Question

A researcher is interested in the difference in the rates of diabetes among people living in rural areas and people living in urban areas. They randomly selected 1023 people living in rural areas and found that 122 had diabetes and of 1332 people living in urban areas found that 167 people had diabetes. Conduct a hypothesis test to determine if there is a difference between the diabetes rates of people living in rural and urban areas. Please include all steps of the test. Use significance level of 0.05

Answer 1

We conclude that there is no significant difference between the diabetes rates of people living in rural and urban areas.

Explanation:

We are given that a researcher is interested in the difference in the rates of diabetes among people living in rural areas and people living in urban areas.

They randomly selected 1023 people living in rural areas and found that 122 had diabetes and of 1332 people living in urban areas found that 167 people had diabetes.

Let
`p_1` = proportion of people living in rural areas having diabetes.

`p_2` = proportion of people living in urban areas having diabetes.

So, Null Hypothesis,
`H_0` :
`p_1-p_2` = 0 {means that there is no significant difference between the diabetes rates of people living in rural and urban areas}

Alternate Hypothesis,
`H_A` :
`p_1-p_2\\eq` 0 {means that there is a significant difference between the diabetes rates of people living in rural and urban areas}

The test statistics that would be used here Two-sample z proportion statistics;

T.S. =
`\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{(\hat p_1(1-\hat p_1))/(n_1)+(\hat p_2(1-\hat p_2))/(n_2) } }` ~ N(0,1)

where,
`\hat p_1` = sample proportion of people having diabetes living in rural areas =
`(122)/(1023)` = 0.119

`\hat p_2` = sample proportion of people having diabetes living in urban areas =
`(167)/(1332)` = 0.125

`n_1` = sample of people living in rural areas = 1023

`n_2` = sample of people living in urban areas = 1332

So, test statistics =
`\frac{(0.119-0.125)-(0)}{\sqrt{(0.119(1-0.119))/(1023)+(0.125(1-0.125))/(1332) } }`

= -0.442

The value of z test statistics is -0.442.

Now, at 0.05 significance level the z table gives critical values of -1.96 and 1.96 for two-tailed test.

Since our test statistics lies within the range of critical values of z, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.

Therefore, we conclude that there is no significant difference between the diabetes rates of people living in rural and urban areas.