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

Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1.52a

Chapter 6, Problem 6.1.52a

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.

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.

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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.

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.

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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

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

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Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

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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

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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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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