The label on 1-liter cans of paint states that the amount of paint in the can is sufficient to paint 50 square feet. However, this number is quite variable. In fact, the amount of coverage is known to be approximately normally distributed with a standard deviation of 6 square feet.

How large a sample should be taken to estimate the true mean coverage of all 1-liter cans to within 1.5 square feet with 95% confidence?



Answer :

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.