Tar and Nicotine Use the results of Problem 16 in Section 12.3 to answer the following questions:
b. Construct a 95% confidence interval for the tar content found in part (a).

Verified step by step guidance

1. Identify the sample mean (\$\bar{x}\$) and the sample standard deviation (\$s\$) for the tar content from part (a) or from the given data in Problem 16 of Section 12.3.
2. Determine the sample size (\$n\$), which is the number of observations used to calculate the sample mean and standard deviation.
3. Since the population standard deviation is unknown and the sample size is likely small, use the t-distribution to construct the confidence interval. Find the appropriate t-value (\$t^*\$) for a 95% confidence level and \$n-1\$ degrees of freedom from the t-table.
4. Calculate the standard error of the mean (SEM) using the formula: \[\text{SEM} = \frac{s}{\sqrt{n}}\]
5. Construct the 95% confidence interval using the formula: \[\bar{x} \pm t^* \times \text{SEM}\] This interval estimates the range in which the true mean tar content lies with 95% confidence.

Key Concepts

Confidence Interval

A confidence interval estimates a range of values within which a population parameter, like the mean tar content, is likely to fall. A 95% confidence interval means we are 95% confident the true mean lies within this range, accounting for sample variability.

Sample Mean and Sample Size

The sample mean is the average tar content calculated from the sample data, serving as the point estimate for the population mean. The sample size affects the precision of the confidence interval; larger samples generally yield narrower intervals.

Standard Error and Critical Value

The standard error measures the variability of the sample mean and is calculated using the sample standard deviation and size. The critical value, from the t-distribution or z-distribution, determines the margin of error for the confidence interval based on the desired confidence level.