assume that S1, . . . , Sn are i.i.d. Bernoulli(π).

a) What distribution does the random variable S = sum of Si from i = 1 to n have? Compute E(S) and var(S). The formulas should involve π and n.

(b) Suppose that n = 30 and π = 0.2. Run a Monte Carlo simulation with m = 10000 replications to verify the formulas for E(S) and var(S) from the previous question. That is, simulate 10000 i.i.d. copies of S and compare the observed average of these to the true mean, and the observed (sample) variance to the true variance. Comment.

(c) Let S¯ = S(n ^−1) = (n ^−1)*sum of Si from i=1 to n. What is the mean and variance of S¯?

(d) Verify your answers to the previous question by a Monte Carlo simulation with m = 10000 replications.

(e) Is S¯ a continuous random variable? Explain.

(f) Run a Monte Carlo simulation to estimate the probability P(S¯− 1/ √ n ≤ π ≤ S¯ + 1/ √ n) when π = 0.2 and n = 10, 20, 80, 160. Hint: For every n considered, do the following m = 10000 times: generate a random variable S˜ with the same distribution as S¯ and record whether |S˜−0.2| ≤ 1/ √ n. The Monte Carlo estimate of the desired probability is the number of times this happened divided by the total number of simulations, m = 10000.

Some students are good in mathematics and others are better at reading or writing. The question is whether there is any relationship between a student’s ability in math and his/her ability in reading or writing. The SAT, a standardized test for college admissions that is widely used in the United States, has three sections, Math, Critical Reading, and Writing. The table below contains SAT Math, Writing, and Critical Reading test scores for 20 randomly chosen students accepted by a university.

Math                         Writing                  Reading

440                           410                           410

550                         570                           520

520                         520                           540

420                         470                           410

550                         620                           530

650                         560                           560

610                         620                           550

610                         520                           600

340                           470                           400

600                           540                           620

680                           580                           580

440                           430                           470

440                           450                           370

390                           430                           390

460                           600                           600

460                           520                           500

520                           570                           580

540                           530                           570

420                           430                           470

550                           480                           530

1. Copy and Paste your data into stat crunch. Label each column.

2. Create a scatter plot for the Math Scores (x) and Writing Scores (y). Go to statàregressionàsimple linear. Select x and y and “compute” click the arrow at the bottom to see the scatter plot. Copy a paste a screen shot of your scatter plot

3. Repeat Step 2, but create a scatter plot for Math Scores (x) and Reading Scores (y). Copy and paste a screen shot of your scatter plot.

4. Looking at the two scatter plots, which appear to be more strongly correlated, Math and Writing or Math and Reading? Explain.

5. Compare the r-values for Math and Writing to the r-value for Math and reading. Which appear to be more strongly correlated Math and Writing or Math and Reading? Explain how you can tell using the r-values.

Charles is a retired bus driver who developed type 2 diabetes over 10 years ago. Since his retirement, his lifestyle has become very sedentary and he has put on a substantial amount of weight. Although he has felt tingling and numbness in his left foot for a while, he has not been worried because he thought his foot was simply “falling asleep.” Recently, a scratch on his foot does not seem to be healing and is becoming increasingly ugly. Because the sore did not bother him much, Charles figured it could not be serious until his daughter noticed a purplish discoloration spreading on the skin and oozing. When he was finally seen by his physician, Charles was rushed to the operating room. His open sore, or ulcer, is the result of a diabetic foot.The concern here is that gas gangrene may have taken hold in the dead tissue. The most likely agent of gas gangrene is Clostridium perfringens, an endospore-forming, gram-positive bacterium. It is an obligate anaerobe that grows in tissue devoid of oxygen. Since dead tissue is no longer supplied with oxygen by the circulatory system, the dead tissue provides pockets of ideal environment for the growth of C. perfringens.A surgeon examines the ulcer and radiographs of Charles’s foot and determines that the bone is not yet infected. The wound will have to be surgically debrided (debridement refers to the removal of dead and infected tissue) and a sample sent for microbiological lab analysis, but Charles will not have to have his foot amputated. Many diabetic patients are not so lucky. In 2008, nearly 70,000 diabetic patients in the United States lost a foot or limb to amputation, according to statistics from the Centers for Disease Control and Prevention.[1]Which growth conditions would you recommend for the detection of C. perfringens?

1-In the long run, what is the theoretical relationship between the rate of growth in real GDP and the real rate of interest?

A) Over the long run, rates of growth in real GDP and the real interest rate should be equal.

B)There is little correlation between rates of growth in real GDP and the real interest rate.

C)Rates of growth in real GDP and the real interest rate have both historically been approximately 5% in the United States.

D)Rates of growth in real GDP and the real interest rate generally move in opposite directions.

2-Thanks to your stellar performance on the job, your supervisor awards you a bonus of $877. However,

things are tight for the company because of the pandemic, so your supervisor informs you that you

may get a bonus of $2,464 next year if you forego your bonus this year. If you accept this offer, what

$MRS have you revealed (in % terms to the nearest tenth, i.e., one decimal place)?

3-Why is a T-bill considered risk-free as long as the US is considered by international markets to be a reliable debtor (i.e., default risk free)?

A)The term to maturity of a T-bill exempts it from any maturity-related risk. Furthermore, because T-bills are backed by the government and are very marketable, they do not carry other forms of risk (as long as international markets consider the US government as a reliable debtor).

B)T-bills are typically indexed for inflation, which allows the present value of their future cash flows to remain the same regardless of market fluctuations.

C)A T-bill is not considered risk free.

D)A T-bill is a long-term investment, so any market fluctuations balance out over time. Because the government issues T-bills, there is no risk of default.