Problem 3: Among private universities in the United States, the mean ratio of st

Question

Problem 3: Among private universities in the United States, the mean ratio of students to professors is 28.8 (i.e., 28.8 students for each professor) with a standard deviation of 12.1.

(i). What is the approximate probability that in a random sample of 36 private universities that the mean student-to-professor ratio is below 26?

(ii) What is the approximate probability that in a random sample of 36 private universities that the mean student-to-professor ratio is between 26 and 30?

(iii) What is the approximate probability that in a random sample of 36 private universities the sample mean of the student-to-professor ratio is below 30, given that the sample mean is larger than 25?

(iv) Suppose a random sample of 36 universities is selected and the observed sample mean student-to professor ratio is 26. Is there evidence that the reported mean ratio actually falls below 28.8, using alpha = 0.05.

Step 1: H0: _____________ Ha: _______________ Significance level = 0.05

Step 2: Verify necessary data conditions, and compute an appropriate test statistic

Compute p-value:

Step 3: Decision rule:

Step 4. Decision [Circle One]: Reject H0 Fail to reject H0

Step 5. Report the conclusion in the context of the problem:

(iiv) Suppose a random sample of 36 universities is selected and the observed sample mean student-to professor ratio is 26.

(a) Compute the 90% confidence interval for the sample mean.

(b) Does the 90% confidence interval include overall population mean

Explanation / Answer

1)std error of mean =std deviation/(n)1/2 where n=36

=2.0167

P(X<26) =P(Z<(26-28.8)/2.0167)=P(Z<-1.3884)=0.0825

2)P(26<X<30) =P(-1.3884<Z<0.5950)=0.7241-0.0825 =0.6416

3)P(X<30|X>25) =P(25<X<30)/P(X>25) =P(-1.884<Z<0.5950)/(1-P(Z<-1.884) =(0.7241-0.0298)/(1-0.0298)=0.7156

4)step 1 ) Ho: mean =28.8 ; Ha: mean <28.8

Step 2 : test stat z =(x-mean)/std error =-1.3884

for which p value =0.0825

step 3: decision rule : for reject null hypothesis p value<0.05

step 4: fail to reject Ho

step 5: we do not have sufficient evidence to accept that mean has fallen from 28.8

(iiv) for 90% CI, z=1.6449

a)hence confidence interval =sample mean +/- z*std error =22.6829 ; 29.3171

b) Yes it contains 28.8 hence  include overall population mean

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