The null hypothesis is represented as
H₀: p ≤ (3/4)
The alternative hypothesis is represented as
Hₐ: p > (3/4)
z = 1.65
p-value = 0.049471
The obtained p-value is less than the significance level at which the test was performed at. Hence, we reject the null hypothesis, accept the alternative hypothesis & say that there is significant evidence to conclude that more than (3/4) of the voting population would vote for the Democrat.
Step-by-step explanation:
The complete question is presented in the attached image to this answer.
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test. It usually maintains that random chance is responsible for the outcome or results of any experimental study/hypothesis testing.
While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test. It usually maintains that other than random chance, there are significant factors affecting the outcome or results of the experimental study/hypothesis testing.
For this question, the null hypothesis would be that there isn't significant evidence that that more than (3/4) of the entire voting population would vote for the Democrat. That is, the proportion of the entire voting population that would vote for the democrat is less than or equal to (3/4).
While the alternative hypothesis is that there is significant evidence that more than (3/4) of the voting population would vote for the Democrat.
Mathematically,
The null hypothesis is represented as
H₀: p ≤ (3/4)
The alternative hypothesis is represented as
Hₐ: p > (3/4)
To do this test, we will use the z-distribution because the sample size (1200) is large enough for the p-value for z-test statistic and the t-test statistic to approximately be equal.
So, we compute the z-test statistic
z = (x - μ)/σₓ
x = sample proportion of the surveyed registered voters that would vote for the democrat = (925/1200) = 0.77
μ = p₀ = The standard proportion that we're comparing against = (3/4) = 0.75
σₓ = standard error = √[p(1-p)/n]
where n = Sample size = 1200
σₓ = √[0.77×0.23/1200] = 0.01215
z = (0.77 - 0.75) ÷ 0.01215
z = 1.65
checking the tables for the p-value of this t-statistic
Significance level = 0.05
The hypothesis test uses a one-tailed condition because we're testing only in one direction (whether the proportion is greater than 0.75)
p-value (for z = 1.65, at 0.05 significance level, with a one tailed condition) = 0.049471
The interpretation of p-values is that
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
So, for this question, significance level = 0.05
p-value = 0.049471
0.171485 < 0.05
Hence,
p-value < significance level
This means that we reject the null hypothesis, accept the alternative hypothesis & say that there is significant evidence to conclude that more than (3/4) of the voting population would vote for the Democrat.
Hope this Helps!!!
