Question

aw data\ to percentages

1. the act is a test to determine readiness for college. the distribution of act scores is approximately

normal with a mean \\( \overline { x } = 21.5 \\) and a standard deviation of \\( s = 5.4 \\).
(a) if anthony got a 28 on his act, what percentile is he in?
(b) if amanda received a 20, how many people scored better than her?

Explanation:

Step1: Calculate z - score for Anthony

The formula for the z - score is \(z=\frac{x-\bar{x}}{s}\). Here, \(x = 28\), \(\bar{x}=21.5\), and \(s = 5.4\).

$$z=\frac{28 - 21.5}{5.4}=\frac{6.5}{5.4}\approx1.20$$

Step2: Find the percentile using the z - score

Using a standard normal distribution table (or a calculator with a normal - distribution function), for \(z = 1.20\), the cumulative probability \(P(Z\leq1.20)\) is approximately \(0.8849\) or \(88.49\%\).

Step3: Calculate z - score for Amanda

For Amanda, \(x = 20\), \(\bar{x}=21.5\), and \(s = 5.4\)

$$z=\frac{20 - 21.5}{5.4}=\frac{- 1.5}{5.4}\approx - 0.28$$

Step4: Find the percentile for Amanda

Using the standard normal distribution table (or a calculator with a normal - distribution function), for \(z=-0.28\), the cumulative probability \(P(Z\leq - 0.28)\) is approximately \(0.3897\) or \(38.97\%\)

Answer:

a) Anthony is in approximately the \(88^{th}\) percentile.
b) The percentage of people who scored better than Amanda is \(100\%-38.97\% = 61.03\%\)