Questions: In a survey of 2085 adults in a certain country conducted during a period of economic uncertainty, 68% thought that wages paid to workers in industry were too low. The margin of error was 4 percentage points with 95% 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. C. The interpretation is flawed. The interpretation provides no interval about the population proportion. D. The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true. (b) We are 91% to 99% confident 68% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.

In a survey of 2085 adults in a certain country conducted during a period of economic uncertainty, 68% thought that wages paid to workers in industry were too low. The margin of error was 4 percentage points with 95% 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.
C. The interpretation is flawed. The interpretation provides no interval about the population proportion.
D. The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true.
(b) We are 91% to 99% confident 68% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.

Transcript text: In a survey of 2085 adults in a certain country conducted during a period of economic uncertainty, $68 \%$ thought that wages paid to workers in industry were too low. The margin of error was 4 percentage points with $95 \%$ 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. C. The interpretation is flawed. The interpretation provides no interval about the population proportion. D. The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true. (b) We are $91 \%$ to $99 \%$ confident $68 \%$ of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.

Solution

Solution Steps

Step 1: Calculate the Confidence Interval

To estimate the population proportion of adults who think that wages paid to workers in industry are too low, we calculate the confidence interval using the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:

$\hat{p} = 0.68$ (sample proportion)
$n = 2085$ (sample size)
$z = 1.96$ (z-score for 95% confidence level)

Calculating the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \sqrt{\frac{0.68(1 - 0.68)}{2085}} \approx 0.02 \]

Thus, the confidence interval is:

\[ (0.68 - 0.02, 0.68 + 0.02) = (0.66, 0.70) \]

Step 2: Interpret the Results

The confidence interval $(0.66, 0.70)$ indicates that we are 95% confident that the true population proportion of adults who believe that wages are too low lies within this interval.

Step 3: Evaluate Interpretations

We evaluate the provided interpretations based on the calculated confidence interval:

Interpretation A: "The interpretation is reasonable."
This is reasonable because it provides a confidence interval.
Interpretation B: "The interpretation is flawed. The interpretation provides no interval about the population proportion."
This is flawed because it does not provide a confidence interval.
Interpretation C: "The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true."
This is flawed because it suggests a standard for all intervals.
Interpretation D: "The interpretation is flawed. The interpretation indicates that the level of confidence is varying."
This is flawed because it suggests a varying confidence level.