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Chapter 7: Problem 31

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.

Short Answer

Expert verified
a) The expected sample proportion is 72%. b) The standard error is 4.56%. c) We expect 72% of the students to have a driver鈥檚 license, give or take 4.56%. d) Increasing the sample size to 500 reduces the standard error to 2.04%

Step by step solution

01

Calculate Expected Value

The population proportion \(p\) is given as 72% or 0.72. For a large sample, the expected proportion should be close to this value. Therefore, the expected value for the sample proportion (also known as the mean of the sampling distribution) is \(p = 0.72\).
02

Calculate the Standard Error

The standard error (SE) for a proportion is calculated using the formula \(\sqrt {p*(1-p)/n}\), where n is the sample size. So, the standard error when n=100 is \(\sqrt {0.72*(1-0.72)/100} = 0.0456\).
03

Interpret the results

We can say that we expect 72% of the sample to have their driver's license, give or take 4.56%.
04

Predict the effect of increasing sample size

According to statistical theory, increasing the sample size decreases the standard error. So, we expect the standard error to be less for n=500.
05

Recalculate the Standard Error with larger sample size

Recalculate the Standard Error using the same formula but now with n=500. Therefore, the standard error when n=500 is \(\sqrt {0.72*(1-0.72)/500} = 0.0204\), which confirms our prediction that the standard error becomes less for a larger sample size.

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Key Concepts

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

Standard Error
Understanding the standard error is crucial when dealing with sampling distributions. It tells us how much the sample proportion is expected to vary from the actual population proportion. When dealing with proportions, the standard error (SE) is calculated with the formula \( \sqrt{\frac{p(1-p)}{n}} \), where:
  • \( p \) is the population proportion
  • \( n \) is the sample size
The standard error provides an indication of the reliability of the sample proportion as an estimate of the population proportion. A smaller SE means that the sample proportion is closer to the population proportion, making it a more reliable estimate.In our example, with a sample size of 100 and a population proportion of 0.72, the standard error calculates to approximately 0.0456. This means the sample proportion is expected to vary from the population proportion by about 4.56% either way.
As the sample size increases, the standard error decreases which signifies a more accurate reflection of the population proportion.
Population Proportion
The population proportion represents the fraction of individuals in a population that have a particular attribute. Understanding the population proportion is essential because it is used to predict the expected value in a sample. In our case, the population proportion \( p \) is given as 72%, or 0.72, indicating that 72% of high school seniors in the entire population have a driver's license.
This is the benchmark against which we measure the sample proportion.When conducting surveys or experiments, we aim to estimate this population proportion as accurately as possible. The sample proportion will approach the actual population proportion more closely with larger samples. As a result, when dealing with proportions, researchers need to pay attention to how closely their sample proportion matches the true population proportion, which is facilitated by understanding the concept of standard error as discussed earlier.
Sample Size
Sample size is a key determinant in the accuracy of your sample's estimate of the population proportion. The larger the sample size \( n \), the more accurate the sample proportion will generally be in estimating the population proportion. Larger sample sizes tend to yield more precise and reliable estimates because they reduce the sampling error, or randomness, that can come from choosing a small portion of the population.In our problem, increasing the sample size from 100 to 500 has a dramatic effect on the standard error. When the sample size is 500, the standard error decreases to 0.0204, compared to 0.0456 with a sample size of 100. This decrease reflects a much tighter estimate around the population proportion.Describing this relationship between sample size and standard error reinforces the principle that scale matters. By opting for larger sample sizes, researchers can achieve more accurate reflections of the true population proportion, helping to make broader inferences with greater confidence.

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Most popular questions from this chapter

Just the Boys Refer to Exercise \(7.77\) for information. This data set records results just for the boys. $$\begin{array}{|lcc|}\hline & \text { Preschool } & \text { No Preschool } \\\\\hline \text { Grad HS } & 16 & 21 \\\\\hline \text { No Grad HS } & 16 & 18 \\\\\hline\end{array}$$ a. Find and compare the percentages that graduated for each group, descriptively. Does this suggest that preschool was linked with a higher graduation rate? b. Verify that the conditions for a two-proportion confidence interval are satisfied. c. Indicate which one of the following statements is correct. i. The interval does not capture 0 , suggesting that it is plausible that the proportions are the same. ii. The interval does not capture 0 , suggesting that it is not plausible that the proportions are the same. iii. The interval captures 0 , suggesting that it is plausible that the population proportions are the same. iv. The interval captures 0 , suggesting that it is not plausible that the population proportions are the same. d. Would a \(99 \%\) confidence interval be wider or narrower?

A 2017 Gallup poll reported that 658 out of 1028 U.S. adults believe that marijuana should be legalized. When Gallup first polled U.S. adults about this subject in 1969 , only \(12 \%\) supported legalization. Assume the conditions for using the CLT are met. a. Find and interpret a \(99 \%\) confidence interval for the proportion of U.S. adults in 2017 that believe marijuana should be legalized. b. Find and interpret a \(95 \%\) confidence interval for this population parameter. c. Find the margin of error for each of the confidence intervals found in parts a and b. d. Without computing it, how would the margin of error of a \(90 \%\) confidence interval compare with the margin of error for the \(95 \%\) and \(99 \%\) intervals? Construct the \(90 \%\) confidence interval to see if your prediction was correct.

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.

Suppose you attend a school that offers both traditional courses and online courses. You want to know the average age of all the students. You walk around campus asking those students that you meet how old they are. Would this result in an unbiased sample?

The Gallup poll reported that \(45 \%\) of Americans have tried marijuana. This was based on a survey of 1021 Americans and had a margin of error of plus or minus 5 percentage points with a \(95 \%\) level of confidence. a. State the survey results in confidence interval form and interpret the interval. b. If the Gallup Poll was to conduct 100 such surveys of 1021 Americans, how many of them would result in confidence intervals that did not include the true population proportion? c. Suppose a student wrote this interpretation of the interval: "We are \(95 \%\) confident that the percentage of Americans who have tried marijuana is between \(40 \%\) and \(50 \%\)." What, if anything, is incorrect in this interpretation?
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