An association of Christmas tree growers in Indiana sponsored a sample survey of 500 randomly selected Indiana households to help improve the marketing of Christmas trees. One question the researchers asked was, "Did you have a Christmas tree this year?" Respondents who had a tree during the holiday season were asked whether the tree was natural or artificial. Respondents were also asked if they lived in an urban area or in a rural area. The tree growers want to know if there is a difference in preference for natural trees versus artificial trees between urban and rural households. Among the 160 who lived in rural areas, 64 had a natural tree. Among the 261 who lived in an urban area, 89 had a natural tree. Construct and interpret a 95% confidence interval for the difference in the proportion of rural and urban Indiana residents who had a natural Christmas tree this year.

And the 95% confidence interval would be given (-0.0362;0.154).  

We are confident at 95% that the difference between the two proportions is between

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

represent the real population proportion for rural

represent the estimated proportion for rural  with natural tree

is the sample size required for rural with natural tree

represent the real population proportion for urban with natural tree

represent the estimated proportion for urban with natural tree

is the sample size required for Brand B

represent the critical value for the margin of error  

Solution to the problem

The population proportion have the following distribution  

The confidence interval for the difference of two proportions would be given by this formula  

For the 95% confidence interval the value of and , with that value we can find the quantile required for the interval in the normal standard distribution.  

And replacing into the confidence interval formula we got:  

And the 95% confidence interval would be given (-0.0362;0.154).  

We are confident at 95% that the difference between the two proportions is between