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?
Here you want to get the area in the center of the normal distribution of 90%, leaving you �}5% at the two tails. Using the table for the normal distribution, find the Z-value for .05.
Using Z = (X - mean)/SE (Not SD. See other post.), find the value of X. That value, subtracted and added to the sample mean, gives you the 90% confidence interval.
I hope this helps. Thanks for asking.
To find the confidence interval estimate for the total number of films rented by households in Wyoming, we can use the formula:
CI = x̄ ± Z * (S/√n)
Where:
- x̄ is the sample mean (2.4 films),
- Z is the Z-value that corresponds to the desired confidence level (90% confidence corresponds to a Z-value of 1.645),
- S is the sample standard deviation (1.6 films), and
- n is the sample size (300 homes).
First, we need to find the Z-value for .05 by looking up the value in the table for the normal distribution. As .05 corresponds to 5% in the tail, we need to find the Z-value that leaves 5% in the tail (i.e., area of 0.05 or 0.025 on each side). From the table, we find that the Z-value for .05 is approximately 1.645.
Next, we substitute the known values into the formula:
CI = 2.4 ± 1.645 * (1.6/√300)
Calculating the values:
CI = 2.4 ± 1.645 * 0.0925
CI = 2.4 ± 0.152
This gives us the confidence interval estimate:
CI = (2.248, 2.552)
Therefore, we can estimate with 90% confidence that the total number of films rented by the 211,000 households in that particular month lies between 2.248 and 2.552 films.
I hope this explanation helps! Let me know if you have any further questions.