Question

Suppose scores on a college entrance exam are normally distributed with a mean of 550 and a standard deviation of 100. Find the probability that a student taking the test will score less than 400. Round to four decimal places.

Answer 1

Probability that a student taking the test will score less than 400 is 0.0668.

Explanation:

We are given that scores on a college entrance exam are normally distributed with a mean of 550 and a standard deviation of 100.

Let, X = scores on a college entrance exam

X ~ N(
`\mu = 550, \sigma = 100^(2)`)

The z score probability distribution is given by;

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

where,
`\mu` = mean score

`\sigma` = standard deviation

So, probability that a student taking the test will score less than 400 is given by = P(X < 400)

P(X < 400) = P(
`(X-\mu)/(\sigma)` <
`(400-550)/(100)` ) = P(Z < -1.50) = 1 - P(Z
`\leq` 1.50)

= 1 - 0.9332 = 0.0668

Therefore, probability that a student taking the test will score less than 400 is 0.0668.

Answer 2

0.0668 = 6.68% probability that a student taking the test will score less than 400.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

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

`Z = (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 problem, we have that:

`\mu = 550, \sigma = 100`

Find the probability that a student taking the test will score less than 400.

This is the pvalue of Z when X = 400. So

`Z = (X - \mu)/(\sigma)`

`Z = (400 - 550)/(100)`

`Z = -1.5`

`Z = -1.5` has a pvalue of 0.0668

0.0668 = 6.68% probability that a student taking the test will score less than 400.