Question

In: Statistics and Probability

1. The SAT Math scores for a recent year are distributed according to the normal distribution...

1. The SAT Math scores for a recent year are distributed according to the normal distribution with mean equal to 506 and standard deviation of 115

A. Determine the range of typical test scores, we consider 2 standard deviation as typical scores.

B. Determine the probability that a student scored higher than 700

C. Determine the probability that a student scored between 450 and 700

D. Determine the probability that a random sample 30 students, that their mean test score is between 450 and 700.

Solutions

Expert Solution

Given that the SAT Math scores for a recent year are distributed according to the normal distribution with mean equal to = 506 and the standard deviation of = 115.

Thus to calculate the probability we use Z statistic.

A) for the range of typical test scores, that we consider 2 standard deviations within the mean is calculated using the Z score formula that is:

and

B) The probability that a student scored higher than 700 is Pr(X>700) is calculated by finding the Z score at X = 700 as:

Thus Pr(X>700) = P(Z>1.687) is computed using excel formula for normal distribution which is =1-NORM.S.DIST(1.687, TRUE), thus P(X>700) is

=1−0.9542=0.0458

C) The probability that a student scored between 450 and 700 is calculated by finding the Z scores as:

So, P(45<X<700) = P(-0.487<Z<1.687) is computed using excel formula =NORM.S.DIST(1.687, TRUE)- NORM.S.DIST(-0.487, TRUE) , now the probability is calculated as:

D) The probability that a random sample n = 30 students, that their mean test score is between = 450 and = 700 is also calculated by finding the Z scores as:

the probability is calculated using the excel formula that is =NORM.S.DIST(9.24, TRUE)- NORM.S.DIST(-2.67, TRUE), now the probability is calculated as:

=1−0.0038=0.9962

orchestra answered 2 years ago
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