Question

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. A survey of factories in five northeastern states found that 10% of the 300 workers surveyed were satisfied with the benefits offered by their employers. The margin of error for the population proportion, rounded to the nearest hundredth of a percent, is ±_______

Answer 1

The margin of error provides the range that the true population proportion is expected to be in, given the sample proportion. A lower margin of error or a narrower confidence interval is achieved by either increasing the sample size or changing the confidence level. The ±3 percent sampling error indicates the possible variation due to the size and selection of the sample.

Step-by-step explanation:

The margin of error in the context of a survey is the range within which the true population proportion is expected to lie, considering the sample proportion. For example, in the survey of factories in five northeastern states with 300 workers, where 10% were satisfied with their benefits, the margin of error would tell us how much we can expect the true proportion of satisfied workers in the total population to vary from this 10% figure. The specific number for the margin of error isn't provided in the question, but is typically calculated using a formula that incorporates the sample size, the sample proportion, and the desired confidence level. The margin of error can be reduced by increasing the sample size or reducing the confidence level.

Regarding the question about how the confidence interval would change with a lower confidence level, it can be stated that the interval would become narrower, implying that there is less certainty about the exact value of the population proportion, but the range is more precise. For instance, if the confidence level is decreased from 99 percent to 90 percent, the confidence interval will be smaller because there's a trade-off between confidence and precision.

To lower the sampling error, one could increase the sample size. The ±3 percent margin mentioned represents the variability due to sampling; this means that the true proportion could be 3 percentage points higher or lower than the sample proportion found in the survey.

Answer 2

The margin of error for the population proportion, rounded to the nearest hundredth of a percent, is ±3.39.

The margin of error (MOE) for a population proportion in a survey can be calculated using the formula for a 95% confidence interval:

`\[ MOE = Z * \sqrt{(p(1 - p))/(n)} \]`

Where:

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`\( Z \)` is the Z-score corresponding to the desired confidence level (1.96 for 95% confidence).

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`\( p \)` is the sample proportion (0.10 in this case).

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`\( n \)` is the sample size (300 in this case).

Since a confidence level isn't specified, the standard 95% confidence level will be assumed. Let's calculate the margin of error using this formula.

The margin of error for the population proportion, rounded to the nearest hundredth of a percent, is ±3.39.