Chapter 7: Problem 5

Use the relatively small number of given bootstrap samples to construct theconfidence interval. In a Consumer Reports Research Center survey, women were asked if theypurchase books online, and responses included these: no, yes, no, no. Letting"yes" = 1 and letting "no" = 0, here are ten bootstrap samples for thoseresponses: \(\\{0,0,0,0\\},\\{1,0,1,0\\}\\{1,0,1,0\\},\\{0,0,0,0\\},\\{0,0,0,0\\},\\{0,1,0,0\\},\\{0,0,0,0\\},\\{0,0,0,0\\},\\{0,1,0,0\\},\\{1,1,0,0\\}.\) Using only the ten given bootstrap samples, construct a \(90 \%\) confidenceinterval estimate of the proportion of women who said that they purchase booksonline.

### Short Answer

Expert verified

The 90% confidence interval is 0 to 0.5.

## Step by step solution

01

## Convert Responses to Numerical Values

Replace 'yes' with 1 and 'no' with 0 for the given responses.

02

## List Bootstrap Samples

Given bootstrap samples are: \( \{0,0,0,0\}, \{1,0,1,0\}, \{1,0,1,0\}, \{0,0,0,0\}, \{0,0,0,0\}, \{0,1,0,0\}, \{0,0,0,0\}, \{0,0,0,0\}, \{0,1,0,0\}, \{1,1,0,0\} \)

03

## Calculate Proportion for Each Sample

Calculate the proportion of 'yes' (1) in each bootstrap sample: Sample 1: 0 Sample 2: 0.5 Sample 3: 0.5 Sample 4: 0 Sample 5: 0 Sample 6: 0.25 Sample 7: 0 Sample 8: 0 Sample 9: 0.25 Sample 10: 0.5

04

## Sort the Proportions

Sorted proportions are: \( \{0, 0, 0, 0, 0, 0, 0.25, 0.25, 0.5, 0.5\} \)

05

## Determine the 90% Confidence Interval

To determine the 90% confidence interval from the sorted proportions, remove the lowest 5% and the highest 5%. Since there are 10 samples, 10% of 10 is 1. Thus, remove the lowest and highest one value. The confidence interval is the range of the remaining proportions: \( \{0, 0, 0, 0, 0, 0.25, 0.25, 0.5\} \).

06

## Identify the Interval

The 90% confidence interval estimates for the proportion of women who purchase books online is from 0 to 0.5.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### sampling distribution

Imagine you asked a group of women whether they purchase books online. Their responses might vary. If you surveyed a new group each time, you’d also get different outcomes. This variation is what we call the sampling distribution. It showcases the range of possible results we might see if we took multiple random samples from the same population.

In our exercise, instead of surveying many groups, we used bootstrap samples (re-sampling with replacement of the original group). The proportions from these samples create an approximate sampling distribution. For example, one bootstrap sample gave us a proportion of 0%, another 25%, and another 50%. This range represents different results we might get if we surveyed different groups of women multiple times. This method helps us understand how variable our estimates can be and explains why no single survey gives a complete picture.

###### proportion estimate

When we talk about a proportion estimate, we're referring to the fraction or percentage of a whole. Here, we're estimating the proportion of women who purchase books online.

From the original responses, we get: 'no', 'yes', 'no', 'no'—numerically, that's 0, 1, 0, 0. If we calculate the proportion of 'yes' responses, which represent book purchases, we'd get \(\frac{1}{4} = 0.25\) or 25%.

However, in our exercise, we go beyond the initial group and use bootstrap samples to estimate. With bootstrap sampling, we re-sample the original responses multiple times to mimic surveying more groups. We then calculate the proportion of 'yes' (purchase) in each sample. For instance, one bootstrap sample might be \[1, 0, 1, 0\] or 50%, and another might be \[0, 0, 0, 0\] or 0%. These varied proportions help us better understand the possible range of our proportion estimate.

###### confidence interval

A confidence interval gives us a range in which we expect the true proportion to lie, with a certain level of confidence (like 90%, 95%, etc.). It reflects how sure we are about where the 'true' proportion falls, based on our sample data.

Let's say we want a 90% confidence interval for our survey question. We first gather several bootstrap samples and calculate the proportion of 'yes' for each. We then order these proportions. To form a 90% confidence interval, we remove the lowest 5% and the highest 5% of our ordered proportions. For our exercise, we had 10 samples, and removing the top and bottom values leaves us with the 90% middle range. This range is our 90% confidence interval.

In our specific case, after sorting the proportions, we removed the extreme values. Our remaining range, from 0 (0%) to 0.5 (50%), is our 90% confidence interval. This suggests that we're 90% confident the true proportion of women who purchase books online lies somewhere between 0 and 50%.

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