Final answer:
To calculate the probability that the mean cost of movie tickets in a sample is between $7.00 and $7.50, we determine the standard error, calculate the z-scores for both price limits, and then find the corresponding probabilities using the standard normal distribution. The probability is the difference between these probabilities.
Step-by-step explanation:
To solve this problem, you can use the concepts of the sampling distribution of the sample mean and the Central Limit Theorem (CLT). Given that the population standard deviation (σ) is $1.39 and the sample size (n) is 50, we can assume that the sample mean will be normally distributed based on the CLT as n ≥ 30.
The first step is to calculate the standard error (SE) of the mean, which is the standard deviation (σ) divided by the square root of the sample size (n):
SE = σ / √ n
SE = $1.39 / √ 50
SE ≈ $0.1964
Next, we can find the z-scores for the lower and upper bounds of the movie ticket cost ($7.00 and $7.50 respectively) using the following formula:
z = (X - μ) / SE
Here X is the value for which you want to find the z-score, and μ is the population mean. For $7.00:
z = ($7.00 - $7.89) / $0.1964
z ≈ -4.53
And for $7.50:
z = ($7.50 - $7.89) / $0.1964
z ≈ -1.98
Now, using the standard normal distribution table, we can find the probabilities corresponding to these z-scores and then find the probability that the sample mean falls between $7.00 and $7.50 by finding the difference between the two probabilities.