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.

검증된 단계별 안내

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 / \sqrt{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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주요 개념

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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.

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.

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.

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.