DA 5 - Confidence Intervals-1

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Oregon State University, Corvallis *

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Economics

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Jan 9, 2024

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Confidence Intervals In this data analysis, you will investigate confidence intervals to gain a deeper understanding of how and why they're constructed. You may find it useful to review the notes from week 5 before proceeding with this assignment. I f you haven’t already done so, work through the tutorial provided on the Data Analysis 5 Canvas page. Once you’ve worked through the tutorial, write up your responses to the questions listed throughout the tutorial. The same questions are included below to help you format your submissions. Submit a PDF copy of your responses to Gradescope by the deadline stated on Canvas. Part 1: Confidence Interval for a Proportion Question 1 (1 point) What proportion of the adults in your sample think climate change affects their local community? Hint: Just like we did with the population, we can calculate the proportion of those in this sample who think climate change affects their local community. 62% of the adults in my sample think climate change affects their local community. Question 2 (2 points) Would you expect another student’s sample proportion to be identical to yours? Would you expect it to be similar? Why or why not? I wouldn ’ t expect another’s student sample proportion to be comparable to mine, because I took a random sample from the population, and because it ’ s random each sample is going tio have a higher chance of being different. Question 3 (2 points) Check that the success-failure conditions needed to apply the Central Limit Theorem to the sample proportion are met in this case. Show your work. Question 4 (3 points) Suppose you instead wanted to construct a confidence interval with a different confidence level. How can you determine the necessary critical values in R for the following confidence levels? Your answers must include the R code used to find the critical value. a. A 95% confidence interval b. An 85% confidence interval

c. A 99% confidence interval Code: Answers: Question 5 (2 points) Using the point estimate from the sample of size n=180 you generated earlier, construct a 95% confidence interval for the true population proportion p. Show your work. Code: Answer: Question 6 (1 point) Does your confidence interval capture the true population proportion of US adults who think climate change affects their local community, p=0.62? My confidence interval does capture the TRUE population of US adults who think climate change affects their local community. Question 7 (1 point) There are 280 students enrolled in ST 314 this term. Each student should have gotten a slightly different 95% confidence interval due to random sampling. How many of 280 confidence intervals constructed do you expect to have captured the true population proportion p=0.62? Using the following reasoning, we can determine the number of confidence intervals that are anticipated to capture the true population proportion, p=0.62. A 95% confidence interval suggests that there is a 95% chance that it contains the true population parameter, so we would anticipate that 95% of the intervals will capture the true proportion. Thus, we would anticipate that about 0.95 × 280 intervals would capture the actual fraction out of 280 intervals.

Code: Answer: Question 8 (2 points) What proportion of your simulated intervals captured the true population parameter p=0.62? Is this proportion what you expected? Why or why not? A proportion that is near to 0.95 indicates that the simulation fits the theoretical model. The Law of Large Numbers implies that the proportion should converge to the predicted value after a sufficient number of simulations, therefore any deviations may be the result of random fluctuation Part 2: Confidence Interval for a Mean Question 9 (1 point) Start by calculating the point estimate for the parameter of interest. You can do this by wrapping the mean function around the appropriate variable from the salinity data set. You’ll need to use this value later on, so it is recommended that you store this value in the object called samp_mean. Report the value of point estimate, 𝑥 . Question 10 (2 points) We’re working towards constructing a confidence interval for the true population mean. To do this, we’ll need to estimate the standard error associated with the point estimate. Calculate this value. Hint: use the sd() function to determine the sample standard deviation, a necessary piece of the standard error estimate.

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Question 11 (2 points) Return to the problem of interest: estimating the average salinity for the entire Bimini Lagoon. Determine the appropriate critical value that will be used to construct a 90% confidence interval. Include the numerical value of the critical value and the R code you used to determine this value. Question 12 (2 points) Using your answers to questions 9, 10, and 11, construct the 90% confidence interval for the true average salinity for the entire Bimini Lagoon. Report the lower and upper bounds of the interval and show your work. Question 13 (2 points) Interpret the 90% confidence interval in the context of the problem. Between the lower and higher boundaries of the computed interval — the values found in Question 12 — we have 90% confidence that the real average salinity for the whole Bimini Lagoon lies. Gradescope Page Matching When you upload your PDF file to Gradescope, you will need to match each question on this assignment to the correct pages. Video instructions for doing this are available in the Start Here module on Canvas on the page “Submitting Assignments in Gradescope” . Failure to follow these instructions will result in a 2-point deduction on your assignment grade. Match this page to outline item “Gradescope Page Matching”.

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