Question

1. in a marathon, finish times are normally distributed with a mean of 4.5 hours and a standard deviation of 0.5 hours.

a. find the z - score for a runner finishing in 3.8 hours.
b. what percentile does this roughly correspond to?

1. sat scores are normally distributed with a mean of 1024 and a standard deviation of 229.

a. find the z - score for a student scoring 1300.
b. find the z - score for a student scoring 900.
c. who performed better relative to the mean?

Explanation:

Problem 6a

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Identify values

We are given that $x = 3.8$ hours, $\mu=4.5$ hours, and $\sigma = 0.5$ hours.

Step3: Substitute into formula

Substitute the values into the z - score formula: $z=\frac{3.8 - 4.5}{0.5}=\frac{- 0.7}{0.5}=-1.4$

A z - score of $z=-1.4$ can be used to find the percentile. We can use the standard normal distribution table (z - table). The z - table gives the area to the left of the z - score. Looking up $z = - 1.4$ in the z - table, the area to the left of $z=-1.4$ is approximately 0.0808 or 8.08%. So this roughly corresponds to the 8th percentile (or approximately the 8th percentile, we can also say around 8% of the data is below this value).

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Identify values

We are given that $x = 1300$, $\mu = 1024$, and $\sigma=229$.

Step3: Substitute into formula

Substitute the values into the z - score formula: $z=\frac{1300 - 1024}{229}=\frac{276}{229}\approx1.205$ (rounded to three decimal places)

Answer:

The z - score is $- 1.4$

Problem 6b