ISYE-3030-Assignment-3

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Georgia Institute Of Technology *

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4031

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Statistics

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Feb 20, 2024

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ISYE 3030 Assignment 3 Lokranjan Lakshmikanthan 10/6/2021 Libraries library (tidyverse) Question 8.2.9 Part A Chapter8 = read.csv ( "ch08.csv" ) data = na.omit (Chapter8 $ EX. 8 . 2.9 ) ggplot ( as.data.frame (data), aes ( sample = data)) + stat_qq () + stat_qq_line () The data appears to be approximately normal. The data mostly falls on the linear line shown in the normal probability plot above. The deviations of the data points from the line
are small enough to assume normality however the population variance is not known. Below is the confidence interval made using T-scores. Part B sMean = mean (data) SD = sd (data) n = length (data) t = qt ( 0.975 , n -1 ) TwoSidedError = t * SD / sqrt (n) Upper = sMean + TwoSidedError Lower = sMean - TwoSidedError Upper ## [1] 2282.516 Lower ## [1] 2237.317 The two-sided 95% confidence interval on the mean strength is (2237.317 <= True Mean <= 2282.516) Part C OneSidedError = qt ( 0.95 , n -1 ) * SD / sqrt (n) Lowerz = sMean - OneSidedError Lowerz ## [1] 2241.477 The one-sided lower 95% confidence interval on the mean strength is (2241.477 <= True Mean). The lower bound of the two-sided interval is less than the lower bound for the one sided interval. This is since the alpha of the one-sided interval is larger and results in a lower T-score. The lower T-score reduces the error on the one-sided interval. Question 8.3.3 data2 = as.numeric (Chapter8 $ EX. 8 . 3 . 3.1 [ 2 : 9 ]) ggplot ( as.data.frame (data2), aes ( sample = data2)) + stat_qq () + stat_qq_line ()
Data from the Average Mean Temperature of the 8 sites appears to be normally distributed. The normal probability plot has few extreme deviations from the line making it reasonable to assume a normal distribution. SQU = qchisq ( 0.975 , 7 ) SQL = qchisq ( 0.025 , 7 ) V1 = var (data2) n1 = length (data2) - 1 LowerC = sqrt ((n1 * V1) / SQU) UpperC = sqrt ((n1 * V1) / SQL) LowerC ## [1] 1.402562 UpperC ## [1] 4.317464 By assuming a normal distribution for the data we can arrive at a 95% two-sided confidence interval for the standard deviation over the sites: (1.402562 <= True Standard Deviation <= 4.317464).
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Question 8.3.6 TumorAppearanceTimes = na.omit (Chapter8 $ EX. 8 . 3.6 ) ggplot ( as.data.frame (TumorAppearanceTimes), aes ( sample = TumorAppearanceTimes)) + stat_qq () + stat_qq_line () The data does not appear to be normally distributed. It does not follow a linear trend on the normal probability plot. n2 = length (TumorAppearanceTimes) - 1 CHSQU_Tumor = qchisq ( 0.975 ,n2) CHSQL_Tumor = qchisq ( 0.025 ,n2) V_Tumor = var (TumorAppearanceTimes) Upper_Tumor = sqrt ( (n2 * V_Tumor) / CHSQL_Tumor ) Lower_Tumor = sqrt ( (n2 * V_Tumor) / CHSQU_Tumor ) Upper_Tumor ## [1] 20.46307 Lower_Tumor
## [1] 13.13045 If the data did have a normal distribution, the 95% two-sided confidence interval for the standard deviation over the sites would be (13.13045 <= True SD <= 20.46307). Question 8.4.3 Part A n_r = 30 phat = 12 / n_r Error_r = qnorm ( 0.975 ) * sqrt ( phat * ( 1 - phat) / n_r ) Upper_r = phat + Error_r Lower_r = phat - Error_r Upper_r ## [1] 0.5753045 Lower_r ## [1] 0.2246955 The 95% two-sided confidence interval for the true proportion of underweight rats is (0.2247 <= p <= 0.5753). We can assume normality due to the large n. Part B Required_SampleSize = phat * ( 1 - phat) / ( 0.02 / qnorm ( 0.975 )) ^ 2 Required_SampleSize ## [1] 2304.875 A sample size greater than or equal to 2305 is needed to keep the error less than 0.02. Part C n = ( 1.96 / 0.02 ) ^ 2 n = n * . 25 n ## [1] 2401 n >= 2401 Question 8.4.6 Part A n_seeds = 200 phat_seeds = 180 / n_seeds Error_seeds = qnorm ( 0.975 ) * sqrt ( phat_seeds * ( 1 - phat_seeds) /
n_seeds ) Upper_seeds = phat_seeds + Error_seeds Lower_seeds = phat_seeds - Error_seeds Upper_seeds ## [1] 0.9415771 Lower_seeds ## [1] 0.8584229 Since N is large enough to assume a normal distribution, the 95% two-sided confidence interval for the true proportion of seeds that germinate is (0.8584 <= p <= 0.9416). Part B Yes, the confidence interval we constructed provides evidence that the claim made by the packet of seeds is reasonable. The true proportion of 93% falls in between the confidence interval we constructed.
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