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.