[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.


a. What is the shape of the distribution of the sample mean? Why?

Verified step by step guidance

1. Step 1: Understand the problem context. We have a sample of 60 married couples, each with a recorded number of children. The sample mean (\(\bar{x} = 2.27\)) and sample standard deviation (\(s = 1.22\)) are given. The question asks about the shape of the distribution of the sample mean.
2. Step 2: Recall the Central Limit Theorem (CLT). The CLT states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.
3. Step 3: Consider the sample size. Since the sample size is 60, which is generally considered large (usually \(n \geq 30\) is sufficient), the sampling distribution of the sample mean will tend to be normal.
4. Step 4: Note the shape of the original data distribution. By looking at the data values (number of children ranging from 0 to 5), the distribution might be skewed or not perfectly symmetric, but this does not affect the sampling distribution of the mean for large samples.
5. Step 5: Conclude that the shape of the distribution of the sample mean is approximately normal due to the Central Limit Theorem and the sufficiently large sample size.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of means calculated from multiple samples of the same size drawn from a population. It helps us understand how the sample mean varies and is crucial for making inferences about the population mean.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for sufficiently large sample sizes, the sampling distribution of the sample mean will be approximately normal regardless of the population's distribution shape. This allows us to use normal probability models for inference even if the original data is not normal.

Shape of the Distribution

The shape of a distribution refers to its symmetry, skewness, and modality. Understanding the shape of the sample data and the sample mean's distribution helps determine the appropriate statistical methods and whether normal approximation is valid.