Question

1. which of the following is the closest z - score to the 90th percentile of the standard normal distribution?

a. ( z = - 1.28 )
b. ( z = - 0.9 )
c. ( z = 0.9 )
d. ( z = 1.28 )

1. mr. tuttle is planning to buy a new car. he’s considering the ford escape, a sport utility vehicle (suv) that gets 28 highway miles per gallon, and the ford fusion, a mid - sized sedan that gets 31 highway miles per gallon. the mean fuel efficiency for all sport utility vehicles is 23, with a standard deviation of 7.6. the mean fuel efficiency of all mid - sized sedans is 27, with a standard deviation of 5.2. which vehicle has a better gas mileage, relative to others of the same style?

a. the ford fusion sedan has a better gas mileage, because its z - score is greater.
b. the ford fusion sedan has a better gas mileage, because its z - score is closer to 0.
c. the ford escape suv has a better gas mileage, because its z - score is greater.
d. the ford escape suv has a better gas mileage, because its z - score is closer to 0.

1. in a 5k race with thousands of participants, the distribution of finish times can be modeled by a uniform distribution on the interval from 16.5 minutes to 40.5 minutes. all participants that finish in less than 25 minutes get a t - shirt. what percent of participants will get a t - shirt?

a. 25.0%
b. 35.4%
c. 42.6%
d. 48.7%

Explanation:

Question 1

Step1: Recall 90th percentile z - score

The 90th percentile of the standard normal distribution (the z - score such that \(P(Z < z)=0.9\)) is approximately \(z = 1.28\) (from standard normal tables or z - score calculators). We need to find which of the given z - scores is closest to \(1.28\).

Step2: Calculate differences

• For option a (\(z=- 1.28\)): The difference from \(1.28\) is \(|1.28-(-1.28)| = 2.56\)
• For option b (\(z = - 0.9\)): The difference from \(1.28\) is \(|1.28-(-0.9)|=2.18\)
• For option c (\(z = 0.9\)): The difference from \(1.28\) is \(|1.28 - 0.9|=0.38\)
• For option d (\(z = 1.28\)): The difference from \(1.28\) is \(|1.28 - 1.28| = 0\)

Since the difference for option d is \(0\), it is the closest.

Step1: Calculate z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step2: Calculate z - score for Ford Fusion

For the Ford Fusion (mid - sized sedan): \(x = 31\), \(\mu=27\), \(\sigma = 5.2\)
\(z_{Fusion}=\frac{31 - 27}{5.2}=\frac{4}{5.2}\approx0.77\)

Step3: Calculate z - score for Ford Escape

For the Ford Escape (sport utility vehicle): \(x = 28\), \(\mu = 23\), \(\sigma=7.6\)
\(z_{Escape}=\frac{28 - 23}{7.6}=\frac{5}{7.6}\approx0.66\)

Step4: Analyze z - scores

A higher z - score indicates a better (more above the mean) gas mileage relative to its vehicle type. The z - score of the Ford Fusion (\(\approx0.77\)) is greater than the z - score of the Ford Escape (\(\approx0.66\)). Also, we can think in terms of how far from the mean in a positive direction. The Ford Fusion's z - score is greater, meaning it is more above the mean of its vehicle class. So the Ford Fusion sedan has a better gas mileage because its z - score is greater.

Step1: Recall uniform distribution probability formula

For a uniform distribution on the interval \([a,b]\), the probability that \(X

Step2: Identify values

Here, \(a = 16.5\) minutes, \(b = 40.5\) minutes, and \(x = 25\) minutes.

Step3: Calculate probability

\(P(X<25)=\frac{25 - 16.5}{40.5 - 16.5}=\frac{8.5}{24}\approx0.3542\)
To convert to a percentage, we multiply by \(100\): \(0.3542\times100 = 35.42\%\approx35.4\%\)

Answer:

d. \(z = 1.28\)

Question 2