Question

PLEASE SHOW CALCULATOR STEPS INCLUDING WHAT YOU INPUT FOR THE TESTS Just before the 2004 Democratic convention, Rasmussen Reports polled 1500 likely voters at random and found that 705 favored John Kerry. Just after the convention, they took another random sample of 1500 likely voters and found that 735 favored Kerry. Did Kerry’s favorability rating increase after the national convention? Use a significance level of a = 0.05.

a) Give the name of the hypothesis test that would be appropriate for this situation. (1 point)

b) State the hypotheses in symbols. (2 points)

c) Use your calculator to perform the appropriate hypothesis test and report the test statistic and p-value. Be sure to write out what you entered in your calculator. (3 points)

d) Make a sketch of the test distribution. Be sure to label the test statistic and p-value. (2 points)

e) Write a full conclusion for this test in the context of the problem. (2 points)

f) Find a 90% confidence interval for the difference in John Kerry’s favorability rating before and after the convention. Do not make these calculations by hand. Instead, use the correct command in your graphing calculator and write out what you entered. (3 points)

g) Does this confidence interval support your conclusion in part (e)? Explain. (2 points)

Answer 1

Before, we have that the sample size is N1​=1500, the number of favorable cases is X1​=705, so then the sample proportion is p^​1​=​X1/N1​​=705/1500​=0.47

For sample 2, we have that the sample size is N2​=1500, the number of favorable cases is X2​=735, so then the sample proportion is p^​2​=​X2/N2​​=735/1500 ​=0.49

The value of the pooled proportion is computed as p¯​=​X1​+X2​​/N1​+N2 =705+735​/1500+1500 =0.48

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:p1​=p2​

Ha:p1​<p2​ LEFT TAILED TEST

This corresponds to a left-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 \alpha = 0.05α=0.05, and the critical value for a left-tailed test is zc​=−1.64.

The rejection region for this left-tailed test is R={z:z<−1.64}

(3) Test Statistics

The z-statistic is computed as follows:

z=p^​1​−p^​2/sqrt(​​p¯​(1−p¯​)(1/n1​+1/n2​)​) =0.47−0.49/sqrt(0.48⋅(1−0.48)(1/1500+1/1500)​) ​=−1.096

(4) Decision about the null hypothesis

Since it is observed that z=−1.096≥zc​=−1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.1365, and since p=0.1365≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected at the 0.05 significance level. Therefore we dont have enough evidence to conclude that Kerry’s favorability rating increase after the national convention.

Confidence Interval

The 95% confidence interval for p1​−p2​ is −0.056<p1​−p2​<0.016.

The 90% confidence interval for p1​−p2​ is:−0.05<p1​−p2​<0.01.

Yes it supports conclusion .

Explaination:If the confidence interval for the difference does contain zero, we can conclude that there is NOT a statistically significant difference in the two population values at the given level of confidence.

NOTE: Unable to sketch graph here.