Show, organize, and label (SOL) your work. Use correct notation as done in class or the book. If you use a calculator function to do a calculation, write down the keys used. You do not have to write down calculator steps if you’re just using a calculator to do arithmetic calculations.

1. At the Shenandoah Valley Produce Auction (SVPA) a box contains 5 ghost pumpkins with the following weights: {5 lbs, 7 lbs, 8 lbs, 11 lbs, 14 lbs}. Let X = weight of ghost pumpkins in the box.

a) Fill in the missing cells in the table below. Calculate the mean and standard deviation for all 3 columns. Note: You should use the population standard deviation formula.

b) How did transforming the original data (X) by subtracting the mean of the original data change the mean? In other words, what is the mean of Y? How did dividing by the standard deviation change the standard deviation of the original data? In other words, what is the standard deviation of Z?

c) Does the variable Z have a standard normal distribution? Why did you answer the way you did? Think about what properties a standard normal distribution has.

d) What is the probability that X is greater than 7 if a pumpkin is randomly chosen from the box (should be very simple to answer)? How does this compare to the probability that the weight of a randomly chosen pumpkin has a Z-score greater than the z-score of 7 (again easy to answer)?

2.. The SVPA sells a box of 6 Blue Hubbard pumpkins. The mean weight of all the pumpkins in the box is 14.5 lbs. The table below shows the distribution of the sample mean weight if 3 pumpkins are selected randomly from the box.

a. Suppose the three pumpkins selected are 3lbs, 9 lbs and 21 lbs. Construct a 60% confidence interval on the population mean using this sample. Explain how you figured out the margin of error.

b. Interpret the 60% confidence interval found in part a. Give the full interpretation and use the context of the problem

c. The SVPA claims that based on years of data the mean weight of the pumpkins in the box is 14.5 lbs. Does the 60% confidence interval from part a contradict this claim?

d. What is the probability the mean weight of the pumpkins in the box is in the 60% confidence interval found in part a. Explain.

e. For the hypotheses H₀: μ = 14.5 lbs and H₁: μ ≠ 14.5 lbs on the mean weight of the pumpkins in the box, what would the p-value be for the hypothesis test using the sample from part a. Use correct notation. Explain what the p-value represents within the context of the problem.

f. For the hypothesis test in part e, what would be the lowest significance level at which we would reject the null hypothesis? Explain.

3. Giant pumpkins frequently don’t contain seeds when cut open because of excessive in-breeding. This is unfortunate since seeds can sell for $30 each or more. In a sample of 50 giant pumpkins, 23 did not contain seeds.

a) What is the point estimate for the proportion of giant pumpkins that don’t have seeds?

b) Show that the necessary conditions are satisfied for the sample proportion to have an approximately normal distribution.

c) Using the sample above an obstreperous giant pumpkin researcher wants to use the unusual confidence level of 75% to create a confidence interval for the true proportion of giant pumpkins not containing seeds

                i) What would be the standard error for the sample proportion? Use p-hat as an estimate of p.

                ii) What is the critical value for this confidence interval?

                iii) What is the margin of error?

                iv) What is the75% confidence interval?

d) Interpret the 75% confidence interval within the context of the problem.

e) What sample size would be necessary to produce a margin of error equal to 5 percentage points assuming the true proportion is equal to the sample proportion from above?

Suppose two independent random samples of sizes n1 = 9 and n2 = 7 that have been taken from two normally distributed populations having variances σ12 and σ22 give sample variances of s12 = 94 and s22 = 13. (a) Test H0: σ12 = σ22 versus Ha: σ12 ≠ σ22 with σ = .05. What do you conclude? (Round your answers to 2 decimal places.) F = 7.231 F.025 = H0:σ12 = σ22 (b) Test H0: σ12 < σ22versus Ha: σ12 > σ22 with σ = .05. What do you conclude? (Round your answers to 2 decimal places.) F = 4.147 F.05 = H0: σ12 < σ22

Discuss similarities and differences among these ways to measure data variation.

Why would it seem reasonable to pair the median with a box-and-whisker plot & to pair the mean with the standard deviation?

What are the advantages and disadvantages of each method of describing data spread?

Comment on statements such as following:

a)      The range is easy to compute, but it doesn’t give much information;

b)      Although the standard deviation is more complicated to compute, it    
      has some significant applications;

c)      The box-and-whisker plot is fairly easy to construct and it gives a
       lot of information at a glance.

The concession stand at the Shelby High School stadium sells slices of pizza during soccer games. Concession stand sales are a primary source of revenue for high school athletic programs, so the athletic director wants to sell as much food as possible. However, any pizza not sold is given away to the players, coaches, and referees, or it is thrown away. The athletic director wants to determine a reorder point that will meet, not exceed, the demand for pizza. Pizza sales are normally distributed, with a mean of 6 pizzas per hour and a standard deviation of 2.5 pizzas. The pizzas are ordered from Pizza Town restaurant, and the mean delivery time is 30 minutes, with a standard deviation of 8 minutes.

1. Currently, the concession stand places an order when it has 1 pizza left. What level of service does this result in?
2. What should the reorder point be to have a 98% service level?

You are part of a marketing team that needs to research the potential of a new product. Your team decides to e-mail an interactive survey to a random sample of consumers. Write a short questionnaire that will generate the information you need about the new product. Select a sample of 200 using an SRS from your sampling frame. Discuss how you will collect the data and how the responses will help your market research.

5. (18 pts) Consider the following situation, and answer each of the following multiple-choice questions. Indicate your answer next to each question in the blank. Possible answers are: A = it increases; B = it decreases; C = it stays the same; D = not enough information to say for sure. No need to restate the question or to justify your answer.

______A. As sample size decreases, what happens to the standard error?   

______B. As a sample becomes less and less representative of the population from which it is drawn, what happens to the value of the sample mean?

______C. As |zobt| increases, what happens to the p value?

______D. As |zobt| increases, what happens to alpha?

______E. As power increases, what happens to the probability of correctly rejecting the null hypothesis?

______F. As N increases, what happens to |zcrit|?

______G. As N increases, what happens to |zobt|?

______H. As the number of tails increases from one to two, what happens to |zcrit|?

______I. As the absolute value of the difference between means decreases, what happens to the absolute value of Cohen’s d?

______J. If the standard error decreases in value only because the sample size changed, what will happen to the absolute value of Cohen’s d?

______K. As N increases, what happens to |tobt|? (Assume the increase in N does not affect the value of s.)  

______L. As degrees of freedom increase, what happens to |tcrit|?

______M. As degrees of freedom increase and alpha decreases in value, what happens to |tcrit|?

______N. As the difference between means increases and the sample standard deviation decreases, what happens to |tobt|?

______O. As degrees of freedom decrease in a single sample t test, what happens to power?

______P. As |tobt| decreases, what happens to |tcrit|?

______Q. As |robt| increases, what happens to the likelihood of rejecting the H0 that ρ = 0?

______R. As N decreases and alpha decreases in value, what happens to |rcrit|?

show that standardized poisson follows poisson with mean 0 and variance 1

Conestoga college recently released a flyer in an attempt to recruit accounting students for their business school. In the flyer, Conestoga college claims that the mean first year graduate salary of their accounting students is $50,000, whereas their biggest competition for accounting students, Fanshawe College, have a first year graduate salary of only $45,000.

After reading Conestoga's recruiting flyer, Fanshawe College executives believe that Conestoga falsely stated the graduate salaries and as a result have conducted their own study. Based on the sample below, at 1% level of significance is there evidence that Conestoga's first year accounting graduates salaries are less than claimed?

BLANK #1: Is this a question involving mean or proportion? ***ANSWER "MEAN" OR "PROPORTION" (WITHOUT THE QUOTATION MARKS)***

BLANK #2: Which type of distribution should be used to calculate the probability for this question? ***ANSWER "NORMAL", "T", OR "BINOMIAL" (WITHOUT THE QUOTATION MARKS)***

BLANK #3: Which of the following options are the appropriate hypotheses for this question: ***ANSWER WITH THE CORRECT LETTER, WITHOUT ANY QUOTATION MARKS OR BRACKETS***

A) H₀: μ = $50,000 H₁: μ > $50,000

B) H₀: μ = $50,000 H₁: μ < $50,000

C) H₀: μ = $50,000 H₁: μ ≠ $50,000

D) H₀: p = $50,000 H₁: p > $50,000

E) H₀: p = $50,000 H₁: p < $50,000

F) H₀: p = $50,000 H₁: p ≠ $50,000

BLANK #4: What is the p-value of this sample? ***ANSWER TO 4 DECIMALS, BE SURE TO INCLUDE LEADING ZERO, EXAMPLE "0.1234"...NOT ".1234"***

BLANK#5: Based on this sample, at 1% level of significance is there evidence that Conestoga's first year accounting graduates salaries are less than claimed? ***ANSWER "YES" OR "NO" (WITHOUT THE QUOTATION MARKS)***

In what follows use any of the following tests/procedures: Regression, confidence intervals, one-sided t-test, or two-sided t-test. All the procedures should be done with 5% P-value or 95% confidence interval.

Use the Weight_vs_IQ data. SETUP: Common sense dictates that a person’s IQ and Weight should not be related. However, one never knows until one examines the data. Given the data (IQ and weight for 29 students) your job is to check if the common sense assumption is reasonable or maybe it is not.

13. What test/procedure did you perform?

• a. One-sided t-test
• b. Two-sided t-test
• c. Regression
• d. Confidence interval

14. What is the P-value/margin of error?

• a. 0.749971
• b. 8.2E-06
• c. 0.711936
• d. 0.355968

15. Statistical interpretation

• a. Since the P-value is very small, we can confirm the common sense assumption that IQ and Weight are not related.
• b. Since the P-value is large, we cannot claim that averages are different.
• c. Since the P-value is large, we cannot claim that IQ and Weight are related.
• d. None of these.

16. Conclusion

• a. I have no statistical evidence to claim that IQ and Weight are related.
• b. I am confident that IQ and Weight are related.
• c. I am confident that IQ and Weight are not related.

DATA:

Weight  IQ
88      99
112     103
107     97
112     99
80      100
86      90
103     118
121     79
116     80
82      88
65      82
107     103
100     116
91      112
110     92
113     115
105     100
108     94
90      118
73      106
102     126
99      95
80      119
114     116
90      79
95      92
106     78
86      111
106     80

Table below shows monthly beer sales at Gordon’s Liquor Store in 2017.

The sales manager had predicted that January 2017 beer sales would be 950.

• Forecast beer sales needed for January 2018 using the moving average with 3 periods.

• Forecast beer sales needed for January 2018 using an exponential smoothing method with α = 0.2 and α = 0.9

• What is the best method to forecast the beer sales according to MSE. (moving average with 3 periods, exponential smoothing method with α = 0.2, or exponential smoothing method with α = 0.9)

Data on pull-off force (pounds) for connectors used in an automobile engine application are as follows: 79.7 75.1 78.2 74.1 73.9 75.0 77.6 77.3 73.8 74.6 75.5 74.0 74.7 75.9 72.5 73.8 74.2 78.1 75.4 76.3 75.3 76.2 74.9 78.0 75.1 76.8 If necessary, round all intermediate calculations to four decimal places (e.g. 12.3456). (a) Calculate a point estimate of the mean pull-off force of all connectors in the population (Round the answer to four decimal places (e.g. 90.2353).) (b) Calculate a point estimate of the pull-off force value that separates the weakest 50% of the connectors in the population from the strongest 50% (Express the answer to two decimal place (e.g. 90.15).) (c) Calculate the point estimate of the population variance (Round the answer to three decimal places (e.g. 3.567).) (d) Calculate the point estimate of the population standard deviation (Round the answer to two decimal places (e.g. 1.23).) (e) Calculate the standard error of the point estimate found in part (a) (Round the answer to two decimal places (e.g. 1.23).) (f) Calculate a point estimate of the proportion of all connectors in the population whose pull-off force is less than 73 pounds (Round the answer to three decimal places (e.g. 0.123).)

Left ventricular mass (LVM) is an important risk factor for subsequent cardiovascular disease. A study is proposed to assess the relationship between childhood blood pressure levels and LVM in children as determined from echocardiograms. The goal is to stratify children into a normal bp group (< 80th percentile for their age, gender, and height) and an elevated bp group (≥ 90th percentile for their age, gender, and height) and compare change in LVM between the 2 groups. Before this can be done, one needs to demonstrate that LVM actually changes in children over a 4-year period.

To help plan the main study, a pilot study is conducted where echocardiograms are obtained from 10 random children from the Bogalusa Heart Study at baseline and after 4 years of follow-up.

Implement an appropriate 2-sided test to test the hypothesis that there is a change in mean LVM over 4 years?

You must clearly write out all 4 steps of the appropriate hypothesis test clearly defining the parameter(s) involved, calculate the value of the test statistic and the p-value, and state your conclusion in the context of the problem.

Considering the following observations for an amount of polymerization of a certain specimen:

418 421 421 422 425 427 431 434 437 439 446 447 448 453 454 463 465

a) Using Minitab, construct and interpret a boxplot.

b) Is it plausible that the given sample observations were selected from a normal distribution?

c) Calculate a two-sided 95% confidence interval for true average amount of polymerization. Does the

interval suggest that 440 is a plausible value for true average? What about 450? (Use Minitab when

necessary)

A journal article reports that 34% of fathers take no responsibility for child care. A group of fathers believes that this estimate is too high. In a random sample of 80 fathers, it was found that 23 of those father take no responsibility for child care. Is there significant evidence at the 1% significance level that less than 34% of father take no responsibility for child care?