/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 According to a 2017 Pew Research... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 32

According to a 2017 Pew Research report, \(40 \%\) of millennials have a BA degree. Suppose we take a random sample of 500 millennials and find the proportion who have a BA degree. a. What value should we expect for our sample proportion? b. What is the standard error? c. Use your answers to parts a and \(\mathrm{b}\) to complete this sentence: We expect ____ \(\%\) to have a BA degree give or take ___ \(\%\). d. Suppose we decreased the sample size from 500 to 100 . What effect would this have on the standard error? Recalculate the standard error to see if your prediction was correct.

Short Answer

Expert verified
a. We should expect our sample proportion to be \(40\%\). b. The standard error is approximately \(2.2\%\). c. We expect \(40\%\) of millennials in our sample to have a BA degree give or take \(2.2\%\). d. Decreasing the sample size from 500 to 100 results in an increased standard error of about \(4.9\%\).

Step by step solution

01

Calculate Expected Value of Sample Proportion

As stated in the question, \(40\%\) (or 0.4) of millennials have a BA degree. So, we would expect our sample proportion to be 0.4 or 40\% since our sample is expected to represent the population.
02

Compute the Standard Error

Standard error for a sample proportion is given by the formula: \(\sqrt{p(1 - p) / n}\), where \(p\) is the population proportion and \(n\) is the sample size. Therefore, \(\sqrt{0.4(1 - 0.4) / 500} = \sqrt{0.24 / 500} = 0.0219\)It's common to round this value, so we can say the standard error is approximately 0.022 or 2.2\%.
03

Interpret the Results

This means we expect the proportion of millennials in our sample who have a BA degree to be \(40\%\) (from part 'a') give or take \(2.2\%\)(from part 'b').
04

Examine the Impact of a Changed Sample Size

If we decrease the sample size, we can expect the standard error to increase because the formula for standard error divides by the sample size \(n\). The smaller \(n\) becomes, the larger the resulting standard error, therefore increasing uncertainty. We can confirm by recalculating the standard error for a sample size of 100 instead of 500: \(\sqrt{0.4(1 - 0.4) / 100} = \sqrt{0.24 / 100} = 0.049\) or \(4.9\%\). As predicted, the standard error has increased.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Standard Error
The concept of the standard error is crucial in statistics as it helps measure how much variability exists in a sample statistic. This term often sounds technical, but it's easier to understand than you might think. The standard error tells us how much the sample proportion could differ from the true population proportion if we took different samples from the same population repeatedly. In mathematical terms, for a sample proportion, this is calculated using the formula:
\[SE = \sqrt{\frac{p(1 - p)}{n}}\]
Here:
  • \(p\) is the population proportion, which, in our problem, is 0.4.
  • \(n\) is the sample size.

So, for a sample size of 500, we find the standard error by substituting into the formula which results in a value of approximately 0.022 or 2.2%. This 2.2% represents the expected variability in the sample proportion due to random sampling.
Population Proportion
Understanding population proportion is key to not only solving our example problem but also to grasping broader statistical analyses. The population proportion is simply the percentage of a total population that shares a particular characteristic. It serves as a parameter that we aim to estimate through sample surveys or studies.
In this exercise, the population proportion was given as 40% or 0.4. This means that for every 100 millennials in the population, 40 are expected to have a BA degree. When we draw a representative sample, like in our problem, we anticipate that the sample will mirror the characteristics of the entire population. Therefore, the expected value of our sample proportion is also 0.4, which reflects the underlying population proportion. Insights about this parameter help us evaluate how accurate our sample results are and how closely they can predict population behaviors.
Sample Size Impact
The sample size plays a vital role in determining the precision of our sample estimates. Any change in sample size directly affects the standard error, which in turn impacts our confidence in the sample's ability to approximate the population.
When you decrease the sample size, as shown in the exercise from 500 to 100, the standard error increases. This increase signifies greater variability and less precision in the sample estimate. Calculating for a sample size of 100, the standard error becomes 0.049 or 4.9%, indicating a higher range of uncertainty.
In simpler terms, the larger the sample size, the more reliable and closer our estimate is to the true population proportion. Larger samples pull down the standard error, giving us tighter and more accurate confidence intervals. This is why statisticians always emphasize the importance of using a sufficiently large sample when aiming for precise statistical inferences.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

According to a 2017 survey conducted by Netflix, \(46 \%\) of couples have admitted to "cheating" on their significant other by streaming a TV show ahead of their partner. Suppose a random sample of 80 Netflix subscribers is selected. a. What percentage of the sample would we expect have "cheated" on their partner? b. Verify that the conditions for the Central Limit Theorem are met. c. What is the standard error for this sample proportion? d. Complete the sentence: We expect ____ \(\%\) of streaming couples to admit to Netflix "cheating," give or take _____ \(\% .\)

In a simple random sample of 1200 Americans age 20 and over, the proportion with diabetes was found to be \(0.115\) (or \(11.5 \%)\). a. What is the standard error for the estimate of the proportion of all Americans age 20 and over with diabetes? b. Find the margin of error, using a \(95 \%\) confidence level, for estimating this proportion. c. Report the \(95 \%\) confidence interval for the proportion of all Americans age 20 and over with diabetes. d. According to the Centers for Disease Control and Prevention, nationally, \(10.7 \%\) of all Americans age 20 or over have diabetes. Does the confidence interval you found in part c support or refute this claim? Explain.

In 2003 and 2017 Gallup asked Democratic voters about their views on the FBI. In \(2003,44 \%\) thought the \(\mathrm{FBI}\) did a good or excellent job. In \(2017,69 \%\) of Democratic voters felt this way. Assume these percentages are based on samples of 1200 Democratic voters. a. Can we conclude, on the basis of these two percentages alone, that the proportion of Democratic voters who think the FBI is doing a good or excellent job has increase from 2003 to \(2017 ?\) Why or why not? b. Check that the conditions for using a two-proportion confidence interval hold. You can assume that the sample is a random sample. c. Construct a \(95 \%\) confidence interval for the difference in the proportions of Democratic voters who believe the FBI is doing a good or excellent job, \(p_{1}-p_{2}\). Let \(p_{1}\) be the proportion of Democratic voters who felt this way in 2003 and \(p_{2}\) be the proportion of Democratic voters who felt this way in 2017 . d. Interpret the interval you constructed in part c. Has the proportion of Democratic voters who feel this way increased? Explain.

According to The Washington Post, \(72 \%\) of high school seniors have a driver's license. Suppose we take a random sample of 100 high school seniors and find the proportion who have a driver's license. a. What value should we expect for our sample proportion? b. What is the standard error? c. Use your answers to parts a and \(\mathrm{b}\) to complete this sentence: We expect _____\(\%\) to have their driver's license, give or take ____\(\%\). d. Suppose we increased the sample size from 100 to 500 . What effect would this have on the standard error? Recalculate the standard error to see if your prediction was correct.

The Perry Preschool Project was created in the early 1960 s by David Weikart in Ypsilanti, Michigan. In this project, 123 African American children were randomly assigned to one of two groups: One group enrolled in the Perry Preschool, and one group did not enroll. Follow-up studies were done for decades. One research question was whether attendance at preschool had an effect on high school graduation. The table shows whether the students graduated from regular high school or not and includes girls only (Schweinhart et al. 2005). $$\begin{array}{lcc}\hline & \text { Preschool } & \text { No Preschool } \\\\\hline \text { HS Grad } & 21 & 8 \\ \text { No HS Grad } & 4 & 17\end{array}$$ a. Find the percentages that graduated for both groups, and compare them descriptively. Does this suggest that preschool was associated with a higher graduation rate? b. Which of the conditions fail so that we cannot use a confidence interval for the difference between proportions?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.