Certainly! To determine the sample size necessary to estimate the true mean coverage of all 1-litre cans to within 1.5 square feet with 95% confidence, we'll follow these steps:
### Step 1: Identify the Given Information
- Standard deviation of coverage (σ): 6 square feet
- Desired margin of error (E): 1.5 square feet
- Confidence level: 95%
### Step 2: Determine the Z-Score
A 95% confidence level corresponds to a Z-score. For a 95% confidence interval, the Z-score (Z) is approximately 1.96.
### Step 3: Use the Sample Size Formula
To estimate the sample size, we use the formula for sample size estimation for the mean:
[tex]\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \][/tex]
Where:
- [tex]\( Z \)[/tex] is the Z-score
- [tex]\( \sigma \)[/tex] is the standard deviation of the population
- [tex]\( E \)[/tex] is the desired margin of error
### Step 4: Plug in the Values
Substitute the given values into the formula:
[tex]\[ n = \left( \frac{1.96 \cdot 6}{1.5} \right)^2 \][/tex]
### Step 5: Perform the Calculation
1. First, calculate the numerator inside the parentheses:
[tex]\[ 1.96 \cdot 6 = 11.76 \][/tex]
2. Next, divide by the margin of error:
[tex]\[ \frac{11.76}{1.5} = 7.84 \][/tex]
3. Finally, square the result:
[tex]\[ 7.84^2 = 61.4656 \][/tex]
### Step 6: Round Up to the Nearest Whole Number
The sample size must be a whole number, so we round up to the nearest whole number:
[tex]\[ n \approx 62 \][/tex]
### Conclusion
Therefore, in order to estimate the true mean coverage of all 1-litre cans to within 1.5 square feet with 95% confidence, a sample size of 62 should be taken.
