# 7.3 A population proportion -- rrc math 1020  (Page 3/24)

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A random sample of 25 statistics students was asked: “Have you smoked a cigarette in the past week?” Six students reported smoking within the past week. Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke.

## Reminder

Remember that the plus-four method assume an additional four trials: two successes and two failures. You do not need to change the process for calculating the confidence interval; simply update the values of x and n to reflect these additional trials.

Using a calculator, we get a confidence interval of (0.113, 0.439).

## Try it

Out of a random sample of 65 freshmen at State University, 31 students have declared a major. Use the “plus-four” method to find a 96% confidence interval for the true proportion of freshmen at State University who have declared a major.

## Solution b

Using a calculator, we get a confidence interval of (0.355, 0.602).

The Berkman Center for Internet&Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of 50 teens, 13 reported having more than 500 friends on Facebook. Use the “plus four” method to find a 90% confidence interval for the true proportion of teens who would report having more than 500 Facebook friends.

## Solution b

Using a calculator, we get a confidence interval of (0.178, 0.378).

## Try it

The Berkman Center Study referenced in [link] talked to teens in smaller focus groups, but also interviewed additional teens over the phone. When the study was complete, 588 teens had answered the question about their Facebook friends with 159 saying that they have more than 500 friends. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens that would report having more than 500 Facebook friends based on this larger sample. Compare the results to those in [link] .

## Solution b

Using a calculator, we get a confidence interval of (0.242, 0.302).

Conclusion: The confidence interval for the larger sample is narrower than the interval from [link] . Larger samples will always yield more precise confidence intervals than smaller samples. The “plus four” method has a greater impact on the smaller sample. It shifts the point estimate from 0.26 (13/50) to 0.278 (15/54). It has a smaller impact on the EPB , changing it from 0.102 to 0.100. In the larger sample, the point estimate undergoes a smaller shift: from 0.270 (159/588) to 0.272 (161/592). It is easy to see that the plus-four method has the greatest impact on smaller samples.

## Calculating the sample size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a population proportion is

• $EBP=\left({z}_{\frac{\alpha }{2}}\right)\left(\sqrt{\frac{{p}^{\prime }{q}^{\prime }}{n}}\right)$
• Solving for n gives you an equation for the sample size.
• $n=\frac{{\left({z}_{\frac{\alpha }{2}}\right)}^{2}\left({p}^{\prime }{q}^{\prime }\right)}{EB{P}^{2}}$

Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones.

From the problem, we know that EBP = 0.03 (3%=0.03) and ${z}_{\frac{\alpha }{2}}$ z 0.05 = 1.645 because the confidence level is 90%.

However, in order to find n , we need to know the estimated (sample) proportion p ′. Remember that q ′ = 1 – p ′. But, we do not know p ′ yet. Since we multiply p ′ and q ′ together, we make them both equal to 0.5 because p q ′ = (0.5)(0.5) = 0.25 results in the largest possible product. (Try other products: (0.6)(0.4) = 0.24; (0.3)(0.7) = 0.21; (0.2)(0.8) = 0.16 and so on). The largest possible product gives us the largest n . This gives us a large enough sample so that we can be 90% confident that we are within three percentage points of the true population proportion. To calculate the sample size n , use the formula and make the substitutions.

$n=\frac{{z}^{2}{p}^{\prime }{q}^{\prime }}{EB{P}^{2}}$ gives $n=\frac{{1.645}^{2}\left(0.5\right)\left(0.5\right)}{{0.03}^{2}}=751.7$

Round the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.

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