Question

Another paper, by Kristin Butcher and Anne Piehl (1998), compared the rates of institutionalization (in jail, prison, or mental hospitals) among immigrants and natives. In 1990, 7.54% of the institutionalized population (or 20,933 in the sample) were immigrants. The standard error of the fraction of institutionalized immigrants is 0.18. What is a 95% confidence interval for the fraction of the entire population who are immigrants? If you know that 10.63% of the general population at the time are immigrants, what conclusions can be made? Explain.

Answer 1

Solution:

Given: In 1990, 7.54% of the institutionalized population (or 20,933 in the sample) were immigrants.

thus  

The standard error of the fraction of institutionalized immigrants is 0.18%

That is:

We have to find  a 95% confidence interval for the fraction of the entire population who are immigrants.

Formula:

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

thus

Thus

If you know that 10.63% of the general population at the time are immigrants, what conclusions can be made? Explain.

Since 10.63% is outside ( above ) the limits of 95% confidence interval, there is not sufficient evidence to conclude that: 10.63% of the general population at the time are immigrants.