In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
The population standard deviation of the weights of the two-year-old males on a pediatrician’s patient list is 2.49 pounds. The mean weight of a sample of 10 of the two–year–old males is 13.68 pounds. Weights are known to be normally distributed.

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Step 1: Identify the type of distribution to use. Since the population standard deviation (σ = 2.49 pounds) is known and the weights are normally distributed, we use the standard normal distribution (Z-distribution) to construct the confidence interval.

Step 2: Write the formula for the confidence interval using the Z-distribution: CI = x̄ ± Z * (σ / √n), where x̄ is the sample mean, Z is the critical value for the desired confidence level (95%), σ is the population standard deviation, and n is the sample size.

Step 3: Determine the critical value (Z) for a 95% confidence level. For a two-tailed test, the critical value corresponds to the area in the tails of the standard normal distribution. Look up the Z-value in a Z-table or use statistical software.

Step 4: Plug the known values into the formula. Substitute x̄ = 13.68, σ = 2.49, and n = 10 into the formula. Calculate the standard error (SE) as SE = σ / √n.

Step 5: Compute the margin of error (ME) as ME = Z * SE. Add and subtract the margin of error from the sample mean (x̄) to find the lower and upper bounds of the confidence interval. Interpret the interval in the context of the problem, explaining that it represents the range in which the true population mean is likely to fall with 95% confidence.

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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, such as 95%. It provides an estimate of uncertainty around the sample mean, indicating how much the sample mean might vary from the true population mean.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, it is important because the weights of the two-year-old males are stated to be normally distributed, allowing the use of the standard normal distribution for constructing confidence intervals.

t-Distribution

The t-distribution is a type of probability distribution that is used when the sample size is small (typically n < 30) and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which provides a more accurate estimate of the confidence interval when dealing with smaller samples.

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