Question 1181114
**(a) Sample size calculation:**

1. **Identify the given information:**

*   Confidence level = 98%
*   Margin of error (E) = 10
*   Population standard deviation (σ) = 26

2. **Find the critical z-value:**

For a 98% confidence level, α = 1 - 0.98 = 0.02.  α/2 = 0.01.  The critical z-value (z*) corresponding to 0.01 in the tail of the standard normal distribution is approximately 2.33.

3. **Use the sample size formula:**

n = (z* * σ / E)²

4. **Plug in the values:**

n = (2.33 * 26 / 10)²
n = (60.58 / 10)²
n = 6.058²
n ≈ 36.7

5. **Round up:** Since the sample size must be a whole number, always round up to the nearest integer. Therefore, the company should take a sample of at least 37.

**(b) Effect of sample size on standard error:**

The standard error of the mean (SEM) is calculated as:

SEM = σ / √n

Where:

*   σ = population standard deviation
*   n = sample size

If the sample size is increased from n₁ = 60 to n₂ = 240, let's see what happens to the SEM.

*   Initial SEM (SEM₁) = σ / √60
*   New SEM (SEM₂) = σ / √240

We can rewrite √240 as √(4 * 60) = 2√60

So, SEM₂ = σ / (2√60) = (1/2) * (σ / √60) = (1/2) * SEM₁

Therefore, when the sample size is increased from 60 to 240 (a four-fold increase), the standard error of the mean is *halved*.