A poll of 1000 registered voters reveals that 480 of the voters surveyed prefer the Democratic candidate for the presidency. At the 0.05 level of significance, do the data provide sufficient evidence to conclude that the percentage of voters who prefer the Democratic candidate is less than 50%? State the null hypothesis and the conclusion.
Question 1 options:
a) H0: p = 0.50; Do not reject H0.
b) H0: p = 0.50; Reject H0.
c) H0: p = 0.48; Do not reject H0.
d) H0: p = 0.48; Reject H0.
To determine whether the data provides sufficient evidence to conclude that the percentage of voters who prefer the Democratic candidate is less than 50%, we can conduct a hypothesis test.
The null hypothesis (H0) represents the assumption or claim that we want to test. In this case, the null hypothesis would be that the percentage of voters who prefer the Democratic candidate is equal to 50%.
The alternative hypothesis (Ha) represents the opposite of the null hypothesis. In this case, the alternative hypothesis would be that the percentage of voters who prefer the Democratic candidate is less than 50%.
To conduct the hypothesis test, we can use a Z-test for proportions. The formula for the test statistic (Z) is:
Z = (p - P) / sqrt(P * (1-P) / n)
Where:
p is the proportion of voters in the sample who prefer the Democratic candidate (p = 480/1000 = 0.48),
P is the proportion specified in the null hypothesis (P = 0.50),
n is the sample size (n = 1000).
By calculating the test statistic Z using the given values, we can compare it to the critical value from the standard normal distribution corresponding to the significance level of 0.05.
If the test statistic Z is less than the critical value, we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis. If the test statistic Z is greater than or equal to the critical value, we do not reject the null hypothesis.
Now we can compare the calculated test statistic Z with the critical value to draw a conclusion:
Given that the null hypothesis is H0: p = 0.50, the conclusion based on the calculated test statistic Z would be:
b) H0: p = 0.50; Reject H0.
Therefore, the data provides sufficient evidence to conclude that the percentage of voters who prefer the Democratic candidate is less than 50%.