CI Interpretation Problems total 50 point
For each problem, be sure to include a sentence with your overall conclusion (eg. "Based on the fact that our null value of __ was included in the confidence interval, we are 95% confident that the sample is ___ compared to ___")
Suppose you want to estimate with 95% confidence the difference between the mean (average) lengths of the cobs of two varieties of sweet corn (allowing them to grow the same number of days under the same conditions). Call the two varieties Corn-e-stats and Stats-o-sweet. Suppose your random sample of 100 cobs of the Corn-e-stats variety averages 7.9 inches, and your random sample of 110 cobs of Stats-o-sweet averages 7.5 inches, and standard deviations for Corn-e-stats and Stats-o-sweet are 0.35 inches and 0.45 inches, respectively.
The confidence interval for the average difference <Δx> between sample from different sets 1 and 2, whose averages are
Where Z depends on the degree of confidence and for 95% we have Z = 1.960. Sp is the pooled standard deviation and is so defined:
By substituting, we get < Δx > = 0.40 ± 0.11.
A study of teenage suicide included a sample of 96 boys and 123 girls between ages of 12 and 16 years selected scientifically from admissions records to a private psychiatric hospital. Suicide attempts were reported by 18 of the boys and 60 of the girls. We assume that the girls constitute a simple random sample from a population of similar girls and likewise for the boys. Construct a 95% percent confidence interval for the difference between the two proportions.
The proportion of boys who attempted suicide is pb = 18/96 = 0.1875 and the one for girls is pg = 0.4878. The formula for the confidence interval of the difference between proportions is the following:
Where p1 and p2 are the proportions for each set and n1 and n2 the respective set sizes. So, the 95% confidence interval for the difference between proportion is 0.300 ± 0.118.
Suppose the Cartoon Network conducts a nation-wide survey to assess viewer attitudes toward Superman. Using a simple random sample, they select 400 boys and 300 girls to participate in the study. Forty percent of the boys say that Superman is their favorite character, compared to thirty percent of the girls. What is the 95% confidence interval for the true difference in attitudes toward Superman?
This exercise is again about difference in proportions, so we can apply the previous formula. We get that the percentage difference for the true difference in attitude toward Superman between boys and girls is 10 ± 7.1%.
Suppose that simple random samples of college freshman are selected from two universities - 25 students from school A and 20 students from school B. On a standardized test, the sample from school A has …