Question

The mean score for a standardized test is 1700 points. The results are normally distributed with a standard deviation of 75 points. What is the probability that a student will score more than 1700 points?

Answer 1

Probability that a student will score more than 1700 points is 0.50.

Explanation:

We are given that the mean score for a standardized test is 1700 points. The results are normally distributed with a standard deviation of 75 points.

Let X = Scores results on a test

So, X ~ N(
`\mu=1700,\sigma^(2) =75^(2)`)

The z-score probability distribution for normal distribution is given by;

Z =
`( X -\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean score = 1700 points

`\sigma` = standard deviation = 75 points

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

So, the probability that a student will score more than 1700 points is given by = P(X > 1700 points)

P(X > 1700) = P(
`( X -\mu)/(\sigma)` >
`(1700-1700)/(75)` ) = P(Z > 0) = 1 - P(Z
`\leq` 0)

= 1 - 0.50 = 0.50

Now, in the z table the P(Z
`\leq` x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 0 in the z table which has an area of 0.50.

Hence, the probability that a student will score more than 1700 points is 50%.