Answer to Question #167477 in Statistics and Probability for kibe

Question

The Dean the School of education in a University wants to determine whether there are statistically significant difference of opinion among the different cadres of academic staff members at the university concerning a proposed curriculum change in which postgraduate students are to be taught research methods on-line. Interviewing a sample of 313 members of the academic staff constituting 104 Lecturers, 131 Senior Lecturers, and 78 Professors, the Dean obtained the results shown in the following table:

 

Rank

 

Response

Lecturer

Senior Lecturer

Professor

Total

Against

47

34

14

95

Not committed

41

49

29

119

In support

16

48

35

99

Total

104

131

78

313

 

Compute a chi-square test for the above data and draw conclusion at α = .05                              

           

4) The mean I.Q of a sample of 1600 children was 99. It is likely that this was a random sample from a population with mean I.Q 100 and standard deviation 15.

 Required:

Using appropriate statistics, deduce whether there is any statistically significant difference between the two means if z critical value at α = .05 is 1.96                                                        

Answer 1

There is statistically significant difference of opinion among the different cadres of academic staff members.

$4)\ N=1600\\ \overline{x}=99\\ a_0=100\\ \sigma=15\text{ --- population standard deviation.}\\ \alpha=0.05\\ H_0:a=a_0=100\\ H_1:a\neq a_0=100\\ a\text{ --- population mean}.\\ \text{We will use the following random variable}:\\ U=\frac{(\overline{X}-a_0)\sqrt{n}}{\sigma}\\ \text{Observed value }u_{obs}=\frac{(99-100)\sqrt{1600}}{15}\approx -2.7\\ \text{Critical value } u_{cr}:\\ \Phi(u_{cr})=\frac{1-\alpha}{2}\\ \Phi(u_{cr})=0.475\\ u_{cr}\approx 1.96\\ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_0^xe^{-\frac{t^2}{2}}dt\text{ --- Laplace function}.\\ \text{Critical region}: (-\infty,-1.96)\cup (1.96,\infty)\\ u_{obs}\approx -2.7 \text{ is in the critical region. So we reject }H_0\\ \text{and accept }H_1.$