Question

A firm wanted to see if there was a significant difference between two neighboring communities in terms of their usage of social media. A random survey of 150 households in the first community showed the 55% of the respondents used social media frequently. In the second community, of a random survey of 200 households, 120 used social media frequently. Assuming a significance level of .01, test the hypothesis that the two communities are statistically equivalent with respect to the usage of social media.

Answer 1

There is no difference between two neighboring communities in terms of their usage of social media.

Explanation:

In this case we need to determine whether there is any difference between two neighboring communities in terms of their usage of social media.

The hypothesis can be defined as follows:

H₀: There is no difference between two neighboring communities in terms of their usage of social media, i.e. p₁ - p₂ = 0.

Hₐ: There is a significant difference between two neighboring communities in terms of their usage of social media, i.e. p₁ - p₂ ≠ 0.

The test statistic is defined as follows:

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

The information provided is:

n₁ = 150

n₂ = 200

`\hat p_(1)` = 0.55

`\hat p_(2)=(120)/(200)=0.60`

Compute the total proportions as follows:

`\hat P=(n_(1)\hat p_(1)+n_(2)\hat p_(2))/(n_(1)+n_(2))=((150* 0.55)+(200* 0.60))/(150+200)=0.579`

Compute the test statistic value as follows:

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

`=\frac{0.55-0.60}{\sqrt{0.579(1-0.579)[(1)/(150)+(1)/(200)]}}\\\\=-0.94`

The test statistic value is -0.94.

The decision rule is:

The null hypothesis will be rejected if the p-value of the test is less than the significance level, α = 0.01.

Compute the p-value as follows:

`p-value=2* P(Z<-0.94)\\=2* 0.17361\\=0.34722\\\approx 0.35`

*Use a z-table.

The p-value of the test is 0.35.

p-value = 0.35 > α = 0.01.

The null hypothesis will not be rejected at 1% significance level.

Thus, it can be concluded that there is no difference between two neighboring communities in terms of their usage of social media.