Question

Exercises 3.112 to 3.115 give information about the proportion of a sample that agree with a certain statement. Use StatKey or other technology to find a confidence interval at the given confidence level for the proportion of the population to agree, using percentiles from a bootstrap distribution. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. Find a \(99 \%\) confidence interval if, in a random sample of 1000 people, 382 agree, 578 disagree, and 40 can't decide.

Short answer

The 99% confidence interval would be the range calculated as [lower limit, upper limit]. To find the precise interval, replace p and SE with their calculated values in the formula to calculate the lower and upper limits.

Step-by-step solution

Calculate the Proportion

Calculate the proportion of people who agreed with the statement by dividing the number of people who agreed by the total number of people who made a decision. So, it will be \(p = \frac{382}{(382 + 578)} = \frac{382}{960}\).

Calculate the Standard Error

Next, calculate the standard error using the formula \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(n\) is the total number of responses (excluding those who couldn't decide). So, it would be \(SE = \sqrt{\frac{p(1-p)}{960}}\).

Find the Confidence Interval

For a 99% confidence interval, we need to find the 0.5% and 99.5% percentile values from a Normal distribution(Normally these values are -2.576 and 2.576). Then, calculate the confidence limits as \(confidence\_interval = p ± Z_{value} * SE\).[lower limit = \(p - 2.576 * SE\) and upper limit = \(p + 2.576 * SE\)]

Key concepts

Bootstrap Distribution

Understanding the bootstrap distribution is essential for estimating the variability of a statistic, like a sample proportion. The bootstrap distribution is generated by repeatedly resampling from the original data, with replacement, and recalculating the statistic for each resample. This process creates a distribution of the statistic that approximates the sampling distribution we would see if we could take many samples from the population.

In practice, this means that to create a bootstrap distribution for our sample proportion, we repeatedly sample 960 responses (the size of the sample without indecisive respondents) and calculate the proportion that agrees each time. Over thousands of such resamples, we build up a bootstrap distribution that we can use to estimate the variability of our sample proportion.

Proportion

The term 'proportion' refers to the fraction of the total sample that exhibits a particular trait or characteristic. In the problem at hand, we calculated the proportion of people who agree with the statement out of those who made a decision. To find this, we divided the number of people who agreed, 382, by the total number of decisions, which is the sum of agreements and disagreements, 960. This proportion is representative of the larger population and is used to draw inferences about the population's sentiment regarding the statement.

Standard Error

The standard error (SE) measures the amount of variability in the sampling distribution of a statistic. In this context, it gives us an idea of how precisely we have estimated the sample proportion. The standard error is calculated using the formula \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion and \(n\) is the sample size. A smaller standard error indicates a more precise estimate, which in turn provides a narrower confidence interval, offering a more precise estimate of the population proportion.

Confidence Level

A confidence level indicates the degree of certainty that the true population parameter lies within the calculated confidence interval. For example, a 99% confidence level means that if we were to take many samples and compute a confidence interval for each sample, we would expect about 99% of those intervals to contain the population parameter. Higher confidence levels correspond to wider confidence intervals, reflecting a higher certainty about the estimate including the true population parameter, but they also mean less precision about where that parameter is located.

StatKey Technology

StatKey technology is a set of interactive web-based statistical tools that provides a convenient way for students to perform bootstrap simulations and other statistical analysis. To solve the problem presented, one would use the 'CI for Single Proportion' tool within StatKey. By entering the data into the tool, it calculates a bootstrap distribution for us, from which we can then find the confidence interval for the population proportion at the specified confidence level. It simplifies the process of statistical analysis, making it accessible to students without advanced programming or statistical skills.