In a survey of 2085 adults in a certain country conducted during a period of economic uncertainty, 57% thought that wages paid to workers in industry were too low. The margin of error was 5 percentage points with 90% confidence. For parts (a) through (d) below, which represent a reasonable interpretation of the survey results? For those that are not reasonable, explain the flaw.(a) We are 90% confident 57% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.Is the interpretation reasonable?(b) We are 85% to 95% confident 57% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.Is the interpretation reasonable?c) We are 90% confident that the interval from 0.52 to 0.62 contains the true proportion of adults in the country during the period of economic uncertainty who believed wages paid to workers in industry were too low.Is the interpretation reasonable?(d) In 90% of samples of adults in the country during the period of economic uncertainty, the proportion who believed wages paid to workers in industry were too low is between 0.52 and 0.62.Is the interpretation reasonable?
(a) The interpretation states that “We are 90% confident that 57% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low
(a) The interpretation states that “We are 90% confident that 57% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.” This interpretation is reasonable. When we conduct a survey, there is always a margin of error associated with our estimate. In this case, with a 90% confidence level and a margin of error of 5 percentage points, it means that we can say with 90% confidence that the true proportion of adults who think wages are too low lies between an interval of 52% (57% – 5%) and 62% (57% + 5%). Therefore, the interpretation correctly reflects the results of the survey.
(b) The interpretation states that “We are 85% to 95% confident that 57% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.” This interpretation is not reasonable. Confidence intervals are constructed based on a specific confidence level, which in this case is 90%. Stating that we are confident in a range from 85% to 95% introduces confusion and does not accurately reflect the confidence level of the survey.
(c) The interpretation states that “We are 90% confident that the interval from 0.52 to 0.62 contains the true proportion of adults in the country during the period of economic uncertainty who believed wages paid to workers in industry were too low.” This interpretation is reasonable. It correctly represents the confidence interval from the survey, which is a statistical range containing the true proportion of adults who believed wages were too low with 90% confidence.
(d) The interpretation states that “In 90% of samples of adults in the country during the period of economic uncertainty, the proportion who believed wages paid to workers in industry were too low is between 0.52 and 0.62.” This interpretation is not reasonable. The statement suggests that in 90% of samples taken from the population, the true proportion of adults who believe wages are too low falls within the range of 0.52 to 0.62. However, the correct interpretation is that we are 90% confident that the given range (0.52 to 0.62) contains the true proportion of adults who felt wages were too low. Confidence intervals provide a range within which the true proportion is likely to fall, but it does not provide a specific probability for individual samples.
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