Question

According to AccuData Media Research, 36% of televisions within the Chicago city limits are tuned to "Eyewitness News" at 5:00 pm on Sunday nights. At 5:00 pm on a given Sunday, 2500 such televisions are randomly selected and checked to determine what is being watched.

a. Find the mean and standard deviation for the number of televisions tuned to "Eyewitness News."
b. Using the range rule of thumb, would it be unusual to find that 840 of the 2500 televisions are tuned to "Eyewitness News"?
c. Using the range rule of thumb, would it be unusual to find that 945 of the 2500 televisions are tuned to "Eyewitness News"?

Answer 1

a) Mean: 900

Standard deviation: 24

b) Very unusual

c) Unusual

Explanation:

We have a population proportion p=0.36 and we are taking a sample of size n=2500. This can be modeled as binomial sampling.

For this sampling distribution, we have a mean and STD that can be calculated as:

`\mu_s=n\cdot p=2500\cdot0.36=900\\\\\sigma_s=√(n\cdot p(1-p))=√(2500*0.36*0.64))=√(576)=24`

b) A value of 840 is a very unusual as is more than 2 standard deviations from the expected value of 900 (more exactly, at 2.5 standard deviations). Approximately 2% of the values are below 2 standars deviations from the mean.

Having 840 or less televisions tuned to "Eyewitness News" would have a probability of P=0.00621.

c) A value of 945 would be also unusual, but not as unusual as 840, as is between 1 and 2 standard deviation from the expected value.

Having 945 or more televisions tuned to "Eyewitness News" would have a probability of P=0.0304.