5 Multiple Choice

1. A health fitness research group wishes to estimate the mean amount of time (in hours) that members of a fitness center spend each week exercising at the center. They want to estimate the mean within a margin of error of 0.5 hours with a 95% level of confidence. Previous data suggests that standard deviation is 2.2. which of the following is the smallest sample size that meets these criteria?

a) 60

b) 75

c) 90

d) 180

e) 190

Answer: b) 75

2. A 90% confidence interval for a population mean is determined to be 800 to 900. If the confidence is increased to 95% confidence while the sample statistics and sample size remaine the same, the confidence interval for mu

a) becomes narrower

b) becomes 0.05

c) does not change

d) becomes wider

e) becomes 0.025

Answer: c) does not change, because confidence interval changes according to sample size. Answer a is wrong because as sample size increases, confidence interval becomes wider. 

3. A researcher is interested in determining the mean energy consumption of a new compact florescent light bulb. She takes a random sample of 41 bulbs and determines that the mean consumption is 1.3 watts per hour with a standard deviation of 0.7. When constructing a 97% confidence interval, which would be the most appropriate value of the critical value?

a) 1.936

b) 2.072

c) 2.250

d) 2.704

e) 2.807

Answer: c) 2.250 because decreasing the confidence level decreases the margin of error.

4. Which of the following would result in the widest confidence interval?

a) Small sample size and 95% confidence

b) Small sample size and 99% confidence

c) Large sample size and 95% confidence

d) Large sample size and 99% confidence

e) This cannot be answered without knowing an appropriate standard deviation

Answer: b) Small sample size and 99% confidence, because small sample size and bigger confidence levels makes the widest confidence interval. All the other ones produce confidence levels that are too narrow. 

5. What is the critical t-value for finding a 96 percent confidence interval estimate from a sample of 18 observations?

a) 2.054

b) 2.205

c) 2.224

d) 2.214

e) 2.235

Answer: c) 2.224 If you look at the tchart you would get 2.224 with df of 17 and at 96% confidence. Everything else is wrong (have the wrong df or at the wrong confidence level). 

FRQs

To increase business, the owner of a restaurant is running a promotion in which a customer’s bill can be randomly selected to receive a discount. When a customer’s bill is printed, a program in the cash register randomly determines whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.2, that is, giving 20 percent of the bills a discount in the long run. However, the owner is concerned that the program has a mistake that results in the program not generating the intended long-run proportion of 0.2.

The owner selected a random sample of bills and found that only 15 percent of them received discounts. A confidence interval for p, the proportion of bills that will receive a discount in the long run, is 0.15 0.06 . All conditions for inference were met.

(a) Consider the confidence interval 0.15 0.06 .

(i) Does the confidence interval provide convincing statistical evidence that the program is not working as intended? Justify your answer.

(ii) Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.2 ? Justify your answer.

A second random sample of bills was taken that was four times the size of the original sample. In the second sample 15 percent of the bills received the discount.

(b) Determine the value of the margin of error based on the second sample of bills that would be used to compute an interval for p with the same confidence level as that of the original interval.

(c) Based on the margin of error in part (b) that was obtained from the second sample, what do you conclude about whether the program is working as intended? Justify your answer.