Respuesta :
Using the z-distribution, as we are working with a proportion, it is found that there is evidence that the phrasing may have influenced responses.
What are the hypotheses tested?
At the null hypothesis, it is tested if there is no difference between the proportions, that is:
[tex]H_0: p_1 - p_2 = 0[/tex]
At the alternative hypothesis, it is tested if there is difference, that is:
[tex]H_1: p_1 - p_2 \neq 0[/tex]
What is the mean and the standard of the distribution of the differences of sample proportions?
For each sample, we have that:
[tex]p_1 = \frac{463}{615} = 0.7528, s_1 = \sqrt{\frac{0.7528(0.2472)}{615}} = 0.0174[/tex]
[tex]p_2 = \frac{403}{585} = 0.6889, s_2 = \sqrt{\frac{0.6889(0.3111)}{585}} = 0.0191[/tex]
For the distribution of differences, we have that:
[tex]\overline{p} = p_1 - p_2 = 0.7528 - 0.6889 = 0.0639[/tex]
[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0174^2 + 0.0191^2} = 0.0258[/tex]
What is the test statistic?
It is given by:
[tex]z = \frac{\overline{p} - \mu}{s}[/tex]
In which [tex]\mu = 0[/tex] is the value tested at the null hypothesis.
Hence:
[tex]z = \frac{\overline{p} - \mu}{s}[/tex]
[tex]z = \frac{0.0639 - 0}{0.0258}[/tex]
[tex]z = 2.47[/tex]
What is the decision?
Considering a two-tailed test, as we are testing if the proportion is different of a value, and a significance level of 0.05, the critical value is [tex]|z^{\ast}| = 1.96[/tex].
Since the absolute value of the test statistic is greater than the critical value, it is found that there is evidence that the phrasing may have influenced responses.
More can be learned about the z-distribution at https://brainly.com/question/26454209
