Question

Q7.The grades in a statistics course for a particular semester were as follows

Grade A B C D F
f 14 18 32 20 16

significance, that the distribution of grades is uniform.

Answer 1

$H_0:$ The distribution of grades is uniform.

$H_1:$ The distribution of grades is not uniform.

Significance level: $\alpha=0.05$

n = number of observations = 14+18+32+20+16 = 100

k = number of intervals = 5

Test statistics:

$$\chi^2=\sum{(f_0-f_e)^2\over f_e} \ with \ v=5-1=4, {n \over k}={100 \over 5}=20$$

Critical Regions

The critical region is given by

$$\chi^2>\chi_{\alpha, v}^2,\ \chi^2>\chi_{0.05, 4}^2, \ \chi^2>9.488$$

Computation

$$\chi^2=\sum{(f_0-f_e)^2\over f_e} ={(14-20)^2\over 20}+{(18-20)^2\over 20}+{(32-20)^2\over 20}+$$

$$+{(20-20)^2\over 20}+{(16-20)^2\over 20}=10$$

Statistical decision

Since $10>9.488,$ then we reject $H_0.$

Conclusion

The distribution of grades is not uniform.