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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 6.2.25c
Chapter 6, Problem 6.2.25c

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.
SAT Scores The SAT scores of 12 randomly selected high school seniors
Table displaying SAT scores of 12 randomly selected high school seniors, arranged in two rows and six columns.

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Step 1: Calculate the sample mean (x̄) by summing all the SAT scores provided in the data set and dividing by the total number of scores (n = 12). Use the formula: x̄ = (Σx) / n.
Step 2: Compute the sample standard deviation (s) using the formula: s = sqrt((Σ(x - x̄)^2) / (n - 1)), where x̄ is the sample mean and n is the sample size.
Step 3: Determine the critical value (t*) for a 99% confidence level using a t-distribution table. The degrees of freedom (df) are calculated as n - 1 (df = 12 - 1 = 11).
Step 4: Calculate the margin of error (E) using the formula: E = t* × (s / sqrt(n)), where s is the sample standard deviation and n is the sample size.
Step 5: Construct the confidence interval for the population mean using the formula: Confidence Interval = x̄ ± E, where x̄ is the sample mean and E is the margin of error.

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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 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population mean.
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Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In constructing confidence intervals, it is often assumed that the population from which the sample is drawn is normally distributed, especially when the sample size is small.
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Sample Mean and Standard Deviation

The sample mean is the average of a set of values, calculated by summing all the observations and dividing by the number of observations. The sample standard deviation measures the amount of variation or dispersion in a set of values. Both the sample mean and standard deviation are crucial for calculating the confidence interval, as they provide the necessary statistics to estimate the range around the population mean.
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Related Practice
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.

b. Find the minimum sample size needed, using a prior survey that found that 34% of U.S. adults disapprove of the U.S Senate’s use of the filibuster. (Source: Monmouth University)

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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 (b) the population standard deviation σ. 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 (b) find the sample standard deviation. Assume the population is normally distributed.

SAT Scores The SAT scores of 12 randomly selected high school seniors

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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.

c. Increase in the population standard deviation

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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.

c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.

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Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population 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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