Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.
Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

Verified step by step guidance

1. Step 1: Calculate the sample mean (x̄) by summing all the data points provided in the table and dividing by the total number of data points (n = 18). Use the formula: $\mathrm{x̄}=\frac{\sum x}{n}$
2. Step 2: Calculate the sample standard deviation (s) using the formula: $s=\sqrt{\frac{\sum (x-\mathrm{x̄}{)}^2}{n-1}}$
3. Step 3: Determine the critical value (t*) for a 99% confidence interval using a t-distribution table. The degrees of freedom (df) are calculated as: $\mathrm{df}=n-1$. For df = 17, find the t* value corresponding to a 99% confidence level.
4. Step 4: Calculate the margin of error (E) using the formula: $E=\mathrm{t*}\times \frac{s}{\sqrt{n}}$
5. Step 5: Construct the confidence interval for the population mean using the formula: $[\mathrm{x̄}-E,\mathrm{x̄}+E]$. This interval represents the range within which the true population mean is likely to fall with 99% confidence.

Key Concepts

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. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population mean.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In constructing confidence intervals, it is often assumed that the population from which the sample is drawn is normally distributed, especially when sample sizes are small.

Sample Mean and Standard Deviation

The sample mean is the average of a set of values, calculated by summing all observations and dividing by the number of observations. The sample standard deviation measures the amount of variation or dispersion in a set of values. Both the sample mean and standard deviation are crucial for calculating confidence intervals, as they provide the necessary statistics to estimate the range within which the population mean is likely to fall.