[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.


b. Compute a 95% 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 (\(\bar{x}\)) and sample standard deviation (\(s\)) from the problem statement. Here, \(\bar{x} = 2.27\) and \(s = 1.22\).
2. Determine the sample size (\(n\)), which is given as 60 married couples.
3. Since the sample size is greater than 30, use the z-distribution to construct the confidence interval. For a 95% confidence level, find the critical z-value (\(z_{\alpha/2}\)) corresponding to 0.025 in each tail. This value is approximately 1.96.
4. Calculate the standard error of the mean (SEM) using the formula: \(\text{SEM} = \frac{s}{\sqrt{n}}\) where \(s\) is the sample standard deviation and \(n\) is the sample size.
5. Construct the 95% confidence interval using the formula: \(\bar{x} \pm z_{\alpha/2} \times \text{SEM}\) This interval estimates the range in which the true mean number of children for all couples married 7 years is likely to fall with 95% confidence.

Key Concepts

Confidence Interval for the Mean

A confidence interval estimates the range within which the true population mean likely falls, based on sample data. For a 95% confidence level, we expect that 95% of such intervals calculated from repeated samples will contain the true mean. It combines the sample mean, variability, and sample size to provide this range.

Sample Mean and Sample Standard Deviation

The sample mean (x̄) is the average value of the data collected, representing the central tendency. The sample standard deviation (s) measures the spread or variability of the data around the mean. Both are essential for calculating the confidence interval and understanding the data distribution.

t-Distribution and Degrees of Freedom

When the population standard deviation is unknown and the sample size is small or moderate, the t-distribution is used instead of the normal distribution to calculate confidence intervals. The shape depends on degrees of freedom (n-1), which adjusts for sample size and ensures more accurate interval estimates.