Fire department response time is the length of time it takes a fire truck to arrive at the scene of a fire starting from the time the call was given to the truck. Response time for Gabble Fire Department follows a mound shaped and symmetric distribution. The response time has a mean of 8.5 minutes with a standard deviation of 1.8 minutes. If a random sample of 34 response times is taken and the mean response x is computed, find P(7 < x < 11)

A mound-shaped symmetrical distribution sounds normal to me.

Z = (score-mean)/SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.

To find the probability P(7 < x < 11), we will use the z-score formula in conjunction with the given mean and standard deviation. The z-score formula allows us to standardize the values and work with the standard normal distribution.

First, we need to calculate the z-scores for both 7 minutes and 11 minutes.

The formula for calculating the z-score is: z = (x - μ) / σ

Where:
- x is the value we want to calculate the z-score for
- μ is the population mean
- σ is the population standard deviation

For 7 minutes:
z1 = (7 - 8.5) / 1.8

For 11 minutes:
z2 = (11 - 8.5) / 1.8

Now, we can look up the corresponding probabilities using a standard normal distribution table (also known as a z-table) or use statistical software.

Let's assume we use a standard normal distribution table and the values we obtain are:
- The z-score for 7 minutes, z1 = -1.39
- The z-score for 11 minutes, z2 = 1.39

To find P(7 < x < 11), we need to find the area under the curve between these two z-scores.

Using the z-table, we find the corresponding probabilities:
- P(z < -1.39) = 0.0823
- P(z < 1.39) = 0.9177

To find the area between two z-scores, we subtract the smaller probability from the larger one:

P(7 < x < 11) = P(z1 < z < z2) = P(z < z2) - P(z < z1) = 0.9177 - 0.0823 = 0.8354

Therefore, the probability that the response time falls between 7 and 11 minutes is approximately 0.8354 or 83.54%.