In a random sample of 36 top-rated roller coasters, the average height is 165 feet and the standard deviation is 67 feet. Construct a 90% confidence interval for μ. Interpret the results. (Source: POP World Media, LLC)

Verified step by step guidance

1. Step 1: Identify the given values from the problem. The sample size (n) is 36, the sample mean (x̄) is 165 feet, the sample standard deviation (s) is 67 feet, and the confidence level is 90%.
2. Step 2: Determine the critical value (z*) for a 90% confidence level. Since the sample size is large (n ≥ 30), we can use the standard normal distribution. For a 90% confidence level, the critical value corresponds to the z-score that leaves 5% in each tail of the distribution. Use a z-table or statistical software to find this value.
3. Step 3: Calculate the standard error of the mean (SE). The formula for the standard error is: $\mathrm{SE}=\frac{s}{\sqrt{n}}$, where s is the sample standard deviation and n is the sample size.
4. Step 4: Compute the margin of error (ME). The formula for the margin of error is: $\mathrm{ME}=z*\mathrm{SE}$, where z* is the critical value and SE is the standard error.
5. Step 5: Construct the confidence interval. The formula for the confidence interval is: $[\mathrm{x̄}-\mathrm{ME},\mathrm{x̄}+\mathrm{ME}]$. Substitute the values of x̄ (sample mean) and ME (margin of error) to find the interval. Finally, interpret the results by explaining that we are 90% confident the true population mean height of top-rated roller coasters lies within this interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter (e.g., the mean) with a specified level of confidence. For example, a 90% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 90% of those intervals would contain the true population mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of a sample, it quantifies how much individual data points deviate from the sample mean. A larger standard deviation indicates greater variability in the data, which affects the width of the confidence interval.

Sample Size

Sample size refers to the number of observations or data points collected in a study. In this case, a sample size of 36 roller coasters is used. Larger sample sizes generally lead to more accurate estimates of the population parameters and narrower confidence intervals, as they reduce the impact of random sampling error.