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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.1.49a
Chapter 6, Problem 6.1.49a

Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.
a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.

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Step 1: Recall the formula for determining the minimum sample size required for estimating a population mean: n = (Z * σ / E)^2, where Z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.
Step 2: Identify the given values from the problem: the confidence level is 95%, so the z-score (Z) for a 95% confidence level is approximately 1.96. The population standard deviation (σ) is 3.10 milligrams, and the margin of error (E) is 0.75 milligrams.
Step 3: Substitute the given values into the formula: n = (1.96 * 3.10 / 0.75)^2.
Step 4: Simplify the expression inside the parentheses first: calculate (1.96 * 3.10) and then divide the result by 0.75.
Step 5: Square the result from Step 4 to find the minimum sample size (n). If the result is not a whole number, always 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 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.
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Introduction to Confidence Intervals

Sample Size Determination

Determining the minimum sample size is crucial for ensuring that the estimate of the population mean is accurate within a specified margin of error. The formula for calculating sample size involves the desired confidence level, the population standard deviation, and the margin of error, which in this case is 0.75 milligrams.
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Coefficient of Determination

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this context, the population standard deviation of 3.10 milligrams indicates how much individual cholesterol content values deviate from the mean, which is essential for calculating the sample size needed for the confidence interval.
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Related Practice
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.

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

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Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

a. Increase in the level of confidence

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Textbook Question

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.

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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.

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Textbook Question

Senate Filibuster You wish to estimate, with 99% confidence, the population proportion of U.S. adults who disapprove of the U.S Senate’s use of the filibuster. Your estimate must be accurate within 2% of the population proportion.

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

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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.

Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)

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