MGMT2263

Worksheet #1

1) A company believes its site receives significantly fewer than 300 hits per day on average. For a sample of 60 days, the average was 283 hits with a standard deviation of 58.

a) Test the hypothesis at a 5% level of significance? (Z = -2.27; conclude the site receives fewer than 300 hits per day on average)

b) If the level of significance were 1% instead, would the conclusion be different? (p-value = 0.0116; since p-value > 1%, conclusion would be different)

c) What would be the power of the test at a 5% level of significance if the average number of hits per day is 283 based on the data provided? (73.57%)

d) What would the alternative mean need to be at a 5% level of significance if they wanted the power to be 99%? Round the value to 1 decimal. (270.3)

2) Based on historical data, the life span of an electronic chip used in a PC is normally distributed with a standard deviation of 20 hours. A sample of 40 chips has an average life span of 625.4 hours.

a) Construct a 90% confidence interval for the mean life of a chip and interpret what it means. (620.2 < m < 630.6; We are 90% confident that the average chip life is between 620.2 and 630.6 hours)

b) If we test the hypothesis that the average life of a chip is 633 hours, what conclusion would we reach using a 10% level of significance? (conclude the average life is not 633 hours since this value does not fall in the 90% confidence interval)

c) What is the highest level of significance in which we would not reject the null hypothesis given the evidence? (1.64%)

d) What would be the power of the test at a 10% level of significance if the average life span of a chip is 625 hours? (81.06%)

3) A weight loss clinic wants to see if the average college student weighs more than 175 pounds. A sample of 70 students had an average weight of 182 pounds with a standard deviation of 28 pounds.

a) Does the average student weigh significantly more than 175 pounds? Test at a 3% level of significance. (Z = 2.09; conclude the average college student weighs more than 175 pounds)

b) Construct a 95% confidence interval of the weight of the average college student. Interpret the interval. (175.44 < m < 188.56; With 95% confidence, the average college student weighs between 175 and 189 pounds)

c) What would be the power of the test at a 3% level of significance if the average weight of students is 182 pounds? (58.32%)

d) What would the power of the test increase to if the level of significance were increased to 10%? (79.1%)

4) In an effort to control costs, a quality control inspector is interested in whether the mean number of ounces of sauce dispensed by bottle-filling machines differs from 16 ounces. From the bottling process, the inspector collects the following measurements: 16.3, 16.2, 15.8, 15.4, 16.0, 15.6, 15.5, 16.1, 15.9, 16.1.

a) Test at a 5% level of significance that the bottle-filling machines need adjusting. Assume that the distribution of the amount dispensed is normally distributed. (t = -1.1326; conclude the machine does not need adjusting)

b) Is there a level of significance between 1% and 10% in which the opposite conclusion would have been reached? (p-value is greater than 0.1; since this provide strong evidence for Ho, the answer is no)

5) For a special-ed class, a new teaching method was introduced to see if it would significantly increase the grade on basic arithmetic from the current grade of 52. For the seven students in the class, these were the results:

52

43

59

68

68

70

40

No assumptions about the normality of the data were made. Test the hypothesis at a 5% level of significance.

(T- = 5; conclude the new teaching method does not significantly increase the grade)

6) According to a money management firm, the standard deviation of the daily closing prices on the NYSE has been 13%. The chair of a mutual fund company wishes to determine if the standard deviation of equity mutual funds differs from 13%. The assumption is that returns on mutual funds are normally distributed. A sample of 25 equity funds shows a standard deviation of 9.8%.

a) Test the hypothesis at a 5% level of significance.(c² = 13.6388; conclude the standard deviation is not significantly different from 13%)

b) Construct a 95% confidence interval of the standard deviation of equity mutual funds. Show why you would have reached the same conclusion as in part a.

(7.6521% < s < 13.6333%; since s = 13% falls in this interval we would not reject Ho at a 5% level of significance)

c) Is there a level of significance between 1% and 10% in which the opposite verdict would have been reached? (If the level of significance were 10%, the critical value on the left side would be 13.8484 and the test statistic would fall in the rejection region)

7) A manager at National Insurance believes that more than half of claims are due to speeding. From a random sample of 75 claims, 40 were found to be associated with speeding.

a) Test the manager’s belief at the 5% level of significance. (Z = 0.58; conclude that no more than half of claims are due to speeding)

b) Suppose a level of significance had not been chosen. Why would the same conclusion be reached? (p-value = 0.281; since 28.1% > 10%, we would not reject Ho and reach the same conclusion)

c) Construct a 95% confidence interval of the percentage of claims that are associated with speeding. Interpret the interval. (42.04% < p < 64.62%; with 95% confidence, the average percentage of claims associated with speeding ranges from 42.04% to 64.62%)

d) What would be the probability of a Type II error at a 5% level of significance if the actual percentage of claims due to speeding is 53%? (87.08%)

8) A researcher wants to determine if the percentage of executives using a particular brand of PDA has significantly changed from 25%. A survey of 800 executives shows that 236 of them use this brand of PDA.

a) Test the hypothesis at a 4% level of significance. (Z = 2.94; conclude the percentage has significantly changed from 25%)

b) Suppose a level of significance had not been chosen. Why would the same conclusion be reached? (p-value = 0.0016; p-value < 1%)

c) What would be the power of the test if the percentage of executives using this brand of PDA is 30% at a 4% level of significance? (87.29%)

From the textbook:

8.32 page 252

8.47 page 257

8.52 page 260

8.73 page 266