Juice Dispensing Machine A beverage company uses a machine to fill half-gallon bottles with fruit juice (see figure). The company wants to estimate the mean volume of water the machine is putting in the bottles within 0.25 fluid ounce.
[IMAGE]
a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 1 fluid ounce.

Verified step by step guidance

Step 1: Recall the formula for determining the minimum sample size required for a confidence interval for the population mean: \( n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \), where \( Z \) is the critical value for the desired confidence level, \( \sigma \) is the population standard deviation, and \( E \) is the margin of error.

Step 2: Identify the given values from the problem: \( \sigma = 1 \) fluid ounce (population standard deviation), \( E = 0.25 \) fluid ounce (margin of error), and the confidence level is 95%.

Step 3: Determine the critical value \( Z \) for a 95% confidence level. For a standard normal distribution, the critical value \( Z \) corresponds to the value where the cumulative probability is 0.975 (since 95% confidence level leaves 2.5% in each tail). Using a Z-table or calculator, \( Z \approx 1.96 \).

Step 4: Substitute the values into the formula: \( n = \left( \frac{1.96 \cdot 1}{0.25} \right)^2 \). Simplify the numerator and denominator before squaring the result.

Step 5: After simplifying, round up the result to the nearest whole number, as the sample size must be an integer. This will give you the minimum sample size required to achieve the desired confidence interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals, approximately 95% of those intervals would contain the true population mean.

Sample Size Determination

Sample size determination involves calculating the number of observations needed to achieve a desired level of precision in estimating a population parameter. In this context, the formula incorporates the desired margin of error, the population standard deviation, and the critical value from the normal distribution corresponding to the confidence level.

Population Standard Deviation

The population standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points differ from the population mean. In this scenario, knowing the population standard deviation (1 fluid ounce) is crucial for calculating the required sample size to ensure the estimate of the mean volume is accurate within the specified margin of error.

Watch next

Master Population Standard Deviation Known with a bite sized video explanation from Patrick

Start learning