Answer:
1) The null hypothesis is represented as
H₀: p ≤ 0.40
The alternative hypothesis is represented as
Hₐ: p > 0.40
2) Test statistic = 1.91
3) p-value = 0.028067
4) Conclusion is Option D.
Reject H_0 and find evidence that the proportion is greater than 40%.
This is because the p-value < significance level for this question.
Explanation:
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.
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.
For this question, the null hypothesis is that the sample provides no significant evidence that the proportion of all American adults going a week without paying cash is greater than 40%. That is, the proportion of all American adults going a week without paying cash is lesser than or equal to 40%.
The alternative hypothesis will now be that the sample provides evidence that the proportion of all American adults going a week without paying cash is greater than 40%.
Mathematically,
The null hypothesis is represented as
H₀: p ≤ 0.40
The alternative hypothesis is represented as
Hₐ: p > 0.40
To do this test, we will use the z-distribution because, the degree of freedom is large enough, it checks out.
So, we compute the z-test statistic
z = (x - μ)/σₓ
x = sample proportion = 0.43
μ = p₀ = standard to be tested against = 0.40
σₓ = standard error of the poll proportion = √[p(1-p)/n]
where n = Sample size = 1000
σₓ = √[0.43×0.57/1000] = 0.0156556699 = 0.0157
z = (0.43 - 0.40) ÷ 0.0157
z = 1.911 = 1.91
checking the tables for the p-value of this z-statistic
p-value (for z = 1.91, at 0.05 significance level, with a one tailed condition) = 0.028067
Note that it is one tailed because we're checking if the proportion is greater than a value; moving in only one direction.
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 = 5% = 0.05
p-value = 0.028067
0.028067 < 0.05
Hence,
p-value < significance level
This means that we reject the null hypothesis and accept the alternative hypothesis that the sample provides evidence that the proportion of all American adults going a week without paying cash is greater than 40%.
Hope this Helps!!