MAT 210 Summer 2003 Exam #4 NAME ______________________

Show all work! 125 points possible, but factored to 150.

1. Definitions (14 pts) :

a) A hypothesis (in statistics) is

b) A type I error is

c) A type II error is

d) An estimator is

e) A point estimate is

f) An interval estimate is

g) The level of significance of a hypothesis test is

2. (2 pts) A sample of 50 drive-through customers at Burger King yielded a 92% confidence interval of 5.0 minutes < m < 6.7 minutes for the mean service time. Interpret this result .(In other words, in a complete sentence, explain what a 92% confidence interval for the mean service time at Burger King actually means.)

page #1 over--->>>

3. a) A random sample of 400 Tennessee voters was taken and 232 of the 400 stated that they supported the lottery.

i) (1 pt) Give the best point estimate for the proportion of Tennessee voters who support the lottery.

ii) (6 pts) Construct a 93% confidence interval for percentage of Tennessee voters who support the lottery.

best point estimate _______

E = _________

___________________

b) (2 pts) Prior to the election, several members of the State Senate stated that the minority (that is, less than 50%) of the voters in Tennessee wanted a lottery. Is this claim consistent with the result in part (a). Give a mathematical reason for your answer. (Circle one.)

Yes No , since

c) (1 pt) Show the distribution that you used in part (a) is appropriate.

d) (5 pts) You want to estimate the percentage of U.S. statistics students who get a grade of B or higher. How many such students must you survey to be 99% confident that the sample percentage is off by no more than 3 percentage points?

__________

page #2

e) The service times for 6 in-house customers at McDonald's restaurants are 2, 5, 7, 8, 4, and 4 minutes. Use this data

i) (2 pts) to calculate the sample mean and standard deviation;

ii) (1 pt) to give the best point estimate of the (population) standard deviation of service time for in-house McDonald's customers; and

iii) (6 pts) to construct a 95% confidence interval for the (population) standard deviation of service time for in-house customers at McDonald's restaurants. It is known that the service times for in-house customers at McDonald's restaurants are approximately normally distributed.

(If you cannot compute the sample mean and standard deviation, you may assume they are 4.0 minutes and 1.5 minutes , respectively, and receive partial credit for ii) and iii) using these values.)

i) mean = _____________; standard deviation = ________

ii) __________ minutes

iii) _____________________

page #3 ---->>>

3. f) (6 pts) Use the data in problem 3 e) to compute a 99% confidence interval for the mean service time of in-house McDonald's customers. (Again, use a sample mean 0f 4.0 and a sample standard deviation of 1.5 if you unable to compute them.)

E = ________

_____________

3. g) (5 pts) In a Gallup poll, 90% of those surveyed indicated that restaurants and bars should refuse service to patrons who have had too much to drink. You wish to conduct a new poll to confirm that the percentage continues to be correct. Assuming you use the results of the Gallup sample, how many randomly selected adults must you survey if you want to be 95% confident that the margin of error is (no more than) 4 percentage points?

_________

page #4

3. h) In a particular society, marriage counselors are interested in the length of time from a first marriage to a separation for divorced couples. A random sample of 80 divorced couples yielded a mean time to separation of 6.3 years with a standard deviation of 2.9 years.

i) (1 pt) Give the best point estimate of the mean time to separation.

ii) (6 pts) Construct a 91% confidence interval for the mean time to separation, first calculating the margin of error E.

i) _______

ii) E = _______

________________

i) (5 pts) Union officials are concerned about reports of inferior hourly wages paid to employees in an industry under their jurisdiction. It is known from past studies that these hourly wages have a standard deviation of $3. Determine the sample size required if the union officials want to be 94% confident that their estimate of the mean hourly wage is in error by no more than $0.50.

________

page #5

4. a) (3 pts) Identify the claim : "The standard deviation of the amount of Coke in a can is at least 12 ounces." as either the null hypothesis or the alternative hypothesis.

_______

Finish the following sentence, explaining how you would make a Type I error with this claim. Do not have symbolism or mention null or alternative hypothesis in your (complete sentence) answer.

We would make a Type I error if we conclude

b) (3 pts) Identify the claim : "The mean weight of UTC basketball players is less than 180 pounds." as either the null hypothesis or the alternative hypothesis.

__________

Finish the following sentence, explaining how you would make a Type II error with the claim. Do not have symbolism or mention null or alternative hypothesis in your (complete sentence) answer.

We would make a Type II error if we conclude

5. (36 points; 4 hypothesis tests at 9 points each) In each part test a hypothesis using the appropriate test statistic and critical region.

a) An experiment was conducted to determine the abrasion resistance of a new type of automobile paint. 12 different strips of metal were painted with the new paint and this sample had a mean abrasion resistance score of 3.20 with a standard deviation of 0.73. At the 0.05 level of significance, test the claim that the new paint has a mean abrasion resistance score greater than 2.90, the mean abrasion resistance score of the paint currently used. Assume that the abrasion resistance scores have a distribution that is approximately normal.

page #6

b) Vell Computers embarked on a serious marketing campaign directed at small businesses. After the campaign 300 small businesses were surveyed and it was found that 48 of the companies favored Vell. Test the claim that the percentage favoring Vell after the advertising campaign is greater than 12%, the percentage that favored Vell previously. Use a 0.10 level of significance.

c) It is claimed that the mean amount of money expended for food (per household per week) in Lafayette is equal to $128, the mean amount spent (per household per week) nationally. A random sample of the weekly food expenditures of100 Lafayette households yielded a mean of $124 and a standard deviation of $10.30. Use this information to test the claim at the 0.04 level of significance.

d) The standard deviation of the waiting times for banks with multiple lines is 6.2 minutes. A bank experimented with a single main waiting line and found that, for a simple random sample of 25 customers, the waiting times had a mean of 15.0 minutes and a standard deviation of 3.8 minutes. Use a 0.05 level of significance to test the claim that the waiting time for banks with a single line has lower variation than the waiting time for banks with multiple lines Assume that the waiting times at banks are normally distributed.

6. A biologist wishes to determine if there is a linear relationship between growth rate y and incubation time x of a bacteria culture. The growth rate and incubation time were determined for 5 different cultures of bacteria, yielding :

x (hours) 1 3 5 7 9

y (# per hour) 10.0 10.3 12.2 12.6 13.9

a) (3 pts) Determine the linear correlation coefficient.

b) (2 pts) Determine the percentage of variation in y that is due to the variation in x.

c) (5 pts) Determine the regression line.

d) (2 pts) Use the regression line to predict the growth rate when the incubation time is 6 hours.

e) (8 pts) Test the claim that there is no linear correlation between incubation time x and growth rate y. Use a 0.05 level of significance.

page #7

Answer Key

2. If we repeatedly take samples of 50 drive-through customers at Burger King and compute 92% confidence intervals for the mean service time from each sample, 92% of these confidence intervals will contain the population mean m and 8% will not.

3. a) i) 0.58 ii) 4.5%; 53.5% < p < 62.5% b) no, since 50% is not in our 93% confidence interval (53.5%, 62.5%)

c) np = 400(0.58) = 232 . 5; nq = 400(0.42) = 168 . 5

d) 1842 e) i) 5.0; 2.2 ii) 2.2 iii) 1.37 < s < 5.37

f) E = 3.6; 1.4 < m < 8.6 g) 217

h) i) 6.3 ii) 0.55 iii) 5.75 < m < 6.85 i) 128

4. a) H0; we conclude the standard deviation of the amount of Coke in a can is less than 12 ounces, when, in fact, it is not.

b) H1; we conclude the mean weight of UTC basketball players is at least 180 pounds, when, in fact, it is not.

5. a) H1: m > 2.90 (right-tailed test) ; CR: t > 1.796; TS: t = 1.423

Conclusion: Fail to reject H0 due to insufficient evidence to the

contrary.

b) H1: p > 0.12 (right-tailed test); CR: z > 1.28; TS: z = 2.13;

Conclusion: Reject H0 with 90% confidence.

c) H1: m _ 128 (2-tailed test); CR: z < - 2.05 or z > 2.05;

TS: z = - 3.88

Conclusion: Reject and conclude m < 128 with 96% confidence.

d) H1: s < 6.2 (left-tailed test); CR: X² < 13.848; TS: X² < 9.016;

Conclusion: Reject H0 with 95% confidence.

6. Summations (x, y, x², y², xy) = (25, 59, 165, 706.9, 315.2)

a) r = 0.976 b) 0.953 c) y = 0.505x + 9.275

d) 12.305 (approximately 12 bacteria per hour)

e) H1: r _ 0 (There is significant linear correlation between incubation time x and growth rate y.) (2-tailed test)

CR: t < - 3.182 or t > 3.182; TS: t = +7.763

Conclusion: Reject H0; we conclude there is a significant positive correlation between incubation time x and growth rate y with 95% confidence.

.