[DATA] Family Size A random sample of 60 married couples who have been married 7 years was asked the number of children they have. The results of the survey are as follows:



Note: x̄ = 2.27, s = 1.22.


c. Compute a 99% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval.

Verified step by step guidance

1. Identify the sample mean (\(\bar{x}\)), sample standard deviation (\(s\)), and sample size (\(n\)) from the problem. Here, \(\bar{x} = 2.27\), \(s = 1.22\), and \(n = 60\).
2. Determine the confidence level, which is 99%, and find the corresponding critical value from the t-distribution because the population standard deviation is unknown and the sample size is less than 30 or the population is not known to be normal. The degrees of freedom (df) is \(n - 1 = 59\).
3. Calculate the standard error of the mean (SEM) using the formula: \(\text{SEM} = \frac{s}{\sqrt{n}}\)
4. Compute the margin of error (ME) by multiplying the critical t-value (\(t^*\)) by the standard error: \(\text{ME} = t^* \times \text{SEM}\)
5. Construct the confidence interval for the population mean number of children using: \(\left( \bar{x} - \text{ME}, \ \bar{x} + \text{ME} \right)\) Interpret this interval as the range in which we are 99% confident that the true mean number of children for all couples married 7 years lies.

Key Concepts

Confidence Interval for the Mean

A confidence interval estimates the range within which the true population mean is likely to fall, based on sample data. It combines the sample mean, variability, and sample size to provide a range with a specified confidence level, such as 99%. This interval helps quantify the uncertainty around the sample mean as an estimate of the population mean.

t-Distribution and Critical Value

When the population standard deviation is unknown and the sample size is small or moderate, the t-distribution is used to calculate confidence intervals. The critical value from the t-distribution depends on the confidence level and degrees of freedom (sample size minus one). It adjusts for extra uncertainty compared to the normal distribution.

Sample Mean and Sample Standard Deviation

The sample mean (x̄) is the average number of children in the sample, serving as a point estimate of the population mean. The sample standard deviation (s) measures the variability of the data around the mean. Both are essential for calculating the standard error and constructing the confidence interval.