In Exercises 21–24, construct the indicated confidence interval for the population mean μ.
c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

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Identify the key components of the problem: the confidence level (c = 0.80), sample mean (x̄ = 20.6), population standard deviation (σ = 4.7), and sample size (n = 100).

Determine the z-score corresponding to the given confidence level (c = 0.80). For an 80% confidence level, the critical z-value can be found using a z-table or statistical software. This z-value corresponds to the middle 80% of the standard normal distribution.

Calculate the standard error of the mean (SE) using the formula:

SE = \frac{σ}{\sqrt{n}}

. Substitute σ = 4.7 and n = 100 into the formula.

Compute the margin of error (ME) using the formula:

ME = z \times SE

. Use the z-value from step 2 and the SE from step 3.

Construct the confidence interval for the population mean μ using the formula:

[x̄ - ME, x̄ + ME]

. Substitute x̄ = 20.6 and the ME from step 4 into the formula to find the interval.

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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 confidence level of 80% means that if we were to take many samples and construct intervals, approximately 80% of those intervals would contain the true population mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of confidence intervals, it helps quantify the uncertainty around the sample mean. A smaller standard deviation indicates that the data points tend to be closer to the mean, leading to a narrower confidence interval.

Sample Size

Sample size refers to the number of observations in a sample. It plays a crucial role in determining the width of the confidence interval; larger sample sizes generally lead to more precise estimates of the population mean, resulting in narrower confidence intervals. In this case, a sample size of 100 provides a solid basis for estimating the population mean.

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.c = 0.80, xbar = 20.6, σ = 4.7, n = 100.