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Chapter 3: Problem 65

What Proportion Believe in One True Love? In Data 2.1 on page 48 , we describe a study in which a random sample of 2625 US adults were asked whether they agree or disagree that there is "only one true love for each person." The study tells us that 735 of those polled said they agree with the statement. The standard error for this sample proportion is \(0.009 .\) Define the parameter being estimated, give the best estimate, the margin of error, and find and interpret a \(95 \%\) confidence interval.

Short Answer

Expert verified
The parameter being estimated is the proportion of all US adults who believe in 'only one true love for each person'. The best estimate of this parameter is 0.28 or 28%. The margin of error for this estimate is 0.018 or 1.8%. Thus, a 95% confidence interval for the proportion of all US adults who agree with the statement is from 26.2% to 29.8%.

Step by step solution

01

Defining the Parameter

The parameter being estimated in this problem is the proportion of all US adults who believe that 'there is only one true love for each person'.
02

Calculating the Best Estimate

The best estimate of the parameter is the sample proportion, \(p\). We can calculate this by dividing the number of people who agreed with the statement (735) by the total number of people surveyed (2625). So, \(p = 735 / 2625 = 0.28\). This suggests that 28% of the surveyed population agree with the statement.
03

Computing the Margin of Error

The margin of error can be calculated using the formula for a 95% confidence interval, which is 卤 1.96 times the standard error. Given that the standard error, \(SE\), is 0.009, our margin of error is \(1.96*SE = 1.96*0.009 = 0.018\). This gives us a margin of error of 0.018 or 1.8%.
04

Constructing and Interpreting a 95% Confidence Interval

A 95% confidence interval is constructed around the best estimate and extends the margin of error on either side. The lower limit of the confidence interval will be \(p - error = 0.28 - 0.018 = 0.262\) and the upper limit will be \(p + error = 0.28 + 0.018 = 0.298\). So, we are 95% confident that the true proportion of all US adults who agree with the statement falls between 26.2% and 29.8%.

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Key Concepts

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

Sample Proportion
The sample proportion is a fundamental concept in statistics, especially in surveys and studies like this one. It represents the fraction of the sample that has a particular trait or characteristic. In our example, we're looking at the proportion of US adults in the sample who agree that "there is only one true love for each person." To find the sample proportion:
  • Count the number of respondents who agree with the statement. In this case, it鈥檚 735 individuals.
  • Divide this number by the total number of respondents. Here, that鈥檚 2625.
So, the calculation is:\[ p = \frac{735}{2625} = 0.28 \]This means that 28% of the sample believes in the notion of one true love. This sample proportion serves as our best estimate of the population proportion鈥攖he characteristic we want to understand for the entire US adult populace.
Margin of Error
The margin of error is a statistical term that provides a range within which we can expect the true population parameter to lie. It is an indication of the precision of your estimate based on the sample data. The smaller the margin of error, the more precise your estimate is likely to be.Calculating the margin of error involves:
  • Identifying the standard error, which is given as 0.009 in this scenario. Standard error quantifies the amount of variation or "spread" in the sample estimate.
  • Using the critical value from the Z-distribution, typically 1.96 for a 95% confidence level.
The margin of error formula is:\[ 1.96 \times 0.009 = 0.018 \]This indicates a margin of error of 1.8% for the sample proportion.With this margin, your sample proportion of 28% can represent a population proportion between 26.2% and 29.8%. This means we're fairly confident that the true proportion lies within this window.
Standard Error
Standard error plays a crucial role when estimating how representative a sample statistic is of a population parameter. It reflects the degree of variability in the sample proportion due to random sampling.The standard error is derived from both the sample proportion and sample size. It鈥檚 calculated using the formula:\[ SE = \sqrt{\frac{p(1-p)}{n}} \]where:
  • \( p \) is the sample proportion (0.28 in our case)
  • \( n \) is the sample size (2625)
However, since we are given the standard error (0.009), this completes our equation parameter.This value is essential for constructing the confidence interval and acts as a building block for calculating the margin of error. A smaller standard error indicates a more precise estimate of the population parameter.

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Most popular questions from this chapter

Who Smokes More: Male Students or Female Students? Data 1.1 on page 4 includes lots of information on a sample of 362 college students. The complete dataset is available at StudentSurvey. We see that 27 of the 193 males in the sample smoke while 16 of the 169 females in the sample smoke. (a) What is the best point estimate for the difference in the proportion of smokers, using male proportion minus female proportion? Which gender smokes more in the sample? (b) Find and interpret a \(99 \%\) confidence interval for the difference in proportions.

Moose Drool Makes Grass More Appetizing Different species can interact in interesting ways. One type of grass produces the toxin ergovaline at levels about 1.0 part per million in order to keep grazing animals away. However, a recent study \(^{27}\) has found that the saliva from a moose counteracts these toxins and makes the grass more appetizing (for the moose). Scientists estimate that, after treatment with moose drool, mean level of the toxin ergovaline (in ppm) on the grass is \(0.183 .\) The standard error for this estimate is 0.016 . (a) Give notation for the quantity being estimated, and define any parameters used. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated. Interpret the interval in context.

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. \(\bar{x}_{1}-\bar{x}_{2}=3.0\) and the margin of error for \(95 \%\) confidence is 1.2

Predicting Election Results Throughout the US presidential election of \(2016,\) polls gave regular updates on the sample proportion supporting each candidate and the margin of error for the estimates. This attempt to predict the outcome of an election is a common use of polls. In each case below, the proportion of voters who intend to vote for each of two candidates is given as well as a margin of error for the estimates. Indicate whether we can be relatively confident that candidate A would win if the election were held at the time of the poll. (Assume the candidate who gets more than \(50 \%\) of the vote wins.) \(\begin{array}{lll}\text { (a) Candidate A: } 54 \% & \text { Candidate }\end{array}\) B: \(46 \%\) Margin of error: \(\pm 5 \%\) (b) Candidate A: \(52 \%\) Candidate B: \(48 \%\) Margin of error: \(\pm 1 \%\) \(\begin{array}{ll}\text { (c) Candidate A: } 53 \% & \text { Candidate }\end{array}\) B: \(47 \%\) Margin of error: \(\pm 2 \%\) \(\begin{array}{lll}\text { (d) Candidate A: } 58 \% & \text { Candidate }\end{array}\) B: \(42 \%\) Margin of error: \(\pm 10 \%\)

Correlation between age and heart rate for patients admitted to an Intensive Care Unit. Data from the 200 patients included in the file ICUAdmissions gives a correlation of 0.037 .
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