Question

according to a recent reporting on a standardized test, the average math score for students in a particular state was 558. assume the scores are normally distributed with a standard deviation of 104. answer parts (a) through (c) below including an appropriately labeled and shaded normal curve for each part.
a. what percentage of the math test takers from this state scored 600 or more?
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 600\), \(\mu=558\), and \(\sigma = 104\).

$$z=\frac{600 - 558}{104}=\frac{42}{104}\approx0.40$$

Step2: Find the probability using the standard normal table

We want to find \(P(X\geq600)\), which is equivalent to \(P(Z\geq0.40)\) since \(P(X\geq x)=P(Z\geq\frac{x - \mu}{\sigma})\).
Using the property \(P(Z\geq z)=1 - P(Z\lt z)\)
From the standard normal table, \(P(Z\lt0.40)=0.6554\)
So \(P(Z\geq0.40)=1 - 0.6554 = 0.3446\)

For the normal curve:
The mean \(\mu = 558\) is in the center. The value \(x = 600\) is to the right of the mean. We shade the area to the right of \(x = 600\) (or \(z=0.40\)) under the normal curve.

Answer:

Approximately \(34.46\%\) of the math test - takers scored 600 or more.