In the context of the sampling distribution of the sample mean, what effect does decreasing the sample size have on the spread ($\sigma$, standard deviation) of the distribution?

• A. The distribution becomes more symmetric.
• B. The spread decreases, making the distribution less variable.
• C. The spread increases, making the distribution more variable.
• D. The spread remains unchanged regardless of sample size.

Answer: (C)

Verified step by step guidance

1. Recall that the sampling distribution of the sample mean has a standard deviation called the standard error, which measures the spread of the distribution.
2. The formula for the standard error (SE) of the sample mean is given by: \( SE = \frac{\sigma}{\sqrt{n}} \) where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
3. Notice that the standard error is inversely proportional to the square root of the sample size \(n\). This means as \(n\) increases, the standard error decreases, and vice versa.
4. Therefore, if the sample size decreases, the denominator \(\sqrt{n}\) becomes smaller, causing the standard error to increase.
5. An increased standard error means the sampling distribution of the sample mean becomes more spread out or more variable when the sample size decreases.