Question

a researcher measured how long 100 kine students from york and 100 u of t kine students could run before quitting. for york students, assume that the mean is 18 miles with a standard deviation of 2.5 miles, and for u of t students assume that the mean is 15 miles with a standard deviation of 4 miles. assume that both distributions are normal.
after 75 york students have stopped running, how many u of t students are still running?
a. 11
b. 12
c. 10
d. 13

Explanation:

Step1: Find York's 75th percentile z-score

We use the standard normal distribution: the z-score for the 75th percentile (top 25% still running) is $z = 0.674$.

Step2: Calculate the cutoff distance

Use $x = \mu + z\sigma$. For York: $\mu=18$, $\sigma=2.5$
$$x = 18 + 0.674 \times 2.5 = 18 + 1.685 = 19.685$$

Step3: Find U of T's z-score for cutoff

Use $z = \frac{x - \mu}{\sigma}$. For U of T: $\mu=15$, $\sigma=4$
$$z = \frac{19.685 - 15}{4} = \frac{4.685}{4} = 1.171$$

Step4: Find percentage of U of T runners

The area to the right of $z=1.171$ is $1 - 0.879 = 0.121$ (12.1%).

Step5: Calculate number of U of T students

Multiply percentage by total U of T students (100):
$$100 \times 0.121 = 12.1 \approx 12$$

Answer:

b. 12