Suppose a random sample of 85 items has been taken from a population and 40 of the sample can be characteristic of interest. Use this information to calculate a 90% confidence interval to estimate the proportion of the population that has the characteristic of interest. Calculate a 95% confidence interval. Calculate a 99% confidence interval. As the level of confidence changes and the other sample information stays constant, what happens to the confidence interval?

**Solution: **The 90% confidence level is

_{}

The sample proportion is

_{}

and the standard error:

_{}

- For a 90% confidence interval we have
_{}, which means that the 90% confidence interval is (0.38153, 0.5596)

- For a 95% confidence interval we have
_{}, which means that the 95% confidence interval is (0.364476, 0.5767)

- For a 99% confidence interval we have
_{}, which means that the 99% confidence interval is (0.331127, 0.61005)

As we can see the width of the confidence interval increases as the confidence level increases.

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