A telephone poll of 1,000 adult Americans was reported in an...

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was What is the main problem facing the country? Twenty percent answered crime. We are interested in the population proportion of adult Americans who feel that crime is the main problem. a. Define the random variables $X$ and $P^{\prime}$ in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 95$\%$ confidence interval for the population proportion of adult Americans who feel that crime is the main problem. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. d. Suppose we want to lower the sampling error. What is one way to accomplish that? e. The sampling error given by Yankelovich Parthers, Inc. (which conducted the poll) is $\pm 3 \% .$ In one to three complete sentences, explain what the $\pm 3 \%$ represents.

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was What is the main problem facing the country? Twenty percent answered crime. We are interested in the population proportion of adult Americans who feel that crime is the main problem.
a. Define the random variables $X$ and $P^{\prime}$ in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95$\%$ confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
d. Suppose we want to lower the sampling error. What is one way to accomplish that?
e. The sampling error given by Yankelovich Parthers, Inc. (which conducted the poll) is $\pm 3 \% .$ In one to three complete sentences, explain what the $\pm 3 \%$ represents.

Key Concepts

Random Variables

A random variable is a function that assigns numerical values to the outcomes of a random process. In the context of survey sampling, one random variable may count how many respondents choose a certain answer, while another (the sample proportion) divides that count by the total number of respondents to provide a proportion of successes.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials with the same probability of success, making it appropriate for dichotomous survey responses (e.g., whether a respondent identifies crime as the main problem). This distribution underlies the theory used to analyze proportions from sample surveys.

Confidence Interval

A confidence interval is a range of values computed from sample data, within which we expect the true population parameter to lie with a given level of confidence, such as 95%. It takes into account the variability in the sample and provides an estimated interval rather than a single point value.

Margin of Error

The margin of error, or error bound, quantifies the maximum expected difference between the sample estimate and the true population parameter. It is determined by factors such as sample size and variability, and it reflects the precision of the estimate when constructing a confidence interval.

Sampling Error

Sampling error refers to the natural discrepancy between a sample statistic and the corresponding population parameter due to the fact that the sample is only a subset of the entire population. It is typically expressed as a percentage and indicates the uncertainty or expected variation inherent in survey results.

Transcript

00:02 So you're doing the statistics for a project for time magazine, and they've asked you to sample some people in the united states, adults from the united states, to ask the question, what is the biggest thing facing us in society? so you take a telephone pole of 1 ,000 people.

00:18 That's your sample space.

00:20 And of the 1 ,000 people 200 have responded that to crime is the most important thing on their minds right now.

00:25 So that's the statistic you choose to study.

00:29 And you need to find the proportion of the entire population that this represents.

00:32 And that would be p hat, which is calculated by taking the number of people who answer favorably divided by the total number in the population sample, which in this case is 200 over 1 ,000, and that will give you 20%.

00:50 You also need to calculate the number of people who did not choose to answer quite favorably.

00:54 Q had q -pad or q -prime, which is 1 minus p -hat or p -prime, or 80%.

01:03 With these two numbers, and in fact that we did sample this in such a way that the number of the number, people in our sample represent the normal population.

01:11 We can use the normal distributions, standard error calculator, calculation from formula, which is a z score from the 95 % confidence interval times the square root of p prime or p hat times q hat 2 prime divided by 1 ,000.

01:33 And if you do the part in the parentheses first, you find that the z score 95 % times, 95 % confidence interval, we multiplied by 0 .0 .126.

01:46 If you look up the z score for 95 % confidence interval, you'll know, you'll find that in the z table that comes out as 1 .96.

01:55 If you multiply that by our 0 .0126, we end up with an error of 0 .025.

02:05 Now what do you do with this number? add it or subtract it to our population proportion...

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