A nationwide census is conducted and it is found that the mean number of hours of television watched per year by Americans is 350 with a standard deviation of 220. Furthermore, the data appears to be normally distributed! Using this information, determine the probability that a group of 4 Americans watch MORE THAN 400 hours of television per year. SHOW ALL WORK.

We are given that a nationwide census is conducted and it is found that the mean number of hours of television watched per year by Americans is 350 with a standard deviation of 220.

A group of 4 Americans is selected.

Let [tex]\bar X[/tex] = sample mean number of hours of television watched per year

The z score probability distribution for sample mean is given by;

Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)

where, [tex]\mu[/tex] = population mean = 350

[tex]\sigma[/tex] = standard deviation = 220

n = sample of Americans = 4

Now, the probability that a group of 4 Americans watch more than 400 hours of television per year is given by = P([tex]\bar X[/tex] > 400 hours)

P([tex]\bar X[/tex] > 400) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] > [tex]\frac{400-350}{\frac{220}{\sqrt{4} } }[/tex] ) = P(Z > 0.45) = 1 - P(Z [tex]\leq[/tex] 0.45)

= 1 - 0.6736 = 0.3264

The above probability is calculated by looking at the value of x = 0.45 in the z table which has an area of 0.6736.

Hence, the probability that a group of 4 Americans watch more than 400 hours of television per year is 0.3264.