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In: Statistics and Probability

In 2016, it was reported that 16% of 500 Utah families had incomes below the poverty...

In 2016, it was reported that 16% of 500 Utah families had incomes below the poverty level. At the same time, 11.3% of 450 Wyoming families had incomes below the poverty level. You need to find out if there was a significant difference in proportion of families with incomes below the poverty level in these two states. Determine the critical values (use 5% significance level).alculate the p-value for the test. (g) Assess the strength of your decision in part (d) using the p-value. (h) Calculate a 95% confidence interval for the difference between the two corresponding population proportions (see pages 280-281 in the Blue Book). (i) Which state has a higher proportion of families with incomes below the poverty level, and by how

Solutions

Expert Solution

For sample 1, N1​=500, the sample proportion is

For sample 2, N2​=450, the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is zc​=1.96.

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=2.083>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0373, and since p=0.0373<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is different than p2​, at the 0.05 significance level.

Confidence Interval

The 95% confidence interval for p1​−p2​ is: 0.003<p1​−p2​<0.09

Utah has a higher proportion of families with incomes below the poverty level, between 0.003<p1​−p2​<0.09.
​​​​

Graphically

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orchestra answered 2 years ago
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