Final answer:
A 95% confidence interval for the mean length of Hollywood movies is constructed using the sample mean, sample size, and sample standard deviation. The resulting interval, using the Z-distribution, is found to be approximately (108.7, 116.3).
Step-by-step explanation:
The question asks to construct a 95% confidence interval for the true mean length of all Hollywood movies made in the last 10 years, based on a sample mean of 112.5 minutes, a standard deviation of 12.6 minutes, and a sample size of 43 movies.
To calculate a 95% confidence interval, you would typically use the formula:
± Z * (σ/√n)
where Z is the Z-score corresponding to the desired confidence level, σ is the standard deviation, and n is the sample size. Because the sample size is below 30 we should use the t-distribution, but since the sample size here is 43 which is sufficiently large, we can use the Z-distribution as an approximation.
To find the 95% confidence Z-score, we look it up in Z-score tables or use a technology tool, and it is approximately 1.96. Then we plug in the values into the equation:
± 1.96 * (12.6/√43)
A calculation yields a margin of error of approximately 3.8. Therefore, we subtract and add this to the sample mean to find the confidence interval. The resulting confidence interval is (112.5 - 3.8, 112.5 + 3.8), which is. (108.7, 116.3).