Questions: Determine the minimum sample size required when you want to be 95% confident that the sample mean is within one unit of the population mean and σ = 5.3. A 95% confidence level requires a sample size of □ (Round up to the nearest whole number as needed.)

Solution Steps

To determine the minimum sample size required for a given confidence level and margin of error, we can use the formula for the sample size in estimating a population mean. The formula is:

\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]

where:

• \( Z \) is the Z-score corresponding to the desired confidence level (for 95% confidence, \( Z \approx 1.96 \))
• \( \sigma \) is the population standard deviation
• \( E \) is the margin of error

Given:

• Confidence level = 95% (Z = 1.96)
• \( \sigma = 5.3 \)
• Margin of error \( E = 1 \)

We will plug these values into the formula and round up to the nearest whole number.

We are given the following values:

• Confidence level: 95%
• Population standard deviation: \( \sigma = 5.3 \)
• Margin of error: \( E = 1 \)

For a 95% confidence level, the Z-score is: \[ Z = 1.96 \]

The formula to determine the minimum sample size \( n \) is: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]

Substituting the given values: \[ n = \left( \frac{1.96 \cdot 5.3}{1} \right)^2 \]

First, calculate the numerator: \[ 1.96 \cdot 5.3 = 10.388 \]

Then, square the result: \[ n = \left( 10.388 \right)^2 = 107.899 \]

Since the sample size must be a whole number, we round up: \[ n = \lceil 107.899 \rceil = 108 \]

Final Answer