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To solve the problem, we are tasked with finding the change in the standard deviation of the sample mean when the sample size increases from \( n = 8 \) to \( n = 52 \). The problem provides the following dataTo solve the problem, we are tasked with finding the change in the standard deviation of the sample mean when the sample size increases from \( n = 8 \) to \( n = 52 \). The problem provides the following 

- Mean tensile strength \( \mu = 75.5 \, \text{psi} \)
- Population standard deviation \( \sigma = 3.5 \, \text{psi} \)
- Initial sample size \( n_1 = 8 \)
- New sample size \( n_2 = 52 \)

### Standard deviation of the sample mean formula:
The formula for the standard deviation of the sample mean (also known as the standard error) is:

\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size

### Step 1: Calculate the standard deviation for \( n_1 = 8 \)
\[
\sigma_{\bar{x_1}} = \frac{3.5}{\sqrt{8}} = \frac{3.5}{2.8284} \approx 1.2374 \, \text{psi}
\]

### Step 2: Calculate the standard deviation for \( n_2 = 52 \)
\[
\sigma_{\bar{x_2}} = \frac{3.5}{\sqrt{52}} = \frac{3.5}{7.2111} \approx 0.4853 \, \text{psi}
\]

### Step 3: Determine the factor by which the standard deviation changes
Now, we calculate the ratio between the two standard deviations to find how much the standard deviation of the sample mean has changed:

\[
\text{Factor} = \frac{\sigma_{\bar{x_1}}}{\sigma_{\bar{x_2}}} = \frac{1.2374}{0.4853} \approx 2.5499
\]

Thus, the standard deviation of the sample mean decreases by a factor of approximately **2.5499** as the sample size increases from 8 to 52.

Would you like more details or have any questions?

Here are 5 follow-up questions related to this problem:
1. How does increasing the sample size generally affect the standard deviation of the sample mean?
2. Why do we divide the population standard deviation by the square root of the sample size?
3. Can the standard deviation of the sample mean ever be larger than the population standard deviation?
4. What is the relationship between the standard error and the central limit theorem?
5. How does this problem illustrate the concept of sampling distributions?

**Tip:** Increasing the sample size reduces the variability of the sample mean, leading to more precise estimates of the population mean.

A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. How is the standard deviation of the sample mean changed when the sample size is increased from n = 8 to n = 52?

Explore how the standard deviation of the sample mean changes in a carpet tensile strength problem as the sample size increases from n = 8 to n = 52. Learn how to apply the standard error formula and understand the effect of sample size on variability. Suitable for students studying statistics at an advanced high school or undergraduate level.

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