Question

Nielsen wants to estimate the percentage that are tuned t_Q the Tonight Show. Assume that they want 95% confidence that their sample percentage has a margin of error 2 percentage points A prior study found that 19% tune to the Tonight Show The number of households must Nielsen survey is

a. 1479,
b. 2555
c, 1042
d. 633
e. 3034

Answer 1

a. 1479

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\hat p` estimated proportion

n represent the sample size

Me =0.02 or 2% points represent the margin of error

Solution to the problem

The population proportion have the following distribution

`p \sim N(p,\sqrt{(\hat p(1-\hat p))/(n)})`

We need to find a critical value in order to estimate the sample size required. In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by
`\alpha=1-0.95=0.05` and
`\alpha/2 =0.025`. And the critical value would be given by:

`t_(\alpha/2)=-1.96, t_(1-\alpha/2)=1.96`

The confidence interval for the true proportion is given by the following formula:

`\hat p \pm z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}`

Wher the margin of error is given by:

`Me= z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}`

And we are interested in find n, solving for n we got:

`((Me)/(z_(\alpha/2)))^2=(\hat p (1-\hat p))/(n)`

`n=(\hat p (1-\hat p))/(((Me)/(z_(\alpha/2)))^2)`

And replacing the values that we have, we got:

`n=(0.19 (1-0.19))/(((0.02)/(1.96))^2)=1478.05`

And if we round up to th nearest integer we got that n=1479