2. The Monitoring the Future Project carried out by the Institute for Social Research at the University of Michigan gathers data annually from a random sample of students at various grade levels attending schools in the United States. Table 2a shows the responses of high school seniors to a question asking them how often they worried about different social problems. The response choices were “never,” “seldom,” “sometimes,” and “often.” The percentages in the table are for those who said they worried sometimes or often about each problem.
Table 2Percentage of High School Seniors Worried About Social Problems: 1986-1994
1986 / 1988 / 1990 / 1992 / 1994(N = 3,073) / (N = 3,326) / (N = 2,595) / (N = 2,736) / (N = 2,664)
Crime and violence / 79.4 / 83.9 / 88.8 / 91.6 / 92.7
Race relations / 43.4 / 53.3 / 57.1 / 68.7 / 71.6
Drub abuse / 69.2 / 78.6 / 82.6 / 77.8 / 76.7
Hunger and poverty / 65.9 / 64.2 / 65.9 / 68.1 / 65.7
Economic problems / 60.6 / 56.2 / 56.8 / 70.6 / 62.6
Nuclear war / 69.1 / 57.3 / 45.1 / 33.4 / 27.9
a)Confidence intervals are often preferred over point estimates for estimating population parameters because they take sampling error into account. Using one of the percentages in Table 2 as an example, explain the difference between a point estimate and an interval estimate. You will need to do some calculations.
b)The percentage of high school seniors who said they were worried about hunger and poverty was the same in 1986 and 1990 (65.9 percent). Calculate the standard error for the percentage in each year, and explain why the values differ.
c)Describe how the size of the standard error affects the precision of confidence intervals. As the standard error increases, do confidence intervals become more precise or less precise?
d)In late 1990 and early 1991, the U.S. economy slipped into recession. Although the recovery began in 1991, the unemployment rate remained above average and the media continued to give extensive coverage to problems associated with global economic competition and corporate downsizing throughout most of 1991. The data in Table 2a show that a higher percentage of students who were surveyed said they were worried about economic problems in 1992 than in 1990 and 1994. Should this higher sample percentage in 1992 be attributed to the fluctuations produced by sampling error, or is it unlikely to be due to sampling error? Use 99 percent confidence intervals to answer this question, and explain the reasoning behind your answer.