An entertainment company owns and operates movie theaters in Wyoming. The president of the company is concerned that film rentals are hurting the business. They directed a staff member to estimate the total number of films rented by households in wyoming in a particular month. A phone survey involving a random sample of 300 homes was conducted with the following results - xbar = 2.4 films. Sx = 1.6 films. Then the 90% confidence interval estimate for the total number of films rented by the 211,000 households in that particular month is?
This has been addressed in a later post.
To estimate the total number of films rented by households in Wyoming in a particular month with a 90% confidence interval, we can use the sample mean and sample standard deviation.
Given:
Sample size (n) = 300
Sample mean (x̄) = 2.4 films
Sample standard deviation (s) = 1.6 films
Total number of households in Wyoming = 211,000
To calculate the 90% confidence interval, we'll use the formula:
Confidence Interval = x̄ ± (Z * s / √n)
First, we need to find the Z-value corresponding to a 90% confidence level. The Z-value is obtained from the standard normal distribution table or by using a statistical calculator. For a 90% confidence interval, the Z-value is approximately 1.645.
Substituting the given values:
Confidence Interval = 2.4 ± (1.645 * 1.6 / √300)
Next, calculate the standard error (SE):
SE = s / √n
SE = 1.6 / √300
Substitute the value of SE in the confidence interval formula:
Confidence Interval = 2.4 ± (1.645 * (1.6 / √300))
Now, compute the lower and upper bounds of the confidence interval:
Lower bound = 2.4 - (1.645 * (1.6 / √300))
Upper bound = 2.4 + (1.645 * (1.6 / √300))
Finally, multiply the lower and upper bounds by the total number of households in Wyoming to obtain the estimate for the total number of films rented by households in that particular month:
Lower estimate = Lower bound * 211,000
Upper estimate = Upper bound * 211,000
The 90% confidence interval estimate for the total number of films rented by the 211,000 households in Wyoming in that particular month is between the lower estimate and the upper estimate.