Questions: What happens to the standard error of M as sample size increases?

Solution Steps

As the sample size increases, the standard error of the mean (M) decreases. This is because the standard error is inversely proportional to the square root of the sample size. To demonstrate this, we can calculate the standard error for different sample sizes using Python.

The standard error (SE) of the mean \( M \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the standard deviation and \( n \) is the sample size. As \( n \) increases, \( SE \) decreases.

Using a standard deviation \( \sigma = 10 \), we calculate the standard error for various sample sizes:

• For \( n = 10 \): \[ SE = \frac{10}{\sqrt{10}} \approx 3.1623 \]

• For \( n = 50 \): \[ SE = \frac{10}{\sqrt{50}} \approx 1.4142 \]

• For \( n = 100 \): \[ SE = \frac{10}{\sqrt{100}} = 1.00 \]

• For \( n = 500 \): \[ SE = \frac{10}{\sqrt{500}} \approx 0.4472 \]

• For \( n = 1000 \): \[ SE = \frac{10}{\sqrt{1000}} \approx 0.3162 \]

The calculated standard errors for the different sample sizes are as follows:

• \( n = 10 \): \( SE \approx 3.16 \)
• \( n = 50 \): \( SE \approx 1.41 \)
• \( n = 100 \): \( SE = 1.00 \)
• \( n = 500 \): \( SE \approx 0.45 \)
• \( n = 1000 \): \( SE \approx 0.32 \)

Final Answer