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In a 2018 survey conducted by Northeastern University, \(28 \%\) of working adults with education levels less than a bachelor's degree worried that their job would be eliminated due to new technology or automation. This was based on a \(95 \%\) confidence interval with a margin of error of 3 percentage points. a. Report the confidence interval for the proportion of adults with education level less than a bachelor's degree who are worried about job loss due to new technology or automation. b. If the sample size were smaller and the sample proportion stayed the same, would the resulting interval be wider or narrower than the one obtained in part a? c. If the confidence level were \(99 \%\) rather than \(95 \%\) and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a? d. In 2018 the population of the United States was roughly 327 million. If 50 million people were added to the population what effect, if any, would this have on the intervals obtained in this problem?

Step by step solution

01

Calculate the Confidence Interval

The sample proportion \(\hat{p}\) is 0.28, and the margin of error is 0.03. We pluck these values into the formula for the confidence interval which looks like: \(\hat{p} \pm\) margin of error. So the confidence interval is \(0.28 \pm 0.03\). This means the confidence interval is: [0.25, 0.31]

02

Understand the Effect of Sample Size

If the sample size decreases and the sample proportion remains the same, the standard error will increase (because the denominator in its calculation is the square root of the sample size). This will result in a larger margin of error, and thus a wider confidence interval.

03

Understand the Effect of the Confidence Level

If the confidence level increases to 99% from 95% while the sample proportion remains the same, the z-value corresponding to the confidence level in the margin of error formula will increase. This will result in a larger margin of error, and thus a wider confidence interval.

04

Understand the Effect of Population Size

Increasing the population size will not affect the confidence interval. This is because the population size is not a factor in the calculation of a confidence interval for a proportion. The confidence interval depends on the sample proportion and the sample size, not the overall population size.

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