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Chapter 7: Problem 10

In a random sample of 1,250 household moves, 822 were moves to a location within the same county as the original residence. a. Give a point estimate of the proportion of all household moves that are to a location within the same county as the original residence. b. Construct a \(98 \%\) confidence interval for that proportion.

Short Answer

Expert verified
Point estimate: \(0.6576\); Confidence interval: \((0.626, 0.689)\).

Step by step solution

01

Identify the Sample Proportion

To find the point estimate of the proportion, we need to calculate the sample proportion \( \hat{p} \). This is done by dividing the number of successful outcomes (household moves within the same county) by the total number of observations (total household moves). Thus: \( \hat{p} = \frac{822}{1250} = 0.6576 \).
02

Determine the Confidence Level Parameters

For a \(98\%\) confidence interval, we first calculate the critical value. The confidence level is \(1 - 0.98 = 0.02\), and since it's two-tailed, we use \(\frac{0.02}{2} = 0.01\) per tail. Looking up \(z_{0.01}\) in the Z-table, we find that the critical value \(z_{0.01}\) is approximately \(2.33\).
03

Calculate the Standard Error

The standard error (SE) is calculated using the formula: \(SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.6576 \times (1 - 0.6576)}{1250}} = 0.0135\).
04

Compute the Margin of Error

The margin of error (ME) is found by multiplying the critical value by the standard error: \(ME = 2.33 \times 0.0135 = 0.031455\).
05

Construct the Confidence Interval

The \(98\%\) confidence interval is constructed by adding and subtracting the margin of error from the point estimate: \( \hat{p} \pm ME = 0.6576 \pm 0.031455 \). Thus, the interval is \( (0.626145, 0.689055) \). This indicates we are \(98\%\) confident the true proportion of moves within the same county is between \(62.6\%\) and \(68.9\%\).

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Key Concepts

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

Sample Proportion
The sample proportion, denoted as \( \hat{p} \), is a fundamental concept when working with data samples. It represents an estimate of a population parameter based on a particular sample we've gathered. In simpler terms, it's the fraction or percentage of the sample that meets a certain condition. For example, in the context of household moves, if we want to know how many families stay within the same county, the sample proportion helps us estimate this.
  • To find \( \hat{p} \), we divide the number of successes (in our case, moves within the same county) by the total sample size.
  • Using the data provided: \( \hat{p} = \frac{822}{1250} = 0.6576 \), meaning 65.76% of the sample moved within the same county.
This measure provides a simple snapshot of the behavior within our data, giving us a base for further analysis like confidence intervals.
Standard Error
Once we have the sample proportion, the next step is often to calculate the standard error (SE). This is a measure of how much the sample proportion would vary if we were to take many different samples. In essence, it provides us with an understanding of the variability of our estimate.
  • The formula for standard error of a sample proportion is: \( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \), where:
    • \( \hat{p} \) is the sample proportion
    • \( n \) is the sample size
  • For our household moves example: \( SE = \sqrt{\frac{0.6576 \times (1 - 0.6576)}{1250}} = 0.0135 \).
This tells us that our proportion could fluctuate slightly if different samples of 1,250 moves were taken, but on average, it would vary by about 1.35%.
Margin of Error
The margin of error (ME) is a crucial aspect of constructing confidence intervals. It combines the concept of the standard error with a critical value from the Z-distribution.
  • The formula for the margin of error is: \( ME = z^* \times SE \), where:
    • \( z^* \) is the critical value, which depends on our desired confidence level (e.g., 98%). For a 98% confidence level, this value is 2.33.
    • \( SE \) is the standard error.
  • In our case: \( ME = 2.33 \times 0.0135 = 0.031455 \).
This margin of error means we expect our true population parameter to be within about 3.1455% of our sample proportion, 98% of the time. It's important because it defines how precise our estimate is. A smaller margin of error indicates a more precise prediction of where the true population parameter lies.

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Most popular questions from this chapter

A machine for making precision cuts in dimension lumber produces studs with lengths that vary with standard deviation 0.003 inch. Five trial cuts are made to check the machine's calibration. The mean length of the studs produced is 104.998 inches with sample standard deviation 0.004 inch. Construct a \(99.5 \%\) confidence interval for the mean lengths of all studs cut by this machine. Assume lengths are normally distributed. Hint: Not all the numbers given in the problem are used.

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