91Ó°ÊÓ

In a survey conducted by the marketing agency 11 mark, 241 of 1000 adults 19 years of age or older confessed to bringing and using their cell phone every trip to the bathroom (confessions included texting and answering phone calls). (a) What is the sample in this study? What is the population of interest? (b) What is the variable of interest in this study? Is it qualitative or quantitative? (c) Based on the results of this survey, obtain a point estimate for the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (d) Explain why the point estimate found in part (c) is a statistic. Explain why it is a random variable. What is the source of variability in the random variable? (e) Construct and interpret a \(95 \%\) confidence interval for the population proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (f) What ensures that the results of this study are representative of all adults 19 years of age or older?

Step by step solution

01

Identifying the Sample and Population

The sample in this study consists of the 1000 adults aged 19 or older who participated in the survey. The population of interest is all adults aged 19 or older.

02

Determining the Variable of Interest

The variable of interest is whether an adult brings and uses their cell phone during every trip to the bathroom. This variable is qualitative because it categorizes individuals based on behavior rather than numerical measurements.

03

Point Estimate Calculation

To find the point estimate for the population proportion of adults who bring their cell phone to the bathroom, use the formula for sample proportion: \( \hat{p} = \frac{x}{n} \). Here, \(x = 241\ and \ n = 1000\): \( \hat{p} = \frac{241}{1000} = 0.241 \). Hence, the point estimate is 0.241.

04

Explanation of Point Estimate as Statistic and Random Variable

The point estimate \( \hat{p} = 0.241 \) is a statistic because it's calculated from sample data. It's a random variable because different samples could yield different sample proportions. The variability arises from sampling variation.

05

Constructing and Interpreting the Confidence Interval

First, determine the standard error for the proportion: \[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } = \sqrt{ \frac{0.241(1-0.241)}{1000} } \.\] This yields \ SE \approx 0.0138. Next, use the Z-value for a 95% confidence interval, which is 1.96: \[ CI = \hat{p} \pm Z \cdot SE = 0.241 \pm 1.96 \cdot 0.0138.\] Calculating gives \ CI = (0.214, 0.268).\ Therefore, the 95% confidence interval is (0.214, 0.268). This means we are 95% confident that the true proportion of adults who bring their cell phones to the bathroom is between 21.4% and 26.8%.

06

Ensuring Representativeness

The results are representative of all adults aged 19 or older if the sample was randomly selected. Random sampling helps ensure that every individual in the population had an equal chance of being included in the survey, which minimizes bias.

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Key Concepts

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

Sample and Population

In any survey, understanding the distinctions between the sample and the population is crucial. The sample refers to the subset of individuals from the population who are actually surveyed. In this exercise, the sample consists of the 1000 adults aged 19 or older who participated in the survey. The population, on the other hand, includes all adults aged 19 years or older, encompassing a much broader group than just the surveyed sample. By studying the sample, we aim to make inferences about the entire population.
Knowing this difference helps us determine how representative our findings are relative to the whole group we are interested in.

Qualitative and Quantitative Variables

Variables in a survey can be either qualitative or quantitative. Qualitative variables categorize or describe attributes that do not have numerical values. In our exercise, the variable of interest is whether an adult brings and uses their cell phone on every trip to the bathroom. This is a qualitative variable because it classifies individuals based on behavior rather than measuring something numerically like age or income.
Understanding the type of variable is important as it dictates the type of analysis we can perform. Qualitative variables are usually summarized using proportions or percentages, while quantitative variables are often summarized using means or standard deviations.

Point Estimate

A point estimate provides a single value estimate for a population parameter based on sample data. To find the point estimate for the population proportion of adults who bring their cell phone to the bathroom, we use the formula for the sample proportion: \[ \hat{p} = \frac{x}{n} \]. Here, \x = 241\ and \ = 1000\, giving \hat{p} = 0.241\. Therefore, the point estimate is 0.241, meaning about 24.1% of the surveyed sample reported using their cell phone on every trip to the bathroom. This estimate helps us make an educated guess about the behavior across the broader population.

Confidence Interval

A confidence interval provides a range of values within which we expect the true population parameter to fall. For our survey, to construct a 95% confidence interval for the population proportion, we first determine the standard error: \[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]. Using \hat{p} = 0.241\ and \ = 1000\, we get \SE \approx 0.0138\. Next, using the Z-value for a 95% confidence interval (1.96): \[ CI = \hat{p} \pm Z \cdot SE\]. Calculations yield \approx 0.241 \pm 1.96 \cdot 0.0138\, which equals approximately (0.214, 0.268). This means we're 95% confident that the true proportion of adults who bring their cell phones to the bathroom falls between 21.4% and 26.8%.

Random Sampling

Random sampling is a technique where each individual in the population has an equal chance of being included in the sample. It ensures that the sample is representative of the entire population, reducing the risk of bias. In the context of this survey, if the sample of 1000 adults was randomly selected, it enhances the reliability of our findings and increases the confidence that the sample proportion reflects the behavior of the broader population of adults aged 19 years or older.
Utilizing random sampling is fundamental in survey analysis because it supports the validity and generalizability of the results.