Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
a. In a random sample of 40 patients, the mean waiting time at a dentist’s office was 20 minutes and the standard deviation was 7.5 minutes. Construct a 95% confidence interval for the population mean.

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Determine which distribution to use: Since the sample size is 40 (greater than 30), the Central Limit Theorem applies, and the sample mean can be approximated by a normal distribution. Additionally, the population standard deviation is not provided, so we use the t-distribution.

Identify the necessary values: The sample mean (\( \bar{x} \)) is 20 minutes, the sample standard deviation (\( s \)) is 7.5 minutes, the sample size (\( n \)) is 40, and the confidence level is 95%.

Find the critical value: For a 95% confidence level and degrees of freedom (\( df = n - 1 = 40 - 1 = 39 \)), use a t-distribution table or software to find the critical value (\( t^* \)).

Calculate the standard error of the mean (SE): Use the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.

Construct the confidence interval: Use the formula \( \bar{x} \pm t^* \cdot SE \), where \( \bar{x} \) is the sample mean, \( t^* \) is the critical value, and \( SE \) is the standard error. This will give the lower and upper bounds of the confidence interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Normal Distribution

The standard normal distribution is a probability distribution that is symmetric about the mean, with a mean of zero and a standard deviation of one. It is used in statistics to determine probabilities and critical values for hypothesis testing and confidence intervals. When sample sizes are large (typically n > 30), the Central Limit Theorem allows us to use this distribution to approximate the sampling distribution of the sample mean.

t-Distribution

The t-distribution is a type of probability distribution that is used when the sample size is small (n < 30) or when the population standard deviation is unknown. It is similar to the standard normal distribution but has heavier tails, which provides a more accurate estimate of the population mean in these cases. The t-distribution is essential for constructing confidence intervals and conducting hypothesis tests when dealing with small samples.

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, typically expressed as a percentage (e.g., 95%). It is calculated using the sample mean, the standard deviation, and the appropriate critical value from either the standard normal or t-distribution. Interpreting a confidence interval involves understanding that if the same sampling process were repeated multiple times, a certain percentage of the intervals would contain the true population mean.

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