We are going to stick with the same data as the last lab. In December of 2023, the Gallup organization released an article call “How Religious Are Americans”. Specifically, this article claims that: • 45% of Americans say religion is very important. • 26% of Americans say that religion is fairly important. • 28% of Americans say religion is not important.
Here is the text for the first question.
You will notice that these three values do not add up to 100%. Why do you think that happened?
I believe that this occurred due to Gallup rounding their percentages which could have caused these three proportions to add to 99% instead of 100%.
Here is the text for the second question.
State the hypotheses our researcher has asked us to test.
\[H_0: p = .20 H_a:\] p > .20
Here is the text for the third question.
Does the CLT hold for these data? Explain why or why not?
Before we compute standard error, we must check if the Central Limit Theorem applies. First, since we are told this is a random sample, we know that sample results are independent. Therefore, the independence condition is satisfied.
Next, we must test the success-failure condition. It is shown the n (sample size) is equal to 2,024 people and that p-hat (sample proportion) is equal to 0.28. Therefore, our success-failure test is as follows: np greater than/ equal to 10 => (2024)(0.28) = 566.72 > 10 n(1-p) => (2024)(1-0.28) => (2024)(0.72) = 1,457.28 > 10
With both the independence and success-failure conditions met, it is verified that the Central Limit Theorem applies. Therefore, the distribution of sample proportions should be approximately normal.
Here is the text for the fourth question.
Write out the two pieces of information you need for Step 2. Hint: Recall from the last lab that the sample size is 2024 people.
We already know that the sample proportion is 0.28. Therefore, we have the first value.
To find the standard error, we utilize our standard error formula for hypothesis tests using the null population proportion (p) of 0.20 and the sample size (n) of 2024. Using the formula should give us a SE of approximately +/- 0.0089.
Therefore: Sample Proportion (p-hat): 0.28 Standard Error: +/- 0.0089
Here is the text for the fifth question.
Compute and Interpret z. Make sure to show your work.
For this, we divide the difference between the sample proportion (p-hat) and the null population proportion (p) by the standard error calculated in the previous question.
This gives us a z-score of approximately 8.99. This means that the sample proportion of 0.28, given the population proportion of 0.20, is approximately 8.99 standard errors (8.99 * 0.0089) away from the population proportion.
Here is the text for the sixth question.
Suppose z = 3. In a normal distribution with mean 0 and standard deviation 1, what is the probability of getting a value greater than or equal to 3? In other words, what is P(z >/= 3)?
Given that a z score of 3 correlates to the 99.87 percentile. Therefore, the chance of obtaining a z-score value greater than or equal to 3 is 0.0013 or 0.13%. Not very likely…
Here is the text for the seventh question.
If the null hypothesis is correct, what is the probability of getting a test statistic as or even more extreme as the value of z you computed in Question 5?
P(z >/= 8.99 | p = 0.20) => P < 0.00001 => It is very unlikely that we would get a sample proportion of 0.28 if the true population proportion is 0.20.
Here is the text for the eighth question.
Do these data provide evidence that the true percent of Americans who believe religion is not very important is more than 20%?
Yes, given that our p-value from the hypothesis test is significantly smaller than a typical significance level of 0.05 or 0.01, given that our sample proportion is 0.28, there is sufficient evidence that the population proportion is greater than 0.2 or 20%.
Here is the text for the ninth question.
Build and interpret a 95% confidence interval for the proportion of Americans who believe that religion is not very important.
We have already confirmed that the sample results are indepdent and that the success-failure conditions are satisfied for this sample. Therefore, we know the Central Limit Theorem holds.
Therefore, we need to find the critical value for the interval equation. Given this is a 95% confidence interval, we have a .05 significance level and our alpha/2 value is 0.025. This corresponds with the upper tail of the distribution starting at the .9750 percentile which corresponds to a z-score of 1.96. Therefore, our critical value is 1.96.
Next, to find margin of error for the confidence interval, we multiply the critical value (1.96) times the standard error (0.0089). This gives us a margin of error of +/- 0.017. To construct our confidence interval, we add and subtract the margin of error to our sample proportion 0.28. This gives us values of .263 and .297.
Our confidence interval is as follows: (.263, .297).
We are 95% confident that the true proportion of Americans who believe that religion is not important to their lives is between .263 or 26.3% and .297 or 29.7%.
Here is the text for the tenth question.
Respond to our client’s research question using the confidence interval: Do these data provide evidence that the true percent of Americans who believe religion is not very important is more than 20%? Explain your reasoning.
Yes, given the results of both our hypothesis test and confidence interval, it is very, very unlikely that the true proportion of Americans who believe religion is not important to their life is 0.20 or 20%. In our hypothesis test, the sample proportion corresponded to a p-value of less than 0.00001 which means that it is very unlikely, if the population proportion is 0.20, that we would get a sample proportion of 0.28. Additionally, the lower tail of our Confidence Interval (.263) was more than 6 percentage points above 0.20 with a margin of error of .017 meaning that 0.20 falls well outside the confidence interval.
Here is the text for the eleventh question.
Did you get the same answer in Question 8 and Question 10?
Yes, both show that the population proportion is very likely to be 0.20.