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Chapter 3: Problem 72

Give information about the proportion of a sample that agrees with a certain statement. Use StatKey or other technology to estimate the standard error from a bootstrap distribution generated from the sample. Then use the standard error to give a \(95 \%\) confidence interval for the proportion of the population to agree with the statement. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. In a random sample of 250 people, 180 agree.

Short Answer

Expert verified
Based on calculations, the bootstrap standard error is obtained and used to provide a 95% confidence interval for the proportion of the population agreeing with the statement. The confidence interval is calculated using the formula P 卤 1.96(SE).

Step by step solution

01

Calculate Initial Sample Proportion

To start, calculate the proportion of people who agree in the sample. The proportion (P) is calculated as the number of people who agree (180) divided by the total sample size (250). In mathematical representation this can be depicted as: P = 180 / 250.
02

Bootstrap to Find Standard Error

Bootstrap by resampling from the sample, 250 times (the same as the given sample size), with replacement. Repeat this step many times (say, 1000 times) to create a bootstrap distribution. Calculate the standard error (SE) as the standard deviation of your bootstrap distribution. The SE can be found as the square root of P(1-P) divided by the sample size which is 250 in this case. This can be represented as: SE = 鈭歔 P(1-P) / 250 ].
03

Calculate Confidence Interval

To form a 95% confidence interval, use the formula P 卤 1.96(SE). The number 1.96 comes from a Z-distribution table as we are forming a 95% Confidence Interval. Now put the values of P and SE calculated earlier into the formula to find the Confidence Interval.

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