Question

A researcher wants to investigate the effects of environmental factors on IQ scores. For an initial study, she takes a sample of 400 people who grew up as the only child. She finds that 49.5% of them have an IQ score over 100. It is known that 50% of the general population has an IQ score exceeding 100. a. Find the mean of p, where p is the proportion of people with IQ scores over 100 in a random sample of 400 people. b. Find the standard deviation of p. C. Compute an approximation for P(p>0.495), which is the probability that there will be 49.5% or more individuals with IQ scores over 100 in a random sample of 400. Round your answer to four decimal places. xn mi so SOO

Answer 1

n = 400, p = 0.5 (a) mean r_p = 0.5 (b) variance = sigma^2 = p(1 - p)/n = 0.5 (1 - 0.5)/400 = 0.00625 standard deviation (sigma) = squareroot 0.000625 = 0.025 therefore p tilde N(0.5, 0.025^2) (c) p(x > 0.495) = ? Normal distribution mu = 0.5, sigma = 0.025 p(x > 0.495 - 0.5) = Area of right of 0.495 z = x - mu/sigma = 0.495 - 0.5/0.025 = -0.20 p(x > 0.495) = P(z > - 0.20) = 1 - P(z < -0.20) = 1 - 0.4207 = 0.5793 = 57.93%