Question

The results listed next are from a survey given to a random sample of the American public. For each sample statistic, construct a confidence interval estimate of the population parameter at the \(95 \%\) confidence level. Sample size \((N)\) is 2987 throughout. a. The average occupational prestige score was 43.87, with a standard deviation of \(13.52\). b. The respondents reported watching an average of \(2.86\) hours of TV per day, with a standard deviation of \(2.20\). c. The average number of children was \(1.81\), with a standard deviation of \(1.67\). d. Of the 2987 respondents, 876 identified themselves as Catholic. e. Five hundred thirty-five of the respondents said that they had never married. f. The proportion of respondents who said they voted for Bush in the 2004 presidential election was \(0.52\). g. When asked about capital punishment, 2425 of the respondents said that they favored the death penalty for murder.

Short answer

a) [43.39, 44.35], b) [2.78, 2.94], c) [1.75, 1.87], d) [0.277, 0.309], e) [0.165, 0.193], f) [0.502, 0.538], g) [0.798, 0.826]

Step-by-step solution

- Confidence Interval Formula for Mean

For parts a, b, and c, the confidence interval for the population mean can be constructed using the formula: \ \( \bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{N}} \) \ where \( \bar{X} \) is the sample mean, \( Z_{\alpha/2} \) is the critical value from the Z-distribution for the desired confidence level (1.96 for 95%), \( \sigma \) is the sample standard deviation, and \( N \) is the sample size (2987).

- Calculate Confidence Interval for a

Given: \( \bar{X} = 43.87 \), \( \sigma = 13.52 \). \ Compute the standard error (SE): \ \( SE = \frac{\sigma}{\sqrt{N}} = \frac{13.52}{\sqrt{2987}} \approx 0.247 \). \ The confidence interval is: \ \( 43.87 \pm 1.96 \times 0.247 \approx 43.87 \pm 0.484 \). \ So, the interval is approximately \( [43.39, 44.35] \).

- Calculate Confidence Interval for b

Given: \( \bar{X} = 2.86 \), \( \sigma = 2.20 \). \ Compute the standard error (SE): \ \( SE = \frac{2.20}{\sqrt{2987}} \approx 0.04029 \). \ The confidence interval is: \ \( 2.86 \pm 1.96 \times 0.04029 \approx 2.86 \pm 0.079 \). \ So, the interval is approximately \( [2.78, 2.94] \).

- Calculate Confidence Interval for c

Given: \( \bar{X} = 1.81 \), \( \sigma = 1.67 \). \ Compute the standard error (SE): \ \( SE = \frac{1.67}{\sqrt{2987}} \approx 0.03059 \). \ The confidence interval is: \ \( 1.81 \pm 1.96 \times 0.03059 \approx 1.81 \pm 0.060 \). \ So, the interval is approximately \( [1.75, 1.87] \).

- Confidence Interval Formula for Proportion

For parts d, e, f, and g, the confidence interval for the population proportion can be constructed using the formula: \ \( \hat{P} \pm Z_{\alpha/2} \sqrt{\frac{\hat{P}(1 - \hat{P})}{N}} \) \ where \( \hat{P} \) is the sample proportion, \( Z_{\alpha/2} \) is the critical value from the Z-distribution for the desired confidence level (1.96 for 95%), and \( N \) is the sample size (2987).

- Calculate Confidence Interval for d

Given: \( \hat{P} = \frac{876}{2987} \approx 0.293 \). \ Compute the standard error (SE): \ \( SE = \sqrt{\frac{0.293 \times (1 - 0.293)}{2987}} \approx 0.0084 \). \ The confidence interval is: \ \( 0.293 \pm 1.96 \times 0.0084 \approx 0.293 \pm 0.016 \). \ So, the interval is approximately \( [0.277, 0.309] \).

- Calculate Confidence Interval for e

Given: \( \hat{P} = \frac{535}{2987} \approx 0.179 \). \ Compute the standard error (SE): \ \( SE = \sqrt{\frac{0.179 \times (1 - 0.179)}{2987}} \approx 0.0073 \). \ The confidence interval is: \ \( 0.179 \pm 1.96 \times 0.0073 \approx 0.179 \pm 0.014 \). \ So, the interval is approximately \( [0.165, 0.193] \).

- Calculate Confidence Interval for f

Given: \( \hat{P} = 0.52 \). \ Compute the standard error (SE): \ \( SE = \sqrt{\frac{0.52 \times (1 - 0.52)}{2987}} \approx 0.00915 \). \ The confidence interval is: \ \( 0.52 \pm 1.96 \times 0.00915 \approx 0.52 \pm 0.018 \). \ So, the interval is approximately \( [0.502, 0.538] \).

- Calculate Confidence Interval for g

Given: \( \hat{P} = \frac{2425}{2987} \approx 0.812 \). \ Compute the standard error (SE): \ \( SE = \sqrt{\frac{0.812 \times (1 - 0.812)}{2987}} \approx 0.007 \). \ The confidence interval is: \ \( 0.812 \pm 1.96 \times 0.007 \approx 0.812 \pm 0.014 \). \ So, the interval is approximately \( [0.798, 0.826] \).

Key concepts

sample mean

In statistics, the sample mean is a critical concept. It represents the average value of a set of data points collected from a sample of a population. Imagine you survey 100 people to find out their average height. The arithmetic mean of all heights recorded is your sample mean.
For instance, in our exercise, the average occupational prestige score of 43.87 is a sample mean. It's calculated by summing all individual scores and dividing by the number of respondents. Understanding the sample mean helps us make inferences about the broader population from which the sample was drawn.
The sample mean formula is represented as: \(\bar{X} = \frac{\text{Sum of all data points}}{N}\).
This is pivotal because it's used in constructing confidence intervals and other statistical analyses.

population parameter

Population parameters are the characteristics of the entire population that you're interested in. These include statistics like the population mean, proportion, and standard deviation. Unlike sample statistics, which are derived from a subset of the population, population parameters serve as the true values you aim to estimate through sampled data.
For example, if we're interested in the average number of children for all Americans, that average number would be a population parameter. We attempt to estimate this by calculating the sample mean (like the 1.81 average children in our exercise) and then build a confidence interval to gauge how close it might be to the true population mean.
Understanding population parameters allows us to use sample data to draw conclusions about the larger group.

standard error

Standard error (SE) is crucial in statistics as it measures the accuracy of a sample mean by considering sample size and variability. It tells us how much our sample mean is expected to fluctuate from the true population mean.
In the context of our exercise, when calculating the confidence intervals, we compute SE using the formula: \(\text{SE} = \frac{\text{Sample Standard Deviation}}{\text{Square Root of Sample Size}}\).
The SE for the occupational prestige score, with a standard deviation of 13.52 and a sample size of 2987, is approximately 0.247. This indicates the average deviation of our sample mean from the population mean if we took multiple samples. The smaller the SE, the more accurate our sample mean is presumed to be in reflecting the population mean.

Z-distribution

The Z-distribution, or standard normal distribution, is a statistical concept used when dealing with data that is normally distributed. It transforms individual data points to how many standard deviations they are from the mean, known as Z-scores.
In constructing confidence intervals, we often use the Z-distribution. For example, in our exercise, a 95% confidence level corresponds to a Z-value of 1.96. This critical value is used to determine the range around our sample mean that likely contains the population mean.
The Z-distribution is essential because it helps us quantify uncertainty and variability. It ensures that our confidence intervals correctly reflect the level of confidence (like 95%) we have in our population parameter estimations.

sample proportion

Sample proportion represents the fraction of the sample exhibiting a particular attribute or characteristic. It’s a way to estimate population proportions.
For instance, in our exercise, the proportion of respondents who identified as Catholic is calculated by dividing the number of Catholic respondents (876) by the total sample size (2987), yielding a sample proportion of approximately 0.293.
This sample proportion is crucial for making inferences about the entire population. When constructing confidence intervals for proportions, we use the formula: \(\text{Sample Proportion} \pm \text{Z-value} \times \text{Standard Error of Proportion}\).
This helps us estimate the population proportion within a range, giving us a quantifiable measure of confidence in our estimations based on the sampled data.