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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.

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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.

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Most popular questions from this chapter

According to a 2017 Pew Research survey, \(60 \%\) of young Americans aged 18 to 29 say the primary way they watch television is through streaming services on the Internet. Suppose a random sample of 200 Americans from this age group is selected. a. What percentage of the sample would we expect to watch television primarily through streaming services? b. Verify that the conditions for the Central Limit Theorem are met. c. Would it be surprising to find that 125 people in the sample watched television primarily through streaming services? Why or why not? d. Would it be surprising to find that more than \(74 \%\) of the sample watched television primarily through streaming services? Why or why not?

Ondansetron (Zofran) is a drug used by some pregnant women for nausea. There was some concern that it might cause trouble with pregnancies. An observational study was done of women in Denmark (Pasternak et al. 2013). An analysis of 1849 exposed women and 7376 unexposed women showed that \(2.1 \%\) of the women exposed to ondansetron and \(5.8 \%\) of the unexposed women had miscarriages during weeks 7 to 22 of their pregnancies. a. Was the rate of miscarriages higher with Zofron, as feared? b. Which of the conditions for finding the confidence interval do not hold?

You need to select a simple random sample of four from eight friends who will participate in a survey. Assume the friends are numbered \(1,2,3,4,5\), 6,7, and 8 . Select four friends, using the two lines of numbers in the next column from a random number table. Read off each digit, skipping any digit not assigned to one of the friends. The sampling is without replacement, meaning that you cannot select the same person twice. Write down the numbers chosen. The first person is number 7 .$$\begin{array}{lll}07033 & 75250 & 34546 \\\75298 & 33893 & 64487\end{array}$$ Which four friends are chosen?

Pew Research reported that \(46 \%\) of Americans surveyed in 2016 got their news from local television. A similar survey conducted in 2017 found that \(37 \%\) of Americans got their news from local television. Assume the sample size for each poll was 1200 . a. Construct the \(95 \%\) confidence interval for the difference in the proportions of Americans who get their news from local television in 2016 and 2017 . b. Based on your interval, do you think there has been a change in the proportion of Americans who get their news from local television? 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.
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