The two states with the highest home foreclosure rates in...

The two states with the highest home foreclosure rates in March 2008 were Nevada and Colorado (realestate.msn.com, April 2008). In the second quarter of 2008 , there were 8 foreclosures in a random sample of 1098 homes in Nevada, and 6 in a sample of 1460 homes in Colorado. a) Is there evidence that the percentage of foreclosures is different in the two states? b) Give a $90 \%$ confidence interval for the difference in proportions.

The two states with the highest home foreclosure rates in March 2008 were Nevada and Colorado (realestate.msn.com, April 2008). In the second quarter of 2008 , there were 8 foreclosures in a random sample of 1098 homes in Nevada, and 6 in a sample of 1460 homes in Colorado.
a) Is there evidence that the percentage of foreclosures is different in the two states?
b) Give a $90 \%$ confidence interval for the difference in proportions.

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Key Concepts

Confidence Interval for the Difference in Proportions

This concept provides a range of values within which the true difference between two population proportions is likely to lie with a certain level of confidence, such as 90% or 95%. The interval is calculated by taking the observed difference between the sample proportions and adding and subtracting the product of a critical value (from the standard normal distribution) and the standard error of the difference. This confidence interval not only indicates the precision of the estimate but also helps assess whether a null difference (zero) is a plausible value.

Two-Sample Proportion Hypothesis Testing

This concept involves comparing the proportions from two independent groups to determine if there is a statistically significant difference between them. The procedure typically involves formulating a null hypothesis that the two population proportions are equal and an alternative hypothesis that they are not. A test statistic (usually a Z-score) is computed based on the difference between the sample proportions and the associated standard error. The resulting p-value helps decide whether there is enough evidence to reject the null hypothesis in favor of the alternative.

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