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Chapter 22: Problem 15

Cannabidiol- (CBD-) based products are widely touted for their therapeutic benefits without any psychoactive effects. A 2019 Gallup Poll asked a random sample of 2543 U.S. adults if they personally use CBD products or not. Suppose that, in fact, \(15 \%\) of all U.S. adults have used CBD products. In repeated samples, the sample proportion \(\widehat{p}\) would follow approximately a Normal distribution with mean a. \(381.45\). b. \(0.15\). c. \(0.007\).

Short Answer

Expert verified
The mean of the sample proportion \(\widehat{p}\) is 0.15.

Step by step solution

01

Identify Given Information

We are provided with a sample size of 2543 U.S. adults, and it's known that 15% of all U.S. adults have used CBD products. Therefore, the population proportion \( p \) is 0.15.
02

Understand the Distribution of Sample Proportion

The sample proportion \( \widehat{p} \) follows a Normal distribution with mean \( \mu_{\widehat{p}} \) equal to the population proportion \( p \).
03

Determine the Mean of the Sample Distribution

Using the information that the mean of the sample proportion \( \mu_{\widehat{p}} \) is equal to the population proportion \( p = 0.15 \), we identify the mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion
The concept of sample proportion is fundamental in statistics when you want to make inferences about a larger population. Imagine you're conducting a survey, like the Gallup Poll on CBD usage among U.S. adults. - The sample proportion, denoted as \( \widehat{p} \), represents the fraction of the sample that meets a particular characteristic or condition. - In the context of the exercise, if 250 out of 2543 adults in the sample reported using CBD products, then \( \widehat{p} \) would be calculated as \( 250/2543 \).Understanding sample proportions helps in estimating the overall population proportion. It's important because it forms the basis for making predictions or decisions about the entire population based on a smaller sample. The closer the sample proportion is to the true population proportion, the more confidence we have in our estimates. Linear properties and simple mathematical operations make working with proportions intuitive, which is particularly helpful for predicting outcomes based on sample data.
Normal Distribution
Normal distribution is a critical concept when dealing with sample proportions. It's a type of continuous probability distribution that's symmetrical and describes how the values of a variable are distributed.- In the context of sample statistics, the sample proportion \( \widehat{p} \) is expected to be approximately Normally distributed, especially when the sample size is large, due to the Central Limit Theorem (CLT). - This theorem states that regardless of the shape of the population distribution, the distribution of the sample means (or proportions) will be Normal if the sample size is sufficiently large.The Normal distribution of the sample proportion allows us to make probabilistic statements and apply inferential statistical techniques. We use the mean and standard deviation of the distribution of the sample proportion to calculate z-scores or confidence intervals, guiding us in making educated guesses about the population proportion.
Population Proportion
The population proportion represents the true fraction of a population that exhibits a particular characteristic, such as the percentage of U.S. adults using CBD products. - Denoted by \( p \), the population proportion in a given context (as in the exercise) is known or assumed to be 0.15.- Knowing the population proportion allows us to evaluate how effective our sample is. In practice, the true population proportion is often unknown.From a practical standpoint, population proportions play a crucial role in determining sample sizes. Before conducting a survey, statisticians often use a hypothesized population proportion to decide how many responses are needed to achieve a desired confidence level and margin of error. When working with proportions, it's essential to differentiate between sample and population proportions to accurately interpret results and measure the reliability of the sample data.

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Most popular questions from this chapter

Black Raspberries and Cancer. Sample surveys usually contact large samples, so we can use the large-sample confidence interval if the sample design is close to an SRS. Scientific studies often use smaller samples that require the plus four method. For example, Familial Adenomatous Polyposis (FAP) is a rare inherited disease characterized by the development of an extreme number of polyps early in life and colon cancer in virtually \(100 \%\) of patients before the age of 40 . A group of 14 people suffering from FAP being treated at the Cleveland Clinic drank black raspberry powder in a slurry of water every day for nine months. The numbers of polyps were reduced in 11 out of 14 of these patients. 19 a. Why can't we use the large-sample confidence interval for the proportion \(p\) of patients suffering from FAP that will have the number of polyps reduced after nine months of treatment? b. The plus four method adds four observations: two successes and two failures. What are the sample size and the number of successes after you do this? What is the plus four estimate \(\tilde{p}\) of \(p\) ? c. Give the plus four \(90 \%\) confidence interval for the proportion of patients suffering from FAP who will have the number of polyps reduced after nine months of treatment.

Do Smokers Know That Smoking is Bad for Them? The Harris Poll asked a sample of smokers, "Do you believe that smoking will probably shorten your life, or not?" Of the 1010 people in the sample, 848 said "yes." a. Harris called residential telephone numbers at random in an attempt to contact an SRS of smokers. Based on what you know about national sample surveys, what is likely to be the biggest weakness in the survey? b. We will nonetheless act as if the people interviewed are an SRS of smokers. Give a \(95 \%\) confidence interval for the percent of smokers who agree that smoking will probably shorten their lives.

Harris Announces a Margin of Error. Exercise \(22.25\) describes a Harris Poll survey of smokers in which 848 of a sample of 1010 smokers agreed that smoking would probably shorten their lives. Harris announces a margin of error of \(\pm 3\) percentage points for all samples of about this size. Opinion polls announce the margin of error for \(95 \%\) confidence. a. What is the actual margin of error (in percent) for the large-sample confidence interval from this sample? b. The margin of error is largest when \(\hat{p}=0.5\). What would the margin of error (in percent) be if the sample had resulted in \(\widehat{p}=0.5\) ? c. Why do you think that Harris announces a \(\pm 3 \%\) margin of error for all samples of about this size?

Usage of the Olympic National Park. U.S. National Parks that contain designated wilderness areas are required by law to develop and maintain a wilderness stewardship plan. The Olympic National Park, containing some of the most biologically diverse wilderness in the United States, had a survey conducted in 2012 to collect information relevant to the development of such a plan. National Park Service staff visited 30 wilderness trailheads in moderate-to high-use areas over a 60-day period and asked visitors as they completed their hike to complete a questionnaire. The 1019 completed questionaires, giving a response rate of \(50.4 \%\), provided each subject's opinions on the use and management of wilderness. In particular, there were 694 day users and 325 overnight users in the sample. a. Why do you think the National Park staff only visited trailheads in moderate-to high-use areas to obtain the sample? b. Assuming that the 1019 subjects represent a random sample of users of the wilderness areas in Olympic National Park, give a \(90 \%\) confidence interval for the proportion of day users. c. The response rate was \(49 \%\) for day users and \(52 \%\) for overnight users. Does this lessen any concerns you might have regarding the effect of nonresponse on the interval you obtained in part (b)? Explain briefly. d. Do you think it would be better to refer to the interval in part (b) as a confidence interval for the proportion of day users or the proportion of day users on the most popular trails in the park? Explain briefly.

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. Only half of all mice complete one particular maze in less than 18 seconds. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures the proportion of 40 mice that completed the maze in less than 18 seconds with noise as a stimulus. The proportion of mice that completed the maze in less than 18 seconds is \(\widehat{p}=0.7\). The hypotheses for a test to answer the researcher's question are a. \(H_{0}: p=0.5, H_{a}: p>0.5\). b. \(H_{0}: p=0.5, H_{a}: p<0.5\). c. \(H_{0}: p=0.5, H_{a}: p \neq 0.5\).
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