The average cost for two people to go to a movie is [tex]$22. The standard deviation is $[/tex]3. Assume the cost is normally distributed.

Find the probability that at any given theater, the cost will be more than [tex]$25 for two people to go to the movie. Use the empirical rule.

The probability that at any given theater, the cost will be more than $[/tex]25 for two people to go to a movie is ______%.



Answer :

Alright, let's tackle this problem using the properties of the normal distribution.

1. Identify Given Information:
- Average (mean) cost ([tex]\(\mu\)[/tex]): [tex]$22 - Standard deviation (\(\sigma\)): $[/tex]3
- Threshold cost: [tex]$25 2. Determine What to Find: - The probability that the cost will exceed $[/tex]25.

3. Calculate the Z-Score:
The Z-score helps us understand how many standard deviations away our threshold value ([tex]$25) is from the mean. The Z-score formula is: \[ Z = \frac{X - \mu}{\sigma} \] where: - \(X\) is the threshold cost ($[/tex]25),
- [tex]\(\mu\)[/tex] is the mean ([tex]$22), - \(\sigma\) is the standard deviation ($[/tex]3).

So, plugging in the values:

[tex]\[ Z = \frac{25 - 22}{3} = \frac{3}{3} = 1 \][/tex]

4. Using the Normal Distribution:
- The Z-score of 1 tells us that the cost of [tex]$25 is 1 standard deviation above the mean. - To find the probability of the cost being more than $[/tex]25, we need to look at the area to the right of Z = 1 in the normal distribution curve.

5. Empirical Rule and the Normal Distribution Table:
- The empirical rule states that approximately 68% of data falls within 1 standard deviation (σ) of the mean (µ). This leaves about 32% outside this range, divided between the two tails of the normal distribution.
- Specifically, about 16% (or 0.15865525393145707 to be exact) lies above 1 standard deviation from the mean in the right tail.

6. Conclusion:
The probability that at any given theater, the cost will be more than [tex]$25 for two people to go to a movie is approximately: \[ \text{Probability} \approx 0.15865525393145707 \] Converting this to a percentage: \[ \text{Probability} \approx 15.87\% \] Therefore, the probability that at any given theater, the cost will be more than $[/tex]25 for two people to go to a movie is approximately 15.87%.