Here is the text for the first question.
What is the population of interest for this survey?
The population we are interested in is americans.
Here is the text for the second question.
How many people were interviewed for this survey, i.e., what is the sample size?
They interviewed 2,024 adults aged 18 and over.
Here is the text for the third question.
What methods were used to gather information?
Gallup gathered information for this survey over the phone. 80% of respondents responded over cell phone and 20% responded over landline.
Here is the text for the fourth question.
According to the survey, what proportion of Americans in 2023 say religion is “very important” in their life?
45% say religion is “very important” in their life.
Here is the text for the fifth question.
Is this value a sample statistic or population parameter?
This value (0.45) is a sample statistic because it is a numerical proportion calculated from the sample, not the true population.
Here is the text for the sixth question.
Based on the collection technique for this Gallup poll, what is one potential source of bias?
Due to how the survey questions were worded and how there were some practical difficulties in conducting surveys, there is a chance of bias due to how these two factors can affect public opinion polls.
Here is the text for the seventh question.
Explain briefly to the researcher why the following statement is not correct: “According to this survey, 45% of all Americans feel religion is very important, and therefore less than 50% of all Americans feel religion is very important.”
Because 0.45 or 45% is a sample statistic drawn from a limited sample of 2,024 randomly selected respondants which indicates it’s not a population parameter, we can not conclude by just looking at the sample statistic that the true proportion of americans who view religion as “very important” is 45% and additionally that the true proportion is less than 50%. We would need to run a confidence interval to state this.
Here is the text for the eighth question.
Write out the conditions we need to check before we can use the results above. Do these conditions hold for these data? Explain why or why not.
Before using the sample statistic and standard error to create our normal distribution, we must verify that the Central Limit Theorem applies to our distribution. For the Central Limit Theorem to hold, all findings must be independent and the sample size must be sufficiently large so that our sample size times the sample statistic (np) is greater than 10 and that sample size times one minus the sample statistic is also greater than 10.
To begin, the survey results contributing to the sample statistic appear to be indepdent. Therefore, that condition is satisfied.
Additionally, our sample size (2,024) multiplied by the sample statistic (0.45) gives us a value of 910.8 which is greater than 10. Also, our sample size (2,024) multiplied by one minus the sample statistic (0.55) gives us a value of 1113.2 which is greater than 10. Therefore, the success-failure condition is satisfied and we now have the green light to move forward with our confidence interval.
Here is the text for our ninth question.
What is the standard error for the sampling distribution of 𝑃̂ ?
Utilizing the given standard error formula gives us a standard error of +/- 0.011 or +/- 1.1%. Therefore, our normal distribution is (0.45, .011).
This tells us that 68% of all the acceptable values for the sample statistic fall within 1.1% of 45% and that about 95% of acceptable values fall within 2.2% (1.1*2) of 45%.
Here is the text for the tenth question.
Based on properties of the normal distribution, 95% of all values of 𝑃̂ should be about how far away (±) from P?
Based on our standard error for the sample statistic being +/- 0.011 or +/- 1.1%, 95% of all acceptable values of 𝑃 should fall within 2.2% (1.1*2) of 45%.
Here is the text for the eleventh question.
Adapt the following code to construct a 95% confidence interval for the proportion of American adults who identified as believing religion is very important in 2023.
# The Lower Bound
0.45 - 0.022## [1] 0.428# The Upper Bound
0.45 + 0.022## [1] 0.472Here is the text for the twelfth question.
Interpret the CI from Question 11.
We are 95% confident that the true population proportion of american adults who believe religion is “very important” in their life is between 42.8% (0.428) and 47.2% (0.472).
Here is the text for the thirteenth question.
Use your confidence interval from Question 11 to answer the researcher: Does the evidence given in the article suggest that that less than 50% of all Americans feel religion is very important? Explain your answer.
According to a 95% confidence interval with a 0.05 significance level, we can conclude that the true proportion of american adults who believe religion is “very important” to their lives is less than 50%. This is because 50% falls outside of our range (0.428, 0.472) calculated by our 95% confidence interval. In other words, our range covers the middle 95% of our normal distribution with 0.428 making up the 2.5%ile and 0.472 making up the 97.5%ile and 0.5 falls well outside of our range making the probability of the true proportion of americans believing religion is “very important” in their lives being 0.50 or 50% less than the significance level. Therefore, assuming the population proportion is less than 50% is a plausible conclusion.
Here is the text for the fourteenth question.
Will a 97% confidence interval be narrower or wider than our 95% confidence interval? Explain.
It would be wider. Increasing the confidence interval also increases the range produced by the confidence interval as running a confidence interval of a greater confidence level means that we are expanding the calculated range of values so that we can be more sure that the true population proportion falls within the range of our calculated values.
Here is the text for the fifteenth question.
What is the critical value for a 97% confidence interval?
A 97% confidence interval gives us a significance level of 0.03. Therefore, our z/2 value would be 0.015 and (1-0.015) gives us 0.9850 which correlates to a z-score of 2.17. This z score of alpha divided by 2 is our critical value. Therefore, our critical value for the 97% confidence interval is 2.17.
Here is the text for the sixteenth question.
# The Lower Bound
0.45 - 0.024## [1] 0.426# The Upper Bound
0.45 + 0.024## [1] 0.474Construct and interpret a 97% confidence interval for the proportion of American adults who identified as believing religion is very important in 2023.
Using the critical value of 2.17 gives the confidence interval a margin of error of +/- 0.024 or +/- 2.4%. Therefore, our 97% confidence interval gives us a range of (0.426, 0.474).
Here is the text for the seventeenth question.
Use your confidence interval from Question 16 to answer the researcher: Does the evidence given in the article suggest that that less than 50% of all Americans feel religion is very important? Has your answer changed from Question 13? Did we expect it to?
Yes, after running a 95% and 97% confidence interval for the sample statistic given by the article, I believe it is safe to conclude that the true proportion of american adults who view relgion as “very important” to their life is less than 50% or 0.50. My answer has not changed from question 13 and I didn’t expect it to as a 95% confidence interval is already widely inclusive of acceptable values and it was more than reasonable to conclude from the 95% confidence that the true proportion was less than 0.50 or 50%. I did not believe bumping up the confidence level slightly would really change anything relating to whether or not the true proportion was less than 0.50 when 0.50 was already pretty far outside the range.
Here is the text for the eighteenth question.
Why don’t we always create 99% or 100% confidence intervals? Why do we bother with different confidence levels?
From the way I see it, constantly using an extremely high confidence level when running a confidence interval doesn’t allow for much precision and accuracy when you’re trying to prove to disprove a hypothesis or null hypothesis. Running a 100% confidence level assures you the true population proportion is within the range you’ve calculated from the value but it’s hard to attach any statistical significance to the range you’ve calculated because it is broad and not precise which makes it impossible to decide if a hypothesis is valid or not. Additionally, running a 99% or 100% confidence interval dramatically increases the likelihood of making a type 2 error.