Let's go to the movies: A random sample of 42 Hollywood movies made in the last 10 years had a mean length of 110.2 minutes, with a standard deviation of 12.3 minutes.Part: 0 / 20 of 2 Parts CompletePart 1 of 2(a) Construct a 90% confidence interval for the true mean length of all Hollywood movies in the last 10 years. Round the answers to at least one decimal place.A 90% confidence interval for the true mean length of all Hollywood movies made in the last 10 years is <<μ.
Question
Let's go to the movies: A random sample of 42 Hollywood movies made in the last 10 years had a mean length of 110.2 minutes, with a standard deviation of 12.3 minutes.Part: 0 / 20 of 2 Parts CompletePart 1 of 2(a) Construct a 90% confidence interval for the true mean length of all Hollywood movies in the last 10 years. Round the answers to at least one decimal place.A 90% confidence interval for the true mean length of all Hollywood movies made in the last 10 years is <<μ.
Solution
To construct a 90% confidence interval for the true mean length of all Hollywood movies in the last 10 years, we first need to understand that a confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. In this case, the population parameter is the true mean length of all Hollywood movies in the last 10 years.
The formula for a confidence interval is:
x̄ ± Z * (σ/√n)
where: x̄ is the sample mean, Z is the Z-value from the standard normal distribution for the desired confidence level (for a 90% confidence level, Z = 1.645), σ is the standard deviation of the population, and n is the size of the sample.
Plugging in the given values:
110.2 ± 1.645 * (12.3/√42)
This simplifies to:
110.2 ± 1.645 * (12.3/6.48)
110.2 ± 1.645 * 1.9
110.2 ± 3.1
So, the 90% confidence interval for the true mean length of all Hollywood movies in the last 10 years is (107.1 minutes, 113.3 minutes).
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