Question

According to Pew Research Center, 71% of Internet users accessed video-sharing sites last month. A random sample of 50 Internet users was selected. Using the normal approximation to the binomial distribution, what is the probability that fewer than 40 people from this sample access video-sharing sites

Answer 1

The probability that fewer than 40 people from this sample access video-sharing sites is P=0.92.

Explanation:

We have a populations proportion of internet users accessing video-sharing sites of p=0.71. We take a sample of n=50. This can be treated as a binomial random variable, but a normal distribution approximation is prefered, as the sample size is big to make all the calculations for the binomial distribution.

To approximate the binomial distribution to a normal distribution, we calculate the parameters as:

`\mu=np=50*0.71=35.5\\\\\\\sigma=√(np(1-p))=√(50*0.71*0.29)=3.2`

To calculate the probability that fewer that 40 people from this sample access video-sharing sites, we apply the continuity factor:

`z=(X-\mu)/\sigma=(40-35.5)/3.2=4.5/3.2=1.406\\\\P_B(X<40)=P_N(X<39.5)=P(z<1.406)=0.92`