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

검증된 단계별 안내

Step 1: Identify the given values from the problem. The confidence level (c) is 0.98, the sample mean (x̄) is 4.3, the sample standard deviation (s) is 0.34, and the sample size (n) is 14.

Step 2: Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1. Substitute n = 14 into the formula to calculate df.

Step 3: Find the critical t-value (t*) corresponding to the confidence level (c = 0.98) and the degrees of freedom (df). Use a t-distribution table or statistical software to find t*.

Step 4: Calculate the margin of error (ME) using the formula ME = t* × (s / √n). Substitute the values of t*, s = 0.34, and n = 14 into the formula.

Step 5: Construct the confidence interval for the population mean (μ) using the formula: Confidence Interval = x̄ ± ME. Substitute x̄ = 4.3 and the calculated ME into the formula to express 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 98% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 98% of those intervals would contain the true population mean.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used instead of the normal distribution when the sample size is small (typically n < 30) and the population standard deviation is unknown, making it ideal for constructing confidence intervals for the mean in such cases.

Sample Mean and Standard Deviation

The sample mean (x̄) is the average of a set of sample observations, providing an estimate of the population mean (μ). The sample standard deviation (s) measures the dispersion of the sample data points around the sample mean. Both are crucial for calculating the confidence interval, as they help determine the range within which the true population mean is likely to fall.

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