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)for std error =std deviation/(n)1/2 =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(-1.8843<Z<0.5950)/P(Z>-1.8843)=(0.7241-0.0298)/0.9702 =0.7156

4)Ho : mean =28.8

Ha:mean <28.8

here test stat z=(X-mean)/std error =(26-28.8)/2.0167 =-1.3884

p value =0.0825

3) decision rule : reject Ho if p value is less then 0.05

4) do not reject Ho

(for 90% CI, z=1.6449

hence confidence interval =mean +/- z*std deviation =22.6829 ; 29.3171

as above CI does contain 28.8 as probable value; we can not reject that 28.8 is mean

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