Question

In determining automobile mileage ratings, it was found that the mpg(X) for a certain model is normally distributed, with a mean of 39mpg and a standard deviation of 1.2mpg. Complete parts a through e below. a. Find P(X<38). P(X<38)= (Round to four decimal places as needed) b. Find P(3741). P(X>41)= (Round to four decimal places as needed.) d. Find P(x>38). P(X>38)= (Round to four decimal places as needed) e. Find the mileage rating that the upper 14% of cars achieve. The mileage rating that the upper 14% of cars achieve is mpg. (Round to two decimal places as needed.)

Answer 1

Answer and Step-by-step explanation:

To solve these problems, we can use the standard normal distribution and z-scores. A z-score tells us how many standard deviations an observation is from the mean.

a. To find P(X < 38), we need to find the z-score corresponding to 38 and then use a standard normal distribution table or calculator to find the corresponding probability.

The z-score formula is:

z = (X - μ) / σ

where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

Plugging in the values:

z = (38 - 39) / 1.2 = -0.83

Using a standard normal distribution table or calculator, we find that the probability corresponding to a z-score of -0.83 is approximately 0.2031.

So, P(X < 38) ≈ 0.2031.

b. To find P(X > 41), we can use the same approach as in part a. First, find the z-score for 41.

z = (41 - 39) / 1.2 = 1.67

Using the standard normal distribution table or calculator, we find that the probability corresponding to a z-score of 1.67 is approximately 0.9525.

Since we want P(X > 41), we subtract this probability from 1 to find the probability of X being greater than 41.

P(X > 41) ≈ 1 - 0.9525 = 0.0475.

d. To find P(X > 38), we can use the same approach as in part a. Calculate the z-score for 38:

z = (38 - 39) / 1.2 = -0.83

Using the standard normal distribution table or calculator, we find that the probability corresponding to a z-score of -0.83 is approximately 0.2031.

Since we want P(X > 38), we subtract this probability from 1 to find the probability of X being greater than 38.

P(X > 38) ≈ 1 - 0.2031 = 0.7969.

e. To find the mileage rating that the upper 14% of cars achieve, we need to find the z-score that corresponds to a cumulative probability of 0.86 (1 - 0.14 = 0.86). This z-score will represent the number of standard deviations above the mean.

Using a standard normal distribution table or calculator, we find that the z-score corresponding to a cumulative probability of 0.86 is approximately 1.08.

Now, we can use the z-score formula to find the mileage rating:

z = (X - μ) / σ

Plugging in the values:

1.08 = (X - 39) / 1.2

Solving for X:

1.08 * 1.2 = X - 39

1.296 = X - 39

X ≈ 40.296

The mileage rating that the upper 14% of cars achieve is approximately 40.30 mpg.