MGMT2263 Tutorial Sheet #1

1) A grocery store was concerned that customers were spending less than $125 on average. Past records indicate the amount that customers spend is normally distributed with a standard deviation of $20.50. A sample of 12 customers was taken and the average of this sample was $110.03. Use the p-value approach to reach your decision.

(Z = -2.53; p-value = 0.0057; reject Ho; conclude that customers spend less than $125 on average)

2) What would be the power of the test at a 5% level of significance if the average amount spent by customers is actually $110? (81.33%)

3) What would the alternative mean need to be if they want the power to be 99%? Round to the nearest cent. ($101.50)

4) A new clock system was tested to see if the mean difference between the system and Greenwich Mean Time (GMT) was significantly different from zero. A sample of 40 measurements was taken. The mean difference for the sample was 0.02 seconds with a standard deviation of 0.0832 seconds. Is there any significant difference between the system and GMT? Test at the 5% level of significance. (Z = 1.52; do not reject Ho; conclude no significant difference)

5) If you were to use a 95% confidence interval in the previous question, show why you would reach the same conclusion (-0.0058 < m < 0.0458; hypothesized mean of zero falls into this interval)

6) In question 4, is there a level of significance between 1% and 10% in which the opposite conclusion would have been reached? (p-value = 0.1286; since this provides strong support for Ho, these is no level of significance between 1% and 10% that could have been chosen in which the opposite conclusion would have been reached)

7) Based on the data in question 4, what is the probability of a Type II error at a 5% level of significance if the actual difference between the two systems is 0.02 seconds? (66.98%)

8) A focus group of 8 people was convened to test the hypothesis that the average number of hours that people watch TV is less than 20 hours a week. They assumed that the distribution is normal. From the group they collected the following data: 25, 23, 15, 19, 12, 27, 9, 18. Based on the evidence provided by this group, do people watch fewer than 20 hours of TV a week on average? Use the p-value method. (t = -0.6708; p-value > 0.1; conclude people do not watch fewer than 20 hours of TV a week on average)

9) As an aside in the study in question 8, they wanted to see if there was any significant difference in the standard deviation of the group from that of the general population of 10 hours per week. Test at a 5% level of significance. (c² = 2.8; do not reject Ho; conclude the standard deviation for the focus group is not significantly different from that of the general population)

10) If you were to use a 95% confidence interval in question 9 to test the hypothesis, show why you would have reached the same conclusion. (4.1816 < s < 12.8721; s=10 falls in this interval)

11) Show why the p-value approach to the hypothesis in question 9 would have reached the same conclusion (p-value > 0.1; do not reject Ho under general rule of thumb)

12) A store wanted to see if the percentage of households in a city that have an annual household income above $50,000 was significantly greater than 50%. A survey of 400 households indicated that 220 of them have an annual household income above $50,000. Test the hypothesis at 5%. (Z = 2; conclude the percentage is greater than 50%)

13) What would be the power of the test at a 5% level of significance if the percentage of households with an annual household income above $50,000 is 55%? (64.06%)

14) Construct a 95% confidence interval of the percentage of households in that city that have an annual household income above $50,000. (50.1% < p < 59.9%)