Paint Can Volumes A paint manufacturer uses a machine to...

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). (a) The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.25 ounce. Determine the minimum sample size required to construct a $90 \%$ confidence interval for the population mean. Assume the population standard deviation is 0.85 ounce. (b) Repeat part (a) using an error tolerance of 0.15 ounce. (c) Which error tolerance requires a larger sample size? Explain.

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure).
(a) The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.25 ounce. Determine the minimum sample size required to construct a $90 \%$ confidence interval for the population mean. Assume the population standard deviation is 0.85 ounce.
(b) Repeat part (a) using an error tolerance of 0.15 ounce.
(c) Which error tolerance requires a larger sample size? Explain.

Key Concepts

Confidence Interval

A confidence interval is a range of values that is likely to contain the true population parameter, such as the mean. It is constructed using the sample estimate, the standard error, and a critical value from a probability distribution, typically the Z-distribution when the population standard deviation is known.

Margin of Error

The margin of error denotes the maximum amount by which the sample estimate is expected to differ from the true population parameter. It helps define the precision of the estimate and is determined by the product of the critical value and the standard error.

Z Critical Value

The Z critical value is a quantile from the standard normal distribution corresponding to the chosen confidence level. It adjusts the width of the confidence interval, ensuring that a specified proportion of the distribution lies within the computed range.

Sample Size Determination

Sample size determination involves rearranging the formula for the margin of error to solve for the number of observations needed to estimate the population parameter within a specific tolerance. It ensures that the resulting confidence interval meets the desired precision level.

Effect of Error Tolerance on Sample Size

A smaller error tolerance, which means a more precise estimate, requires a larger sample size, whereas a larger error tolerance allows for a smaller sample size. This relationship highlights the trade-off between the precision of the estimate and the resources needed for data collection.

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