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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 6.1.22
Chapter 6, Problem 6.1.22

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.
c = 0.95, xbar = 31.39, σ = 0.80, n = 82.

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Step 1: Identify the components needed for constructing the confidence interval. These include the sample mean (x̄ = 31.39), population standard deviation (σ = 0.80), sample size (n = 82), and the confidence level (c = 0.95).
Step 2: Determine the critical value (z*) corresponding to the confidence level of 0.95. For a 95% confidence level, the z* value can be found using a standard normal distribution table or calculator. The z* value is approximately 1.96.
Step 3: Calculate the standard error of the mean (SE). The formula for SE is: σ/n. Substitute σ = 0.80 and n = 82 into the formula.
Step 4: Compute the margin of error (ME). The formula for ME is: z*SE. Use the z* value from Step 2 and the SE calculated in Step 3.
Step 5: Construct the confidence interval. The formula for the confidence interval is: [x̄-ME,x̄+ME]. Substitute x̄ = 31.39 and the ME from Step 4 into the formula.

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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 (like the mean) with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals

Sample Mean (x̄)

The sample mean, denoted as x̄, is the average of a set of observations from a sample. It serves as a point estimate of the population mean (μ). In the given question, x̄ = 31.39 indicates the average value calculated from the sample of size n = 82.
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Standard Deviation (σ) and Sample Size (n)

The standard deviation (σ) measures the dispersion or variability of a set of data points around the mean. In this case, σ = 0.80 indicates how spread out the sample values are. The sample size (n) refers to the number of observations in the sample, which is crucial for determining the reliability of the confidence interval; here, n = 82 provides a basis for estimating the population mean.
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Related Practice
Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.99, s^2 = 0.64, n = 7

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Textbook Question

In Exercises 7 and 8, find the margin of error for the values of c, s, and n.

c = 0.99, s = 3, n = 6

133
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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


μ<33

49
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Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.95, σ = 2.5, E = 1.

100
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Textbook Question

In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.

c = 0.98, xbar = 4.3, s = 0.34, n = 14

107
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Textbook Question

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.88

101
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