What is the critical value in this question?
Assume that we randomly sample 200 residents of Indiana and ask them if they favor having the government issue marriage licenses to same-sex couples. 104 say that the do; 96 don’t. Test at the 5% level whether the population support for this proposition is greater than 50%.
In order to determine the critical value for this question, we need to conduct a hypothesis test. A hypothesis test allows us to assess whether an observed sample result is statistically significant.
In this case, the null hypothesis (H0) would be that the population support for issuing marriage licenses to same-sex couples is equal to 50%. The alternative hypothesis (H1) would be that the population support is greater than 50%.
To conduct the hypothesis test, we need to calculate the test statistic. In this case, we can use a one-sample proportion test, as we are comparing the proportion of support in our sample (104 out of 200 residents) to a predetermined value (50%).
The test statistic for a one-sample proportion test is calculated as:
z = (p - P) / √(P * (1 - P) / n)
where p is the sample proportion (104/200), P is the assumed population proportion under the null hypothesis (0.5), and n is the sample size (200 residents).
By plugging in the values, we can calculate the test statistic:
z = (104/200 - 0.5) / √(0.5 * (1-0.5) / 200)
Now, we can use a z-table or statistical software to find the critical value for a one-tailed test at a significance level of 5%.
The critical value represents the threshold beyond which we would reject the null hypothesis. In this case, since we are testing whether the population support is greater than 50%, we will be using a one-tailed test.
For a 5% significance level, the critical value is typically 1.645 for a one-tailed test. If the calculated test statistic (z) is greater than this critical value, we will reject the null hypothesis.
Therefore, the critical value in this question is 1.645.