A certain type of thread is manufactured with a mean tensile strength of 71.8 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample nean changed when the sample size is a) increased from 49 to 256? b) decreased from 196 to 64? a) The sample variance from Type integers or decimals. Do not round.) b) The sample variance from Type integers or decimals. Do not round.) for n = 49 to for n = 196 to for n=256. for n = 64.

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### Variance of Sample Means

A certain type of thread is manufactured with a mean tensile strength of 71.8 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is:

(a) Increased from 49 to 256?
(b) Decreased from 196 to 64?

---

#### Calculations:

**(a) The sample variance (σ²)**

...changes from \( \frac{5.6^2}{49} \) for \( n = 49 \) to \( \frac{5.6^2}{256} \) for \( n = 256 \).

(Type integers or decimals. Do not round.)

**(b) The sample variance (σ²)**

...changes from \( \frac{5.6^2}{196} \) for \( n = 196 \) to \( \frac{5.6^2}{64} \) for \( n = 64 \).

(Type integers or decimals. Do not round.)

---

These calculations illustrate how the sample variance changes inversely with the sample size. As the sample size increases, the sample variance decreases, and vice versa. This is an essential concept in statistics, particularly when understanding the distribution of sample means.

This example can help students visualize and calculate how changes in sample size affect the variance of sample means in practical applications.
Transcribed Image Text:### Variance of Sample Means A certain type of thread is manufactured with a mean tensile strength of 71.8 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is: (a) Increased from 49 to 256? (b) Decreased from 196 to 64? --- #### Calculations: **(a) The sample variance (σ²)** ...changes from \( \frac{5.6^2}{49} \) for \( n = 49 \) to \( \frac{5.6^2}{256} \) for \( n = 256 \). (Type integers or decimals. Do not round.) **(b) The sample variance (σ²)** ...changes from \( \frac{5.6^2}{196} \) for \( n = 196 \) to \( \frac{5.6^2}{64} \) for \( n = 64 \). (Type integers or decimals. Do not round.) --- These calculations illustrate how the sample variance changes inversely with the sample size. As the sample size increases, the sample variance decreases, and vice versa. This is an essential concept in statistics, particularly when understanding the distribution of sample means. This example can help students visualize and calculate how changes in sample size affect the variance of sample means in practical applications.
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