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.
b. In a random sample of 15 cereal boxes, the mean weight was 11.89 ounces. Assume the weights of the cereal boxes are normally distributed and the population standard deviation is 0.05 ounce. Construct a 90% confidence interval for the population mean.

Verified step by step guidance

1. Step 1: Determine which distribution to use. Since the population standard deviation is known and the sample size is less than 30, the standard normal distribution (Z-distribution) is appropriate for constructing the confidence interval.
2. Step 2: Identify the given values. The sample mean (x̄) is 11.89 ounces, the population standard deviation (σ) is 0.05 ounce, the sample size (n) is 15, and the confidence level is 90%.
3. Step 3: Find the critical value (Z*) for a 90% confidence level using the standard normal distribution. The critical value corresponds to the middle 90% of the distribution, leaving 5% in each tail. Look up the Z* value in a Z-table or use statistical software.
4. Step 4: Calculate the standard error of the mean (SE). The formula for SE is: $\sigma /\sqrt{n}$, where σ is the population standard deviation and n is the sample size.
5. Step 5: Construct the confidence interval using the formula: $\mathrm{x̄}\pm \mathrm{Z*}\times \mathrm{SE}$. Substitute the values for x̄, Z*, and SE to find the lower and upper bounds of 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. In this context, the assumption of normality allows for the use of the standard normal distribution to calculate confidence intervals for the population mean.

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 90%. It is calculated using the sample mean, the standard deviation, and the critical value from the appropriate distribution (normal or t-distribution). Understanding how to construct and interpret confidence intervals is essential for making inferences about population parameters based on sample data.

t-Distribution vs. Standard Normal Distribution

The t-distribution is similar to the standard normal distribution but has heavier tails, making it more appropriate for smaller sample sizes (typically n < 30) when the population standard deviation is unknown. In this scenario, since the sample size is 15 and the population standard deviation is known, the standard normal distribution is used to construct the confidence interval. Knowing when to apply each distribution is vital for accurate statistical analysis.