Question

In a certain elementary school, 52% of the students are girls. A sample of 65 students is drawn. Would it be unusual for more than 70% of them to be girls? Clearly explain your reason. (Note that your random variable is the sample proportion).

Answer 1

p = 52% = 0.52
SE = sqrt(p(1 - p)/n) = sqrt(0.52(1 - 0.52)/65) = sqrt((0.52 x 0.48) / 65) = sqrt(0.00384) = 0.0620

Let x be a random variable representing the percent of girls in the sample, then
P(x > 0.70) = 1 - P(x < 0.70) = 1 - P(z < (0.70 - 0.52) / 0.0620) = 1 - P(z < 2.905) = 1 - 0.99816 = 0.00184

The probability that there will be more than 70% of girls in the sample is 0.00184. Therefore, it will be unusual for more than 70% of them to be girls.