Understanding Confidence Intervals: STAT 3090 Lab Activity Guide

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 N AME : Megan McCauley O BJECTIVES : Upon completion of this assignment, you should be able to:  Identify and verify the conditions needed to perform a confidence interval  Use the t-distribution to calculate probabilities and to find critical values  Calculate and interpret a confidence interval for a proportion, for the difference in two proportions and a mean.  Understand that confidence intervals vary from sample to sample and may miss the true population parameter  Realize how the changing sample size or the confidence level affects the width of a confidence interval  Determine the minimum sample size needed for estimating a population parameter Part 1 (27 points): 1. Contact Lenses (17 Points) Delectable Delights is evaluating the vision benefits it offers to its employees. They wonder if they should add some additional benefits to assist with the purchase of contact lenses. According to the Vision Council of America (VCA), 11% of American adults wear contact lenses. Delectable Delights wonders if the percentage of its employees who wear contact lenses is about the same as the percentage reported by VCA. Ken Ronald from Human Resources asks you to take a sample of size 150 from the employees at Delectable Delights and calculate a 95% confidence interval for the proportion of employees who wear contact lenses. D IRECTIONS : Please provide any appropriate output and/or screenshots from JMP. Instructions for creating several types of graphs or tables and statistics can be found on Canvas in the file JMP Instructions.docx . Paste your answers and any output into this document. (a) Define the parameter of interest, p , in the context of the problem. ( 2 points) p = the true proportion of Delectable Delights employees who wear contact lenses (b) To select the sample, you will use a JMP applet found by selecting Help >> Sample Index >> Teaching Scripts >> Interactive Teaching Modules >> Sampling Distribution of Sample Proportion . 1

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 Open the Sampling Distribution of Sample Proportion applet. In the Population Characteristics section, change the population proportion to 0.11 and the category name to Contacts . Under Demo Characteristics , change the sample size to 150 and the number of samples to 1. Then click 'Draw Additional Samples' to draw 1 random samples of size n = 150. JMP will calculate the sample proportion of employees who wear contacts for you in the Sample Summary Table . Place the results from the Sample Summary Table from this one sample below. Be careful when reading the table as JMP sometimes puts the Row headings one row below the data. (3 points) (c) According to your output what is the value of the sample proportion of employees who wear contact lenses, p ? (1 points) p = 0.11 (d) Have the conditions for calculating a confidence interval for a population proportion been met? Explain. Note – if they do not meet the conditions, repeat part (b) and take another sample. (2 points) - Randomly selected sample - n p  5 = 150(0.11) = 17 - n(1-p) = 134 Yes, the conditions are met. 2

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 (e) Use the statistic you reported in part (c) to calculate a 95% confidence interval for the proportion of employees who wear contact lenses. Show your work by filling in the correct values into the formula. Round the lower and upper bounds of the interval to 2 decimal places. (3 points) (f) Interpret the 95% confidence interval you found in part (e) for Ken. (2 points) We are 95% sure that the proportion of employees who wear contact lenses is between 0.06 and 0.16. (g) Does your confidence interval contain the 11% that was reported by the VCA? (1 pts) Yes, 11% falls within the confidence interval range. (h) There are approximately 1300 students taking STAT 3090. Approximately what proportion or percentage of them will calculate a 95% confidence interval in part (e) that does not contain the value 11%? Note: This is an example of the interpretation of the Confidence Level found in your lecture guide. (1 pts) ~5% of the confidence intervals will not contain the 11%. 3

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STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 (i) Would the margin of error for a 95% confidence interval increase or decrease if we increased the sample size to 600 (assume the value of p is the same as part (c))? Why? (2 pts) The margin of error would decrease because the increasing sample size decreases the confidence interval. 2. WebAssign questions 1,3 (10 points) Part 2 (29 points): 3. Medical malpractice lawsuits (10 points) An insurance company wants to develop a better understanding of its claims paid out for medical malpractice lawsuits. The file medicalmalpractice.jmp contains information 110 claim payments made. We will study two variables described below: Private Attorney Whether the claimant was represented by a private attorney Gender Patient Gender Let p 1 be the proportion of male claimants with a private attorney. Let p 2 be the proportion of female claimants with a private attorney. a) Create a contingency table using JMP. Step 1: Open the file medicalmalpractice.jmp . Select Analyze >> Fit Y by X . 4

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 Step 2: Select Private Attorney and click Y, Response . Select Gender and click X, Factor . Click OK . Step 3: Copy and paste your Contingency Table below. You can use screenshot. (3 points) 5

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 b) Using the table found in a), what is the sample proportion p 1 of male claimants with a private attorney? What is the sample proportion p 2 of female claimants with a private attorney? (2 points) p 1 = 0.70 p 2 = 0.65 c) Calculate a 99% confidence interval for the difference of the proportions of claimants with a private attorney between genders. Set up the difference to be p 1 − p 2 . Show your work by filling in the correct values into the formula. Round the lower and upper bounds of the interval to 2 decimal places. (3 points) 6

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STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 d) Interpret the confidence interval you found in part (c). (2 points) We are 99% confident that the difference in the difference of the true proportion of all claimants with a private attorney between genders is lies within -0.18 and 0.28. 4. Inspections (7 points) The department of code enforcement of a county government issues permits to general contractors to work on residential projects. For each permit issued, the department inspects the result of the project and gives a “pass” or “fail” rating. A failed project must be re-inspected until it receives a pass rating. The department had been frustrated by the high cost of re-inspection and decided to publish the inspection records of all contractors on the web. It was hoped that public access to the records would lower the re-inspection rate. A year after the web access was made public, two samples of records were randomly selected. One sample was selected from the pool of records before the web publication and one after. The proportion of projects that passed on the first inspection was noted for each sample. The results are summarized below. Let p 1 be the population of contractors’ records prior to the web publication. Let p 2 be the population of contractors’ records after the web publication. No public web access n 1 = 500 p 1 = 0.67 Public web access n 2 = 100 p 2 = 0.80 7

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 a) Calculate a 95% confidence interval for the difference in the passing rate on first inspection between the two time periods. Set up the difference to be p 1 − p 2 . Round the lower and upper bounds of the interval to 2 decimal places. Remember to check the conditions for calculating a confidence interval for the difference in two proportions before calculating your confidence interval. (3 points) b) Interpret the confidence interval you found in part (a). (2 points) We are 95% confident that the population of contractors’ records after the web publication is greater than the population of contractors’ records before the web publication by 0.22. c) Using the results of your confidence interval for the difference in proportions, do you believe that publishing the inspection records of all contractors has resulted in an increase in the proportion of projects that pass on the first inspection? (2 points) 8

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 Yes. 5. WebAssign questions 2,4,5 – 7 (12 pts) Part 3 (44 points): 6. Noodle Dilemma (AKA the Too Long Spell) (14 Points) The process that makes the spaghetti noodles for Delectable Delights is supposed to produce noodles with an average length of 252.5 mm. Allison, the Quality Control Manager, is concerned that Machine #13 is not working properly and the noodles it produces are the wrong length (they seem to be too long). You are asked to calculate a 90% confidence interval for the average length of Machine #13’s noodles using the results of a random sample of 25 noodles. The measurements from these 25 noodles can be found in the file The Too Long Spell.jmp (a) Define the parameter of interest, μ , in the context of the problem. (2 pts) μ = the mean length of Machine 13’s noodles (b) Have the conditions for calculating a confidence interval for a population mean been met? Be careful. Your sample size is less than 30 thus you will need another way to verify whether or not you can consider the sampling distribution for the sample mean to be approximately normally distributed. Review page 185 in the lecture guide. The instructions on how to do this in JMP is found in the file JMP Normal 9

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STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 Probability Plot Instructions . When you answer this question, include the graphics provided by JMP. (5 pts) (c) The output from part (b) includes a Statistics Summary Table which reports a 95% confidence interval for the average length of Machine #13’s noodles. To change this to a 90% confidence interval select the red triangle next to the Summary Statistics Table and select Customize Summary Statistics. At the bottom of the list in the drop-down menu change the confidence level from .95 to .90. Copy and paste the resulting Summary Statistics table showing the 90% confidence interval. (3 pts) (d) Interpret the confidence interval you found in part (c). Round the lower and upper bounds of the interval to 1 decimal place. (2 pts) 10

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 We are 90% sure that the mean length of Machine 13’s noodles is between 253.2 and 255.3. (e) Using the confidence interval that you found, do you believe that Machine #13 is making noodles that have an average length different than the 252.5mm (do they appear too long)? (2 pts) Yes, Machine 13 is making noodles with an average length that’s longer than the 252.5mm and probably appear too long. Now there is something the employees of Delectable Delights do not know . Arthur Weasley of the Ministry of Magic has twin sons, Fred and George, who are working on perfecting the Too Long Spell for their joke shop. The spell would make things, well, too long. They wanted to test it out and used the spell on Machine #13 to see if the Too Long Spell would cause the machine to make the spaghetti noodles too long. It was a success! To be continued … 7. Cheerio Box Weights (10 Points) A cereal manufacturer claims that their machine, which fills boxes of Cheerios, is calibrated to achieve an average box weight of 20.5 oz, with a standard deviation of 0.4 oz. To verify the manufacturer's claim, a random sample of boxes filled by this machine is selected. The task is to calculate a 99% confidence interval for the true mean weight of the Cheerios boxes filled by this machine. The summary statistics from this sample are shown below. It should be noted that the manufacturing processes are generally known to yield data that approximates a normal distribution. 11

STAT 3090 L AB 8 C ONFIDENCE I NTERVALS S PRING 2024 (a) Define the parameter of interest in the context of the problem. (2 pts) The parameter of interest is the true mean weight of the Cheerio boxes filled by the machine. (b) State and verify the conditions for the confidence interval. (2 pts) - Random sample - Sample size larger than 30 - (c) Use the information given in the Summary Statistics Table to compute the 99% confidence interval for the true mean weight of the Cheerios boxes filled by this machine. Use two decimal places in the calculation process and the final result. (3 pts) Lower bound: 20.36 Upper bound: 20.55 (d) Interpret the confidence interval. (2 pts) We’re 99% confident the mean weight of all Cheerio boxes filled by the machine lies between 20.36 and 20.55. (e) Using the confidence interval that you found, do you believe the manufacturer's claim that the machine is filling boxes with the mean weight of 20.5 oz? (1 pt) Yes 8. WebAssign questions 8-11 (20 pts) 12

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Section 8 Lab Activity - Megan McCauley

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