STAT-2000---Assignment-2---Shell

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University of Manitoba *

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2000

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Statistics

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Feb 20, 2024

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Uploaded by huongnguyenthu307

STAT 2000 - Assignment 2 Thu Huong (Rosie) Nguyen 2023-04-12 Instructions To properly view the assignment questions, knit this file to .PDF and view the output. To enter your answers, add code as needed into the R code chunks given below, and, where applicable, replace the “Delete me; . . . ” and add in your own text response. Be sure when adding in text responses to never copy-paste symbols from outside of the document. Only use the symbols on your keyboard. Do not delete the question text, or modify any other part of the code except for the “author” in Line 3. All numerical and graphical answers must be done using R, unless stated otherwise. You will have a link in your email that takes you to the Crowdmark submission page. Once you have completed the worksheet, knit it to .PDF and upload your output to Crowdmark. Also, upload your .Rmd file to Crowdmark where prompted. To see where your .Rmd file is saved, click File > Save As in the top-left of your screen. Make sure you set your Name and Student Number in the Author section of this document (Line 3). Do not alter the title or the date. Please note that if you do not submit a knit .PDF file, you will be given a grade of zero. After you knit your assignment to PDF, check your code chunks. If your code at any point runs off the page, find the nearest comma, click to the right of it, and press Enter (or Return if you are on a Mac). This will force a break in the code so that it goes onto the next line. All of your code must be readable in the final submission. All calculations and output must be visible in the final document, and all text responses should be in complete English sentences. Your work should be done using the same formatting, functions, and packages as in your labs and course notes, unless otherwise specified. You may speak to your class mates about ideas and what functions/optional arguments you may need to use but you may not directly show your code/output to your classmates. Your full submission is due by 11:59 p.m. on Friday, March 3. Crowdmark may allow you to submit late, but you will be given an automatic grade of zero if you do. If you have an issue that you can’t resolve without someone looking at your work (e.g., you get an error when knitting your document), please see the Help Centre in 311 Machray Hall. Setup [1 mark] 0. Import the Temps dataset, available on the UMLearn page. Make sure you have “Heading” set to “Yes” when you import the data, and make sure you name the object Temps . [1 mark] Temps <- read.csv( "~/Downloads/Temps.csv" ) This dataset contains the CPU temperature information for a sample of 53 PC’s, each approximately three years old. Two readings are made: first, an initial temperature reading is made (after playing a 30 minute benchmark video). Then, the PC is partially disassembled, and a fresh amount of dose of thermal paste 1

is applied to the CPU. After this, another temperature reading is made (again after playing a 30 minute benchmark video). These are the Baseline and Paste variables, respectively. All measurements are made in degrees Celsius. Also, the cooling system is recorded (either “Air” or “Liquid”). This is the System variable. The line of code below will shuffle your data, and make the code unique to you. After importing the data, replace 1111111 with your seven-digit student id number in the set.seed function below, and click the green arrow at the top-right hand side of the code chunk. This part is not worth marks, but you will receive a 0 on your assignment if it is not completed correctly. set.seed( 7959133 ) if (exists( "Temps" )) { Temps$Baseline = round(Temps$Baseline + rnorm(NROW(Temps), 0 , 0.1 ), 1 ) Temps$Paste = round(Temps$Paste + rnorm(NROW(Temps), 0 , 0.1 ), 1 ) } Make sure you import your data and shuffle it (click the green arrow in the top-right) before beginning the assignment questions. Questions [24 marks] The first goal is to investigate whether the re-application of thermal paste reduces the temperature of the CPU. 1. Make a side-by-side boxplot comparing the Baseline measurements to the Paste measurements. Use the names argument to properly name each boxplot, and use the main and ylab arguments to set meaningful titles for the plot and the y-axis. [2 mark] boxplot(Temps$Baseline, Temps$Paste, names = c( "Baseline" , "Paste" ), main = "Comparation between Baseline and Paste" , ylab = "Temperature (Celcius)" , col = "pink" ) 2

Baseline Paste 50 55 60 Comparation between Baseline and Paste Temperature (Celcius) 2. Comment on what you see in the previous boxplot. Does it appear that the the reapplication of thermal paste results in a decrease in CPU temperature? Why? [2 marks] Based on the box plot above, we can see the median of the distribution of the Baseline is a bit higher than the median of the Paste. However, we can see that the sample mean of this data set is positive, indicating that it appear that the the reapplication of thermal paste results in a decrease in CPU temperature. 3. We wish to conduct a hypothesis test to see if the mean temperature is decreasing, at the 1% level of significance. Use TeX formatting to produce the hypotheses for this test. Assume that the differences are measured as d = Baseline − Paste. [1 mark] H 0 : µ d = 0 vs H a : µ d < 0 4. Use t.test to conduct the appropriate hypothesis test for this problem. [3 marks] t.test(Temps$Baseline, Temps$Paste, paired = TRUE, alternative = "less" , conf.level = 0.99 ) ## ## Paired t-test ## ## data: Temps$Baseline and Temps$Paste ## t = 3.6635, df = 52, p-value = 0.9997 ## alternative hypothesis: true mean difference is less than 0 3

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## 99 percent confidence interval: ## -Inf 0.8869225 ## sample estimates: ## mean difference ## 0.5358491 5. Give a fully worded conclusion to this test. [2 marks] Since the p-value is above the alpha, at the 1% level of significance, we fail to reject H0. We have insufficient evidence to conclude that the mean temperatures is decreasing. 6. Use t.test to produce a 99% confidence interval representing the mean decrease in CPU temperatures. [1 mark] t.test(Temps$Baseline, Temps$Paste, paired = TRUE, alternative = "two.sided" , conf.level = 0.99 ) ## ## Paired t-test ## ## data: Temps$Baseline and Temps$Paste ## t = 3.6635, df = 52, p-value = 0.0005837 ## alternative hypothesis: true mean difference is not equal to 0 ## 99 percent confidence interval: ## 0.1447703 0.9269278 ## sample estimates: ## mean difference ## 0.5358491 7. Could the confidence interval in Q6 have been used to conduct the hypothesis test in Q3-5? If so, what would the conclusion be, and why? If not, why not? [2 marks] Since the test is not two-sided test, and the confident level is not add up to 100%, the confidence interval in Q6 couldn’t be used to conduct the hypothesis test in Q3-5. We also wish to determine if Air cooling and Liquid cooling are significantly different, at the 1% level of significance. We will use the Baseline measurements for the following questions. 8. Make a side-by-side boxplot comparing the temperature readings for Air cooling systems to Liquid cooling systems. Use the main and ylab arguments to set meaningful titles for the plot and the y-axis. [1 mark] boxplot(Baseline ~ System, Paste ~ System, data = Temps, main = "Comparation between Air Cooling vs Liquid" , ylab = "Temperature" , col = "lightblue" ) 4

Air Liquid 50 55 60 Comparation between Air Cooling vs Liquid System Temperature 9. Use TeX formatting to produce the hypothesis statements for this test. [1 mark] H 0 : µ 1 = µ 2 vs H a : µ 1 ̸ = µ 2 10. Use aggregate to check the standard deviations of the baseline CPU temperatures, for each cooling type [1 mark] aggregate(Temps$Baseline ~ Temps$System, FUN = sd, data = Temps) ## Temps$System Temps$Baseline ## 1 Air 4.449125 ## 2 Liquid 3.668582 11. Use t.test to conduct the appropriate hypothesis test for this problem. [3 marks] t.test(Temps$Baseline ~ Temps$System, alternative = "two.sided" , var.equal = TRUE, data = Temps) ## ## Two Sample t-test ## ## data: Temps$Baseline by Temps$System ## t = 2.2256, df = 51, p-value = 0.03049 5

## alternative hypothesis: true difference in means between group Air and group Liquid is not equal to 0 ## 95 percent confidence interval: ## 0.2429851 4.7185092 ## sample estimates: ## mean in group Air mean in group Liquid ## 55.10833 52.62759 12. Give a fully worded conclusion to this test. [2 marks] As the p-value is above the level of significance, we fail to reject the null hypothesis. That is, we have insufficient evidence at this level of significance that the true mean temperatures differ for group Air and group Liquid. 13. Use t.test to produce a 99% confidence interval representing the difference in mean temperatures between Air and Liquid cooling systems. [1 mark] t.test(Temps$Baseline ~ Temps$System, alternative = "two.sided" , var.equal = TRUE, data = Temps, conf.l ## ## Two Sample t-test ## ## data: Temps$Baseline by Temps$System ## t = 2.2256, df = 51, p-value = 0.03049 ## alternative hypothesis: true difference in means between group Air and group Liquid is not equal to 0 ## 99 percent confidence interval: ## -0.5017584 5.4632527 ## sample estimates: ## mean in group Air mean in group Liquid ## 55.10833 52.62759 14. Could the confidence interval in Q13 have been used to conduct the hypothesis test in Q9-12? If so, what would the conclusion be, and why? If not, why not? [2 marks] Since this is a two-sided test, and since the confidence level (97%) and the level of significance (1%) add up to 100%, the interval in Q13 could be used to conduct the test in Q9-12. Because zero is include in the confident, we fail to reject the null hypothesis. That is, we have insufficient evidence at this level of significance that the true mean temperatures differ for group Air and group Liquid. 6

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STAT-2000---Assignment-2---Shell

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