2) In your own words, what does "m>4.76" mean?
The population mean score today is higher than 4.76, more than it was 10 years ago, the attitudes toward the Cafeteria have improved
3) In your own words, what is the type I error here? Find three possible consequences of the type I error.
"Reject H0 although it is true": there has been no improvement in the Cafeteria, but we think it has gotton better
• University will not spend more money on Cafeteria, or at least not as much
• Food will not get better there, or at least not as much
• Students might go someplace else where it is more expensive
4) In your own words, what is the type II error here? Find three possible consequences of the type II error.
"Fail to reject H0 although it is false": there has been some improvement in the Cafeteria, but we think it has staid the same
• University spends money on improvements that are not needed, and may not work
• Cafeteria changes the food although the students already like what they get
• Any changes might actually make the Cafeteria worse rather than better.
5) Say they interview 150 students, and the standard deviation turns out to be 1.35. If it were true that today m=5.02, what would be the power of the test?
Calc > Power and Sample Size command > 1-sample t, Sample Size: 150, Differences 0.26, standard deviation: 1.35 gives power=0.65. The power is the probability to reject the null hypothesis if it is false, so here there is a 65% chance that the university will find out that the Cafeteria has improved.
6) Using the same numbers as in part 5) (except the 5.02), what would the mean score today have to be so that the type II error probability b=0.05?
6) First power is 1-b=1-0.05=0.95. Now run the command with different values of "Differences" until the power is close to 0.95. Here are some examples:
Difference Power
0.30 0.7716
0.35 0.8839
0.40 0.9501
0.45 0.9820
0.50 0.9946
so with Differences =0.4 we have power=0.95, or if m=4.76+0.4=5.16
7) If in fact m=5.02, what sample size would be needed so that b=0.05?
Again power is 1-b=1-0.05=0.95. Now run the command with different values of Sample Sizes until the power is close to 0.95. Here are some examples:
Sample Sizes Power
200 0.7735
250 0.8585
300 0.9139
350 0.9488
400 0.9701
so a sample size of about 350 will have b=0.05