Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1.51a

Chapter 6, Problem 6.1.51a

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 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 0.75 ounce.

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size for a confidence interval. The formula is: n = (Z * σ / E)^2, where n is the sample size, Z is the z-score corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error.

Step 2: Determine the z-score for a 90% confidence level. For a 90% confidence interval, the z-score is approximately 1.645. This value is derived from standard normal distribution tables.

Step 3: Substitute the given values into the formula. The population standard deviation (σ) is 0.75 ounces, and the margin of error (E) is 0.5 ounces. Plug these values into the formula: n = (1.645 * 0.75 / 0.5)^2.

Step 4: Simplify the expression inside the parentheses first. Calculate the product of the z-score and the standard deviation, then divide by the margin of error.

Step 5: Square the result from Step 4 to find the minimum sample size. Round up to the nearest whole number, as sample size must be an integer.

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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 90% confidence interval means that if we were to take many samples and construct intervals, approximately 90% of those intervals would contain the true population mean. This concept is crucial for estimating the mean volume of paint in the cans.

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 case, the sample size must be large enough to ensure that the margin of error (0.5 ounces) is met when estimating the mean volume of paint. The formula incorporates the population standard deviation and the desired confidence level.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this scenario, the assumption that the population of paint volumes is normally distributed allows for the use of specific statistical methods to calculate confidence intervals and sample sizes, making it easier to draw conclusions about the population.

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

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Textbook Question

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Textbook Question

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a. No preliminary estimate is available. Find the minimum sample size needed.

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Textbook Question

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Textbook Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (a) find the sample mean

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Textbook Question

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