Choosing a Distribution 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.
Body Mass Index In a random sample of 50 people, the mean body mass index (BMI) was 27.7 and the standard deviation was 6.12.

Verified step by step guidance

1. Step 1: Determine whether to use the standard normal distribution (Z-distribution) or the t-distribution. Since the sample size is 50, which is greater than 30, and the population standard deviation is not provided (only the sample standard deviation is given), the t-distribution is appropriate for constructing the confidence interval.
2. Step 2: Identify the necessary components for constructing the confidence interval. These include the sample mean (\( \bar{x} = 27.7 \)), the sample standard deviation (\( s = 6.12 \)), the sample size (\( n = 50 \)), and the confidence level (95%).
3. Step 3: Calculate the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is \( df = n - 1 \). Substitute \( n = 50 \) into the formula to find \( df \).
4. Step 4: Find the critical t-value (\( t^* \)) corresponding to a 95% confidence level and the calculated degrees of freedom. Use a t-distribution table or statistical software to find \( t^* \).
5. Step 5: Construct the confidence interval using the formula \( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \). Substitute the values for \( \bar{x} \), \( t^* \), \( s \), and \( n \) into the formula to calculate the confidence interval.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is crucial in statistics because many statistical methods, including confidence intervals, assume that the data follows this distribution. When the sample size is large (typically n > 30), the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal, allowing for the use of the normal distribution.

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 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 hypothesis testing when dealing with smaller 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 95%. It is calculated using the sample mean and standard deviation, along with the appropriate distribution (normal or t) based on the sample size and known parameters. Interpreting a confidence interval involves understanding that if the same sampling process were repeated multiple times, a certain percentage of those intervals would contain the true population mean.