Question

The ACT is an achievement test given nationally with normally distributed scores. Tim

scored a 24 on the mathematics portion of his ACT. The mean for the mathematics portion of

the ACT was 22.0 and the standard deviation was 5.1. What percent of the population scored

higher than Tim on the mathematics portion of the ACT?

Answer 1

Answer:

34.83% of the population scored higher than Tim on the mathematics portion of the ACT

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

$\mu = 22, \sigma = 5.1$

Tim scored 24. What percent of the population scored higher than Tim on the mathematics portion of the ACT?

The proportion is 1 subtracted by the pvalue of Z when X = 24. The percentage is the proportion multiplied by 100.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{24 - 22}{5.1}$

$Z = 0.39$

$Z = 0.39$ has a pvalue of 0.6517

1 - 0.6517 = 0.3483

34.83% of the population scored higher than Tim on the mathematics portion of the ACT