(1 point) In a study of retention rates of those using the Platinum Program at Jenny Craig in May 2001-May 2002, it was found that about 25%25% of those who began the program dropped out in the first four weeks. If we have a random sample of 246 people at the beginning of the program, what is the probability that at least 195 people in the sample will still be in the Platinum Program after the first four weeks?

Answer:The required probability is calculated as 0.052Solution:As per the question:The probability that people dropped out in the first 4 weeks of the program, p = 0.25The size of the sample, n = 246Now, To calculate the Probability of at least 195 people being in the program after first 4 weeks:[tex]P(X\geq 195) = P(\frac{X - \mu}{\sigma})[/tex]               (1)where[tex]\mu[/tex] = mean[tex]\sigma[/tex] = standard deviationX = No. of people still part of the programNow,Mean can be given as:[tex]\mu = np = 0.25\times 246 = 61.5 = 62\ (approx)[/tex]The mean no. of people still part of the program = 246 - 62 = 184Standard deviation is given by:[tex]\sigma = \sqrt{npq} = \sqrt{np(1 - p)}[/tex]whereq = 1 - p = 1 - 0.25 = 0.75[tex]\sigma = \sqrt{246\times 0.25\times 0.75} = 6.79[/tex]Now, using the appropriate values in eqn (1):[tex]P(X\geq 195) = P(\frac{X - \mu}{\sigma}\geq \frac{195 - 184}{6.79})[/tex][tex]P(X\geq 195) = P(Z\geq 1.62)[/tex][tex]P(X\geq 195)[/tex] = 1 - P(Z < 1.62)Using Z-table:[tex]P(X\geq 195) = 1 - 0.94738 = 0.052[/tex]Thus the required probability is calculated as 0.052