Problems 20 and 21: In 2012, the National Science Foundation commissioned a survey that revealed about 1 in 4 American adults do not know the earth revolves around the sun. The survey of 2200 people in the United States asked “Does the Earth go around the Sun, or does the Sun go around the Earth? 74% of the people surveyed knew the earth revolves around the sun and not the other way around. Problems 22 Let us say four times as many American adults were surveyed by the National Science Foundation. Refer to questions 20 and 21. (a) Calculate the standard deviation using our new formula. Standard deviations must be written as a percent. Round to one decimal point. DO NOT round until the very end when finding the percent. (b) Assuming this was a random sample of all American adults, find a 95% confidence interval if the change if four times as many American adults were surveyed. (c) What happens to the confidence interval when you have four times as many American adults surveyed? The confidence interval is wider because the margin of error is cut in about half when you quadruple the sample size. The confidence interval is narrower because the margin of error is cut in about half when you quadruple the sample size. Nothing happens to the confidence interval because the margin of error is so small.
The Correct Answer and Explanation is:
To answer this set of questions, we will first define the relevant statistical terms and calculations, then walk through each part clearly.
Given:
- Original sample size = 2,200
- Proportion (p̂) of correct responses = 74% = 0.74
- Therefore, q̂ = 1 – p̂ = 0.26
- New sample size = 4 × 2,200 = 8,800
(a) Calculate the standard deviation (as a percent)
Standard deviation (SD) for a proportion is calculated using the formula: SD=p^(1−p^)n=0.74×0.268800=0.19248800≈0.0000218636≈0.004676SD = \sqrt{\frac{p̂(1 – p̂)}{n}} = \sqrt{\frac{0.74 \times 0.26}{8800}} = \sqrt{\frac{0.1924}{8800}} ≈ \sqrt{0.0000218636} ≈ 0.004676SD=np^(1−p^)=88000.74×0.26=88000.1924≈0.0000218636≈0.004676
To express this as a percent: 0.004676×100≈0.5%0.004676 \times 100 ≈ \boxed{0.5\%}0.004676×100≈0.5%
(b) Find the 95% confidence interval
To find the 95% confidence interval, use the formula: CI=p^±z×SDCI = p̂ \pm z \times SDCI=p^±z×SD
Where:
- p^=0.74p̂ = 0.74p^=0.74
- zzz (for 95% confidence) = 1.96
- SD=0.004676SD = 0.004676SD=0.004676
CI=0.74±1.96×0.004676≈0.74±0.009167CI = 0.74 \pm 1.96 \times 0.004676 ≈ 0.74 \pm 0.009167CI=0.74±1.96×0.004676≈0.74±0.009167 Lowerlimit=0.74–0.0092≈0.7308Lower limit = 0.74 – 0.0092 ≈ 0.7308Lowerlimit=0.74–0.0092≈0.7308 Upperlimit=0.74+0.0092≈0.7492Upper limit = 0.74 + 0.0092 ≈ 0.7492Upperlimit=0.74+0.0092≈0.7492
Expressed as percentages: 95% CI: 73.1% to 74.9%\boxed{95\% \text{ CI: } 73.1\% \text{ to } 74.9\%}95% CI: 73.1% to 74.9%
(c) What happens to the confidence interval when the sample size is quadrupled?
Correct Answer:
✔ The confidence interval is narrower because the margin of error is cut in about half when you quadruple the sample size.
Explanation (300+ words):
When conducting a survey or poll, one of the most important statistical measures is the confidence interval (CI). This range estimates where the true population proportion lies, based on the sample data. The width of this interval depends largely on the standard deviation (SD), which in turn is influenced by the sample size. The formula for SD shows it is inversely proportional to the square root of the sample size. That is: SD∝1nSD \propto \frac{1}{\sqrt{n}}SD∝n1
So, if you quadruple the sample size (from 2,200 to 8,800), the square root of 4 is 2, meaning the standard deviation—and hence the margin of error—is cut in half. The margin of error is the amount you add and subtract from the sample proportion to create your confidence interval. A smaller margin of error means the interval becomes narrower, giving you a more precise estimate of the population proportion.
In this problem, when the sample size increases from 2,200 to 8,800, the standard deviation decreases from about 0.0093 (original) to about 0.0047 (new). As a result, the 95% confidence interval becomes narrower: instead of covering a wider range (e.g., 72.6% to 75.4%), it now covers a tighter one (73.1% to 74.9%).
In summary, increasing the sample size improves the precision of your estimate by reducing variability. This helps researchers and decision-makers draw more reliable conclusions from the data, especially in surveys involving large populations.