Question

Online Buying In a Consumer Reports Research Centre survey, women were asked if they purchase books online, and responses included these: no, yes, no, no. Letting “yes” = 1 and letting “no” = 0, here are ten bootstrap samples for those responses: {0, 0, 0, 0}, {1, 0, 1, 0}, {1, 0, 1, 0}, {0, 0, 0, 0},{0, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 1, 0, 0}, {1, 1, 0, 0}. Using only the ten given bootstrap samples, construct a 90% confidence interval estimate of the proportion of women who said that they purchase books online.

Short answer

The 90% confidence interval of the population proportion of women who purchased the books online is 0<$\hat{p}$<0.5

Step-by-step solution

Given Information

The sample of women purchasing online books is taken from the consumer reports research center survey. If they purchase books online, then the response will be yes=1, and if they do not, then it will be no=0. These are 10 bootstrap samples.

 Requirement Check

The only requirement that needs to be satisfied is that the sample should be randomly selected with replacement. And the given sample satisfied this requirement. There are 10 bootstrap samples with 2 outcomes 0 and 1.

Population proportion

In the Bootstrap sampling method, it is important to find population proportion for all 10 samples to find confidence intervals of the proportion of women who purchased the books online. After finding the proportions, they need to be sorted as low to high.

				proportion p	sort
0	0	0	0	0	0
1	0	1	0	0.5	0
1	0	1	0	0.5	0
0	0	0	0	0	0
0	0	0	0	0	0
0	1	0	0	0.25	0.25
0	0	0	0	0	0.25
0	0	0	0	0	0.5
0	1	0	0	0.25	0.5
1	1	0	0	0.5	0.5

Find confidence interval

The confidence interval is computed using the percentile of the sample. As the requirement of this problem is 90% confidence level, remove 5% from the lower tail and remove 5% from upper tail. Hence, the 90% confidence interval for proportion$\hat{p}$ is calculated with $P_{5}and{P}_{95}$. Here the percentiles are$P_5=0and{P}_{95}=0.5.$

Hence the 90 % confidence interval for the population $\hat{p}$is 0<$\hat{p}$<0.5.