Question

3. Last year, Lipton Tea Company conducted a mall intercept study at six regional malls around the country and found that 20% of the public preferred tea over coffee as a midafternoon hot drink. This year, Lipton wants to have a nationwide telephone survey performed with random digit dialing. What sample size should be used in this year’s study to achieve an accuracy level of ±2.5% at the 99% level of confidence? What about at the 95% level of confidence? (4 points)

Answer 1

a)
`n=(0.2(1-0.2))/(((0.025)/(2.58))^2)=1704.04`

And rounded up we have that n=1705

b)
`n=(0.2(1-0.2))/(((0.025)/(1.96))^2)=983.45`

And rounded up we have that n=984

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".

Solution to the problem

Part a

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 99% of confidence, our significance level would be given by
`\alpha=1-0.99=0.01` and
`\alpha/2 =0.005`. And the critical value would be given by:

`z_(\alpha/2)=-2.58, z_(1-\alpha/2)=2.58`

The margin of error for the proportion interval is given by this formula:

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

And on this case we have that
`ME =\pm 0.025` and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=(\hat p (1-\hat p))/(((ME)/(z))^2)` (b)

And replacing into equation (b) the values from part a we got:

`n=(0.2(1-0.2))/(((0.025)/(2.58))^2)=1704.04`

And rounded up we have that n=1705

Part b

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:

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

`n=(0.2(1-0.2))/(((0.025)/(1.96))^2)=983.45`

And rounded up we have that n=984