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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.

User Funkju
by
4.2k points

2 Answers

1 vote

Answer:

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.

User Apparatix
by
4.0k points
4 votes

Answer:

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.

User RonnBlack
by
4.6k points