Hurricanes Use the results of Problem 14 in Section 12.3 to answer the following questions:
b. Construct a 95% confidence interval for the mean wind speed found in part (a).

Verified step by step guidance

1. Identify the sample mean \( \bar{x} \) and the sample standard deviation \( s \) from the results given in Problem 14, Section 12.3, part (a). These values represent the estimated average wind speed and its variability.
2. Determine the sample size \( n \), which is the number of hurricane wind speed observations used in the sample.
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-critical 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 for the mean wind speed using the formula: \[ \bar{x} \pm t^* \times \text{SEM} \] This interval estimates the range in which the true mean wind speed lies with 95% confidence.

Key Concepts

Confidence Interval

A confidence interval estimates the range within which a population parameter, like the mean, is likely to fall. A 95% confidence interval means we are 95% confident the true mean lies within this range, based on sample data.

Sample Mean and Standard Deviation

The sample mean is the average value calculated from the sample data, serving as a point estimate of the population mean. The sample standard deviation measures the variability of the data and is used to assess the precision of the mean estimate.

t-Distribution and Degrees of Freedom

When the population standard deviation is unknown and the sample size is small, the t-distribution is used to construct confidence intervals. Degrees of freedom, typically sample size minus one, affect the shape of the t-distribution and the critical value for the interval.