Questions: A paint manufacturer uses a machine to fill gallon cans with paint (1 gal = 128 ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.7 ounce. Assume the population of volumes is normally distributed (a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.84 ounce. (b) The sample mean is 127 ounces. With a sample size of 7 a 90% level of confidence, and a population standard deviation of 0.84 ounce, does it seem possible that the population mean could be exactly 128 ounces? Explain.

Solution Steps

To solve these questions, we need to use statistical formulas related to confidence intervals and sample sizes.

(a) To determine the minimum sample size required for a 90% confidence interval, we use the formula for the sample size \( n \): \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] where \( Z \) is the Z-value corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( E \) is the margin of error.

(b) To determine if the population mean could be exactly 128 ounces, we need to construct the confidence interval using the sample mean, sample size, confidence level, and population standard deviation. We then check if 128 ounces falls within this interval.

For a 90% confidence level, the Z-value is calculated as: \[ Z = 1.6449 \]

Using the formula for the sample size \( n \): \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] where \( \sigma = 0.84 \) and \( E = 0.7 \): \[ n = \left( \frac{1.6449 \cdot 0.84}{0.7} \right)^2 \approx 3.678 \] Rounding up to the nearest whole number: \[ n = 4 \]

Given the sample mean \( \bar{x} = 127 \), sample size \( n = 7 \), and population standard deviation \( \sigma = 0.84 \), the confidence interval is calculated as: \[ \text{Confidence Interval} = \left( \bar{x} - Z \cdot \frac{\sigma}{\sqrt{n}}, \bar{x} + Z \cdot \frac{\sigma}{\sqrt{n}} \right) \] \[ \text{Confidence Interval} = \left( 127 - 1.6449 \cdot \frac{0.84}{\sqrt{7}}, 127 + 1.6449 \cdot \frac{0.84}{\sqrt{7}} \right) \] \[ \text{Confidence Interval} = (126.4778, 127.5222) \]

Check if 128 ounces falls within the confidence interval: \[ 126.4778 \leq 128 \leq 127.5222 \] Since 128 is not within the interval, it is unlikely that the population mean is exactly 128 ounces.

Final Answer