A polling company conducts an annual poll of adults about political opinions. The survey asked a random sample of 406 adults whether they think things in the country are going in the right direction or in the wrong direction. 45% said that things were going in the wrong direction. Complete parts a and b below. Calculate the margin of error for the proportion of all adults who think things are going in the wrong direction for 90% confidence. Would the margin of error be larger or smaller for 85% confidence? Explain.

The margin of error or the proportion of all adults who think things are going in the wrong direction for 90% confidence is 0.0406, which is 4.06 percentage points.

At the 85% confidence level, the margin of error would be smaller due to the lower value of Z.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of , and a confidence level of , we have the following confidence interval of proportions.

In which

z is the zscore that has a pvalue of .

The margin of error is:

90% confidence level

So , z is the value of Z that has a pvalue of , so .

In this problem, we have that:

So

The margin of error or the proportion of all adults who think things are going in the wrong direction for 90% confidence is 0.0406, which is 4.06 percentage points.

85% confidence level

So , z is the value of Z that has a pvalue of , so .

With the lower confidence level, z is lower, which means that the margin of error would be smaller.

yes. we do proportions in probabilities.