8.123 Opinions over time about the death penalty For many years, the General Social Survey has asked respondents whether they favor the death penalty for persons convicted of murder. Support has been quite high in the United States, one of few Western nations that currently has the death penalty. The following figure uses the 26 General Social Surveys taken between 1975 and 2012 and plots the 95% confidence intervals for the population proportion in the United States who supported the death penalty in each of the 26 years of these surveys.

Twenty-six 95% confidence intervals for the population proportions supporting the death penalty.

a. When we say we have “95% confidence” in the interval for a particular year, what does this mean?

b. For 95% confidence intervals constructed using data for 26 years, let X = the number of the intervals that contain the true parameter values. Find the probability that x = 26, that is, all 26 inferences are correct. (Hint:
You can use the binomial distribution to answer this.)

c. Refer to part

b. Find the mean of the probability distribution of X.

d. What could you do differently so it is more likely that all 26 inferences are correct?

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Unknown population proportion, which changes over time p.90 0.60 0.70 0.80 1975 1976 -1977 1978 -1980 -1982 1983 1984 1985 1986 1987 -1988. 1989 1990 1991 1993 1994 1996 1998 -2000 2002 2004 2006 2008 2010 2012 P