A professor of Statistics feels that the performance level o

Solution

let u be the mean of the examinations scores of students.

the proffesor feels that the performance level of students has gone down recently. earlier it was 75.

so here the null hypothesis is H₀: u=75       vs H1:u<75

to test this we have a sample of size=n=32 and with sample mean=xbar=71

the standard deviation of the population is given as 8

then the test statistic is given as T=(xbar-75)/(8/sqrt(n)) which under H0 follows a N(0,1) distribution [assuming that the mean scores in examination follows a N(u,8²/n) distribution]

we reject H0 iff t<-tao_0.05 where t is the observed value of T and tao_0.05 is the upper alpha=0.05 point of a N(0,1) distribution. alpha is the level of significance.

now xbar=71 n=32

so t=-2.828 < -tao_0.05=-1.64[suing minitab]

hence at 5% level of significance H0 is rejected and it is concluded that the professor\'s concern is true. performance level of students has really gone down recently.