Answer to Question #324417 in Statistics and Probability for Gaby

Question

6. According to a study from 2012, it was found that 19% of people use their browser in private mode. For a new study on privacy policies, you will consider a random sample of 200 people. Let X be the number of people who use their browser in private mode.

(a) What is the expected value (mean) and standard deviation for X?

(b) Use a normal approximation to estimate the probability that you will find 50 or more people using their browser in private mode in your sample.

(c) Use a normal approximation to estimate the probability that you will find more than 35 and up to 45 people (inclusively) using their browser in private mode in your sample.

Answer 1

$a)\;\mu =n p=200(0.19)=38,\\ \sigma= \sqrt{np(1-p)}= \sqrt{200(0.19)(0.81)}\approx5.55\\ Using\; normal\; approximation\;\\ b)\;P(x\geq 50)=P(x\geq 49.5)\\ =P(z\geq \frac{49.5-38}{5.55})\\ =P(z\geq 2.07)=0.5-P(0<z<2.07)\\ =0.5-0.4808=0.0192\\ c)\;P(35<x< 45)=P(34.5<x<45.5)\\ =P(\frac{34.5-38}{5.55}<z<\frac{45.5-38}{5.55})\\ =P(-0.63<z<1.35)\\ =P(0<z<0.63)-P(0<z<1.35)\\ =0.2357+0.4115=0.6472\\$