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. the percentage that scored 600 or more is 34.5%. (round to one decimal place as needed.) b. what percentage of the math test takers from this state scored between 600 and 652?

Explanation:

Step1: Calculate the z - score for \(x = 600\)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 558\) and \(\sigma=104\).
For \(x = 600\), \(z_1=\frac{600 - 558}{104}=\frac{42}{104}\approx0.40\)

Step2: Calculate the z - score for \(x = 652\)

Using the same formula \(z=\frac{x-\mu}{\sigma}\)
For \(x = 652\), \(z_2=\frac{652-558}{104}=\frac{94}{104}\approx0.90\)

Step3: Find the probabilities

We use the standard normal distribution table (or a calculator with a normal - distribution function).
The probability \(P(Z\lt0.90)\) is approximately \(0.8159\) and \(P(Z\lt0.40)\) is approximately \(0.6554\)

Step4: Calculate the probability between the two z - scores

\(P(0.40\lt Z\lt0.90)=P(Z\lt0.90)-P(Z\lt0.40)\)
\(P(0.40\lt Z\lt0.90)=0.8159 - 0.6554=0.1605\)

Answer:

The percentage of the math test - takers from this state who scored between 600 and 652 is \(16.1\%\) (rounded to one decimal place).

For the Normal curve:

• The x - axis is labeled with the test scores (with mean \(\mu = 558\)).
• The curve is symmetric about \(x = 558\).
• The area between \(x = 600\) and \(x = 652\) is shaded.