MGMT2262

Worksheet #6

1) A school superintendent wanted to see if the percentage of generally satisfied parents was significantly below 85%. Out of 120 parents who returned the survey, 95 of them were generally satisfied. Test the hypothesis at a 5% level of significance. (Z = -1.7896; reject Ho; conclude that the percentage of generally satisfied parents is significantly below 85%)

2) Is there a level of significance between 1% and 10% in which the opposite verdict would have been reached? (p-value = 0.0367; any level of significance between 1% and 3.67%)

3) Construct a 95% confidence interval of the percentage of generally satisfied parents and interpret what it means. (0.7190 < p < 0.8644; we are 95% confident that the percentage of satisfied parents is between 71.9% and 86.44%)

4) Suppose they wanted to conduct another survey. If they wanted the margin of error to be 2% and the level of confidence is 95% and they used the percentage from the first study as an estimate of the percentage of generally satisfied parents, what should the sample size be? (1584)

5) Suppose the school board will not pay for a study with a sample size of more than 1000. What would be the margin of error they would have to live with if everything else is the same as in question 4? (2.52%)

6) 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 a 3% level of significance (Z = 2; conclude the percentage is greater than 50%)

7) For which levels of significance between 1% and 10% would the opposite conclusion be reached for the test in question 6? (1% and 2.28%)

8) A labour union researcher wants to determine if the percentage of people working overtime at least once a week is significantly different than 13%. Of 800 surveyed workers, 120 work overtime at least once a week. Test the hypothesis at a 5% level of significance. (Z = 1.68; the percentage is not significantly different than 13%.)

9) Suppose a level of significance had not been chosen for the test in question 8. What conclusion would be reached? (results are inconclusive)

10) A box-cutting machine must be adjusted if the standard deviation of the box width angle is more than 0.5 mm. It has been noted that the box width is normally distributed. They would sample 20 boxes at a time and measure the width. In one sample, the standard deviation was 0.6 mm. If they test the hypothesis at a 5% level of significance, does the machine need to be adjusted? (c² = 27.36; conclude the machine does not need to be adjusted)

11) In focus groups it is important that the age spread of the participants is approximately equal to that of the general population which is normally distributed. According to records from Stats Canada, the standard deviation of ages of adult Canadians is 12 years. For one such focus group of 8 participants, the standard deviation was 10.2 years. Is the standard deviation of the focus group significantly less than that of the general population? Test at a 5% level of significance. (c² = 5.0575; conclude the standard deviation of the focus group is not below that of the general population)

12) An economist wanted to see if the wage spread of small factories (those with fewer than 50 employees) was significantly different than that of larger factories (those with more than 500 employees). Government records indicated that the wage distribution of large factories is normally distributed with a standard deviation of $1200/month. For a sample of 16 small factories, the standard deviation was $1050/month. Is there any significant difference in the wage spread of small and large factories? Test at 5%. (c² = 11.4844; conclude there is no significant difference between small and large factories)

13) Construct a 95% confidence interval of the standard deviation of small factories. If this interval was used to test the hypothesis in question 12, why would we reach the same conclusion? (775.65 < s < 1625.09; the hypothesis standard deviation of 1200 falls in the interval)

14) Compute the approximate p-values for questions 10-12. (for question 10, the p-value is between 5% and 10%, though closer to 10%, question 11 p-value > 10%, question 12 p-value > 20% )