solve it---The following information can be used to answer Questions 9-11.
In 1985, television shows about teams of crack commandos who promptly escape from maximum security stockades after being convicted for crimes they did not commit had mean Nielsen television ratings of 75%. Today, still wanted by television viewers, a sample of new television shows about crack commandos (N=5) have a mean Nielsen rating of 85%. The estimated population standard deviation is 25.
9. What is the standard error of the mean?
a. 6.55
b. 19.96
c. 11.16
d. 14.03
10. Approximately what is the value of t-statistic?
a. 2.11
b. 0.90
c. 0.44
d. 3.77
11. Based on the statistical evidence, do the current crop of TV shows have the same Nielsen ratings as the shows from 1985?
a. No
b. Yes, if sample size were to increase dramatically
c. Probably, but the evidence is inconclusive
d. Yes
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Let's solve question 9. We need to find the standard error of the mean. The formula is SEM equals sigma divided by the square root of N. With sigma equals 25 and N equals 5, we get SEM equals 25 divided by square root of 5, which is approximately 25 divided by 2.236, giving us 11.18. The closest answer is option c, 11.16.
Now for question 10, we calculate the t-statistic. The formula is t equals sample mean minus population mean, divided by standard error. Substituting our values: t equals 85 minus 75, divided by 11.16. This gives us 10 divided by 11.16, which equals approximately 0.896. The closest answer is option b, 0.90.
For question 11, we perform a hypothesis test. The null hypothesis is that the mean equals 75 percent. With 4 degrees of freedom and alpha equals 0.05, the critical t-value is 2.776. Since our calculated t-statistic of 0.90 is less than 2.776, we fail to reject the null hypothesis. However, with such a small sample size of only 5 shows, the test has low power. Therefore, the evidence is inconclusive. The answer is option c.
Let's summarize our solutions. For question 9, we calculated the standard error of the mean as 11.16, which is answer c. For question 10, we found the t-statistic to be 0.90, which is answer b. For question 11, since our t-statistic is less than the critical value, we cannot conclude the ratings are different, but the small sample size makes the evidence inconclusive, so answer c is correct.
To conclude, we've covered three key statistical concepts. The standard error of the mean measures how much sample means vary from the population mean. The t-statistic standardizes the difference between our sample and population means. In hypothesis testing, we compare our calculated t-value with the critical value. Small sample sizes lead to low statistical power, making it difficult to detect true differences. Our final answers are: question 9 answer c, question 10 answer b, and question 11 answer c.