Answer to Question #233198 in Statistics and Probability for Sos

Question

A recent study claims that girls and boys do not perform the same on math tests taken from a grade 12. Suppose that in a representative sample, 344 of 430 girls and 369 of 450 boys score at proficient or advanced levels on a standardized math test. The principal wants to examine whether the proportion of girls who score at proficient or advanced levels differs from the proportion of boys at 5% level of significance. Which one of the following statements is incorrect?

Answer 1

Let 1 denotes girls and 2 denotes boys.

$N_1=430 \\ x_1=344 \\ \hat{p_1}= \frac{x_1}{N_1}= \frac{344}{430}= 0.8 \\ N_2=450 \\ x_2= 369 \\ \hat{p_2}= \frac{369}{450}=0.82$

The value of the pooled proportion

$\bar{p}= \frac{x_1+x_2}{N_1+N_2}=\frac{344+369}{430+450} = 0.8102 \\ α=0.05$

i)

$H_0: p_1-p_2=0 \\ H_1: p_1-p_2≠0$

ii) Rejection region:

Using Z-table for α=0.05 we get $Z_c=1.96$ . Since the test is two-tailed, the rejection region is |Z|>1.96

iii) Test-statistic:

$Z = \frac{0.8-0.82}{\sqrt{ 0.8102(1-0.8102)(1/403 -1/450) }} \\ = \frac{-0.02}{\sqrt{0.0006993429}} \\ = -0.7563$

iv) Decision rule:

$|Z|=0.7563 < Z_c=1.96$

We accept H₀.

v) Conclusion: There is no difference in the proportions of boys and girls who score at proficient or advanced levels at 0.05 level of significance.

We find that all options are correct except Option 3. because test-statistic is -0.7563.

Answer: Option 3.