Question

The serious delinquency rate measures the proportion of homes that are well behind on their mortgage payments. A real estate investor is trying to decide in which of two neighborhoods in Phoenix to buy an investment property. He is trying to decide if the serious delinquency rate differs between the two neighborhoods. He took a sample of 70 homes in neighborhood 1 and found 14 homes were seriously delinquent. He took a sample of 80 homes in neighborhood 2 and 10 homes were seriously delinquent. Specify the null and alternate hypotheses to determine if the delinquency rates in the two neighborhoods are equal and make a conclusion at the 10% significance level. How should the delinquency rate influence the investor’s decision? (Make sure to follow all procedures for hypothesis testing)

Answer 1

`z=\frac{0.2-0.125}{\sqrt{0.16(1-0.16)((1)/(70)+(1)/(80))}}=1.25`

The p value for this case is given by:

`p_v =2*P(Z>1.25)= 0.211`

Since the p value is higher than the significance level given of 0.1 or 10% we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the two delinquency rates in the two neighborhood ar enot significantly different.

And is important to remark that if the delinquency rate is a big number the investor probably would overthink about the investment

Explanation:

Information given

`X_(1)=14` represent the number of homes were seriously delinquent in neighborhood 1

`X_(2)=10` represent the number of homes were seriously delinquent in neighborhood 2

`n_(1)=70` sample 1 selected

`n_(2)=80` sample 2 selected

`p_(1)=(14)/(70)=0.2` represent the proportion estimated of homes were seriously delinquent in neighborhood 1

`p_(2)=(10)/(80)=0.125` represent the proportion estimated of homes were seriously delinquent in neighborhood 1

`\hat p` represent the pooled estimate of for the proportion p

z would represent the statistic

`p_v` represent the value for the test

`\alpha=0.05` significance level given

System of hypothesis

We want to analyze if the two proportions for the delinquency rates are different , the system of hypothesis are:

Null hypothesis:
`p_(1) = p_(2)`

Alternative hypothesis:
`p_(1) \\eq p_(2)`

The statistic for this case is given by:

`z=\frac{p_(1)-p_(2)}{\sqrt{\hat p (1-\hat p)((1)/(n_(1))+(1)/(n_(2)))}}` (1)

Where
`\hat p=(X_(1)+X_(2))/(n_(1)+n_(2))=(14+10)/(70+80)=0.16`

Now we can calculate the statistic:

`z=\frac{0.2-0.125}{\sqrt{0.16(1-0.16)((1)/(70)+(1)/(80))}}=1.25`

The p value for this case is given by:

`p_v =2*P(Z>1.25)= 0.211`

Since the p value is higher than the significance level given of 0.1 or 10% we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the two delinquency rates in the two neighborhood ar enot significantly different.

And is important to remark that if the delinquency rate is a big number the investor probably would overthink about the investment