The 2003 Statistical Abstract of the United States reported...

The 2003 Statistical Abstract of the United States reported the percentage of people 18 years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of $30 .$ $$ \begin{array}{l}{\text { a. How large a sample should be taken to estimate the proportion of smokers in the popu- }} \\ {\text { lation with a margin of error of } .02 ? \text { Use } 95 \% \text { confidence. }}\end{array} $$ $$ \begin{array}{l}{\text { b. Assume that the study uses your sample size recommendation in part (a) and }} \\ {\text { finds } 520 \text { smokers. What is the pointe of the proportion of smokers in the pop- }} \\ {\text { ulation? }} \\ {\text { c. What is the } 95 \% \text { confidence interval for the proportion of smokers in the population? }}\end{array} $$

The 2003 Statistical Abstract of the United States reported the percentage of people 18 years
of age and older who smoke. Suppose that a study designed to collect new data on smokers
and nonsmokers uses a preliminary estimate of the proportion who smoke of $30 .$
$$
\begin{array}{l}{\text { a. How large a sample should be taken to estimate the proportion of smokers in the popu- }} \\ {\text { lation with a margin of error of } .02 ? \text { Use } 95 \% \text { confidence. }}\end{array}
$$
$$
\begin{array}{l}{\text { b. Assume that the study uses your sample size recommendation in part (a) and }} \\ {\text { finds } 520 \text { smokers. What is the pointe of the proportion of smokers in the pop- }} \\ {\text { ulation? }} \\ {\text { c. What is the } 95 \% \text { confidence interval for the proportion of smokers in the population? }}\end{array}
$$

Key Concepts

Confidence Interval for Proportions

A confidence interval for a proportion is a range of values, derived from sample statistics, that is likely to contain the true population proportion with a specified level of confidence. It reflects the uncertainty inherent in estimating a proportion from a sample and is generally constructed using the normal approximation to the binomial distribution when sample sizes are sufficiently large.

Margin of Error

The margin of error represents the maximum expected difference between the true population proportion and the sample estimate within a specified confidence level. It quantifies the precision of an estimate and is influenced by both the sample size and the variability of the data. A smaller margin of error indicates a more precise estimate of the population parameter.

Sample Size Determination for Proportions

Sample size determination for estimating a population proportion involves calculating the minimum number of observations required so that the estimate has a desired margin of error at a given confidence level. This calculation takes into account the variability in the population, represented by the estimated proportion, and the critical value associated with the desired confidence level.

Point Estimate

A point estimate is a single numerical value that serves as an estimate of a population parameter, such as a proportion. It is derived from sample data and represents the best guess of the true parameter value without accounting for uncertainty, which is later addressed through confidence intervals.

Transcript

00:02 Once again, welcome to a new problem.

00:06 This time we're dealing with proportions.

00:09 And when you think about proportions, you could have population.

00:15 And the population would have a population proportion, population proportion.

00:22 And the population proportion would be something like the proportion of females in a college campus, the proportion of females in a college campus, and the population proportion is based on capital p, as opposed to the sample proportion, which is p hat.

00:57 So p hat is x over n.

01:00 This is the sample proportion where you take the number of subjects of interest divided by the sample size.

01:21 And for the most part, there's always a likelihood that you want to do an estimation of the population proportion based on the sample proportion.

01:32 And the estimation can have a margin of error and a confidence level.

01:44 So for example, if you're 95 % confident, then it means your error is 5%, or sometimes you call that alpha is 5%.

01:55 And from distribution point of view, what you're saying is that this middle part is 0 .95 in probability.

02:04 And then you split the tails, you get 0 .025 on the left tail and then 0 .025 in the right tail.

02:11 And so the formula for confidence interval is the same as p hat plus minus margin of error, where the margin of error is the same as z alpha over 2, radical p hat, 1 minus p hat, all over n.

02:33 Remember z alpha over 2 for 95 % confidence from the tables would be 1 .96.

02:41 So that's this z score, these two z scores.

02:45 This would be the negative z score, and then this would be the positive z score.

02:49 That's why we have plus and minus over there.

02:55 So coming back to the problem, coming back to the problem, we have a situation where we're thinking about the report.

03:09 So given the 2003 statistical abstract.

03:19 Of the us, we see that the percentage of smokers above 18, and what we're doing in this statistical abstract is to a design study, a designed study collecting data on smokers and non -smokers, on smokers and non -smokers and there's an estimate of the proportion who smoke of 30 there's an estimate of 30 so in part a determine the sample size to estimate the sample size to estimate the proportion of smokers in the population with 95 % confidence level and 0 .02 error or margin of error.

05:32 So that's part a and then part b using sample size requirements or suggestions if you want to call it that from part a the study finds 520 smokers what is the point estimate of proportion of smokers in the population in the population and then in part c, determine the 95 % confidence interval for smokers in the population.

07:05 95 % confidence interval of smokers in the population.

07:10 So we want to jump right into it.

07:12 We have the numbers to work with this problem.

07:16 So we'll start by saying, well, this is.

07:18 This was actually 30%.

07:20 So i just remember that.

07:22 We start by saying the p hat in part a, the p hat is 30%, which is the same as 0 .30.

07:30 The margin of error is 0 .02.

07:33 Remember if you have p hat plus minus margin of error, you're saying 0 .30 plus minus 0 .02.

07:41 And then your confidence level is 95%.

07:45 And that means that if you're looking at a distribution, this middle part is 0 .95.

07:51 And then we split the two sides.