Question

N the past 100 years, there have been more than 250 successful

breakouts from Didsbury Prison. Mike is a researcher who has been
hired by the prison governor to investigate the phenomenon. Details
are available only for those breakouts that took place in the past ten
years—a total of 30. Using the records of these 30 breakouts as a sam-
ple, Mike decides to break the figures down to see whether breakouts
were more common in certain wings of the prison than in others. It
transpires that of the 30 breakouts, 4 have been from A-Wing, 8 from
B-Wing, 15 from C-Wing, and 3 from D-Wing.
a. Does Mike have enough evidence to conclude that, over the 100-
year period, breakouts were more (or less) likely to occur from cer-
tain wings than from others? Use a 5% level of significance and out-
line each of the steps required in a test of statistical significance.
b. Would your answer be any different if a 1% level of significance
were used?
c. Are there any problems with Mike’s choice of a sample? Explain
your answer.

Answer 1

Mike would use a chi-square test of independence at a 5% level of significance to determine if there's enough evidence of a difference in breakout frequencies between prison wings. If a 1% significance level is used, the outcome might differ. Also, Mike's sample may suffer from sampling bias as it only represents the last ten years and might not reflect long-term trends.

Step-by-step explanation:

To assess whether there is enough evidence to conclude that, over the 100-year period, breakouts were more likely to occur from certain wings of Didsbury Prison than from others, Mike would need to perform a chi-square test bindependence. Using the data for the past ten years with a total of 30 breakouts and the distribution across the wings, Mike can conduct this test at a 5% level of significance. The steps would include:

• Setting up the null hypothesis (H0) that there is no difference in breakout frequency across the wings.
• Setting up the alternative hypothesis (H1) that there is a difference in breakout frequency across the wings.
• Calculating the expected frequencies for each wing based on uniform distribution.
• Computing the chi-square statistic using observed and expected frequencies.
• Comparing the chi-square statistic to the critical value from the chi-square distribution, or obtaining the p-value.
• Determining if the evidence is strong enough to reject the null hypothesis.

If a 1% level of significance were used, the critical value would be higher, which could potentially change the conclusion if the chi-square statistic was borderline significant at the 5% level.

As for problems with Mike's choice of a sample, it may not be representative of the entire 100-year period due to changes in prison security, administration, or the physical state of the wings over time. This could lead to a sampling bias. Furthermore, only considering the last ten years might not accurately reflect the overall trends and variances that could have taken place over the entire century.