Grade point averages of math majors at a large distance educ

Grade point averages of math majors at a large distance education university are normally distributed with a mean of 2.85 and a standard deviation of 0.30 . If a random sample of 25 math majors is selected from that university, what is the probability that the sample mean grade point average will be

a. either less than 2.709 or more than 2.955?

b. at least 2.757?

Solution

t(2.709) = (2.709-2.85)/[0.3/sqrt(25)] = -2.35
P(x < 2.709) = P(t < -2.35) = tcdf(-100,-2.35,24) = 0.0137
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t(2.9557) = (2.9557-2.85)/[0.3/sqrt(25)] = 1.7617
P(x > 2.9557) = P(z > 1.7617) = tcdf(1.7617,100,24) = 0.0454

Thus the total probability is .0137+.0454

                                  =.0591

b)P(X>=2.757)=1-P(X,=2.757) ,zscore =(2.757-2.85)/(.3/sqrt25)=-.093/.06 =-1.55

from the z table =.0668

Thus probability of at least 2.757=1-.0668 =.9332