Question-30
There are a \(40\%\) proportion of girls in a school. The principal of the school suspects that girls are more likely to score a GPA more than 8. In a sample of 100 students who scored more than 8 GPA, 49 were found to be girls. Is the data enough, at a significance level of 0.05, to conclude that girls are more likely to score more than 8 GPA \(?\)
Use \(F_Z^{-1}(0.95) = 1.64\)
Null and alternative hypothesis will be:
\(H_0 : p = 0.4\) and \(H_1 : p > 0.4\)
Define \(T\) as ``number of girls who scored more than 8 GPA”. Thus, when null hypothesis is true, \(T \sim \text{Binomial}\left(100, 0.4\right)\)
Therefore, \(E[T] = 100 \times 0.4 = 40\) and \(\text{Var}(T) = 100 \times 0.4 \times 0.6 = 24\)
Using CLT, we can write \(\dfrac{T-40}{24} \sim N(0,1)\)
Test will be : Reject \(H_0\) if \(T > c\)
Use \(\alpha = P\left(\text{Reject}~~ H_0 ~|~ H_0 \right)\).
Find the value of \(c\) by solving the above part and then make conclusion.