Question

Pauly’s Pizza claims that the mean time it takes for them to deliver a pizza to dorms at Nat’s college is 31 minutes. After a long wait one night, Nat decides to test this claim. He randomly selects 15 dormitory residents and asks them to record the time it takes for Pauly’s to deliver the next time they order pizza. Here are the results (in minutes). The sample mean is x=33.8 and the sample standard deviation is s= 7.72 .

31 38 39 25 26 45 42 32 23 38 42 21 40 37 28

a. We want to use this information to construct a 90% confidence interval to estimate the true mean delivery time. State the parameter our confidence interval will estimate (in context).

b. Identify the conditions that must be met to use this procedure and explain how you know that each one has been satisfied. For the Nearly Normal Condition, include a picture of the histogram of the sample data. Make sure to include labels. Hint: Start the distribution from 20 and use a bin width of 4.

c. Find the appropriate critical value () and the standard error of the sample mean (). SHOW YOUR WORK! Round the standard error to two decimal places.

d. Use the formula shown in your notes to get the 90% confidence interval by hand. SHOW YOUR WORK! Round to one decimal places.

e. Interpret the confidence interval constructed in part (d) in the context of the problem.

f. Interpret the confidence level in the context of the problem.

g. Suppose you wanted to estimate the mean delivery time to Nat’s college with 90% confidence to have a margin of error no more than 5 minutes. Calculate how large a sample you would need. Assume min. SHOW YOUR WORK! Remember to round your final answer UP to the nearest whole number.

h. Recall that Pauly’s Pizza claimed the average time it would take to deliver to Nat’s college is 31 minutes. Does your 95% confidence interval support this claim?

i. What is the name of the significance test that can we perform to test the claim made?

j. What hypotheses would we use if we wished to conduct a two-sided test?

k. Calculate the t-score using the formula shown in class. Round to two decimal places. SHOW YOUR WORK! l. Use your t-score (with corresponding degrees of freedom) to estimate the p-value with our t-table. m. It turns out the exact p-value is 0.182. Interpret the p-value in context.

n. What decision would you draw based on the size of the p-value?

o. Are our confidence interval and significance test results in agreement?

Answer 1

a.

The parameter that our confidence interval will estimate is the true mean delivery time of pizza to dorms at Nat’s college.

b.

1) The sample must be drawn randomly which is given in this case.

2) The sample values have to be independent of each other. Since the sample is drawn randomly, we can be sure that they are independent of each other.

3) If the sample is drawn without replacement (which must be the case here), the sample size, n should not be more than 10% of the population. We have n =15 and we can be sure enough that our population of delivery times is for more than 150 deliveries.

4) The sample size must be large enough for the Central Limit Theorem to hold good so that we can use normal distribution. If n > 30, it is considered large enough. Here, n =15 which is a small sample. So, we need to check for nearly normal condition.

On X-axis (horizontal axis): Delivery time (in minutes)

On Y-axis (vertical axis): Frequency

The histogram shows that the data is nearly normal.

c.

Since n =15 < 30, we use t-score.

Critical value of t at 90% confidence level for a two-tailed case at n-1 =14 degrees of freedom is: t-critical =1.7613

Standard Error of sample mean, SE(​​​​​​) =s/ =7.72/ =1.99

d.

Sample mean, =33.8

90% confidence interval for the population mean, is:

[t-critical*SE()] =33.8(1.7613*1.99) =[30.3, 37.3]

e.

Interpretation of confidence interval:

We are 90% confident that the interval [30.3, 37.3] contains the true mean delivery time of pizza to dorms at Nat’s college.

f.

Interpretation of 90% confidence level:

If we drew many random samples of size 15 and constructed 90% confidence intervals, then we would expect 90% of such intervals contain the true delivery time of pizza to dorms at Nat’s college and 10% of such intervals do not contain it. This 10% is called the significance level.