Final answer:
In constructing confidence intervals for the ratio of population variances, one must use the F-distribution and the degrees of freedom from the sample sizes to obtain the critical values. For population proportions, the Z-distribution and standard error are used.
Step-by-step explanation:
Constructing Confidence Intervals for Population Variances
In this scenario, a student is tasked with constructing confidence intervals for the ratio of population variances based on sample statistics using an F-distribution, as the question involves variances rather than means.
The degrees of freedom (df) for the two samples would be df1 = n1 - 1 = 12 - 1 = 11 for the first sample, and df2 = n2 - 1 = 7 - 1 = 6 for the second sample. Without a given alpha level for significance, we cannot directly calculate the confidence interval; however, if the question refers to an α of 0.05 for a 95 percent confidence interval, you would use an F-distribution table or appropriate software to find the F-values for the upper and lower critical values corresponding to the dfs and confidence level.
Typically, the formula to calculate the confidence interval for the ratio of two variances would be: ((n2 - 1) * s22)/ (F * (n1 - 1) * s12)) up to ((n2 - 1) * s22)/ (F₁ * (n1 - 1) * s12)), where F and F₁ are the critical values from the F-distribution table.
For the exercise regarding confidence intervals for a population proportion, following a poll, the sample proportion (p-hat) would be 280/500 = 0.56. The sample standard deviation (s) can be calculated using the formula for standard deviation of a proportion s = √[(p-hat)(1 - p-hat)/n], which can then be used to construct the confidence intervals. The 95 percent confidence interval would involve the Z-distribution and the standard error of the proportion.