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Chapter 7: Problem 72

In the 2018 study Closing the STEM Gap, researchers wanted to estimate the percentage of middle school girls who planned to major in a STEM field. a. If a \(95 \%\) confidence level is used, how many people should be included in the survey if the researchers wanted to have a margin of error of \(3 \%\) ? b. How could the researchers adjust their margin of error if they want to decrease the number of study participants?

Short Answer

Expert verified
a. To achieve a \(3\%\) margin of error at a \(95\%\) confidence level, the researchers should survey at least 1067 middle school girls. b. If the researchers want to decrease the number of study participants, they could increase their margin of error, which will decrease the necessary sample size but also decrease the precision of the study.

Step by step solution

01

Identify the given values and relevant formula

For this survey, the confidence level is set at \(95\%\) which corresponds to a Z-score of approximately \(1.96\) (Z-scores are related to the standard normal distribution). The margin of error is \(3\%\), or \(0.03\) when expressed as a decimal. The formula for sample size (n) of a proportion is given by: \[n = \frac{{Z^2*p*q}}{{E^2}}\] where p is the estimated proportion of the population which has the desired attribute (in this case, girls planning to major in STEM fields), q=1-p, Z is the z-score for the chosen confidence level, and E is the margin of error.
02

Handle the unknown proportion

In this case, we do not have a value for p. When we do not know the value of p, we can take p as \(0.5\). This is because the sample size calculated will be the maximum possible, providing a safe estimate. So, take p as \(0.5\). Consequently, q=1鈥損 will also be \(0.5\). Now, we can put these values into the sample size formula.
03

Calculate the sample size

By substituting these values into our formula, we would get: \[n = \frac{{(1.96)^2*(0.5)*(0.5)}}{{(0.03)^2}} \] After calculating, we find that n 鈮� 1067. This means that the researchers should survey at least 1067 middle school girls to achieve their desired margin of error at a \(95\%\) confidence level.
04

Consider the effect of changing the margin of error

For part b, if the researchers want to decrease the number of study participants, they would have to adjust their margin of error to be larger. This is because the margin of error (E) is inversely proportional to sample size (n), as can be seen in the formula. Therefore, a larger margin of error would lead to a smaller required sample size, but would also decrease the precision of the study.

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Most popular questions from this chapter

Suppose that, when taking a random sample of 4 from 123 women, you get a mean height of only 60 inches ( 5 feet). The procedure may have been biased. What else could have caused this small mean?

From Formula \(7.2\), an estimate for margin of error for a \(95 \%\) confidence interval is \(m=2 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) where \(\mathrm{n}\) is the required sample size and \(\hat{p}\) is the sample proportion. Since we do not know a value for \(\hat{p}\), we use a conservative estimate of \(0.50\) for \(\hat{p}\). Replace \(\hat{p}\) with \(0.50\) in the formula and simplify.

Suppose a political consultant is hired to determine if a school bond is likely to pass in a local election. The consultant randomly samples 250 likely voters and finds that \(52 \%\) of the sample supports passing the bond. Construct a \(95 \%\) confidence interval for the proportion of voters who support the bond. Assume the conditions are met. Based on the confidence interval, should the consultant predict the bond will pass? Why or why not?

The Centers for Disease Control and Prevention (CDC) conducts an annual Youth Risk Behavior Survey, surveying over 15,000 high school students. The 2015 survey reported that, while cigarette use among high school youth had declined to its lowest levels, \(24 \%\) of those surveyed reported using e-cigarettes. Identify the sample and population. Is the value \(24 \%\) a parameter or a statistic? What symbol would we use for this value?

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?
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