Statistic assignment !!!

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Institute of Health Profession Education & Research, KMU Peshawar *

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Statistics

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Nov 24, 2024

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1. The number of Wildland fires (in thousands) and wildland acres burned (in millions) in the US for eight years is in the table below, Fires (x) 84.1 73.5 63.6 65.5 66.8 96.4 85.7 79.0 Acres (y) 3.6 7.2 4.0 8.1 8.7 9.9 9.3 5.3 a. Calculate the sample correlation coefficient, r. Round to the nearest 1000 th . Fires (X) Acres (Y) XY X 2 Y 2 84.1 3.6 302.76 7072.81 12.96 73.5 7.2 529.2 5402.25 51.84 63.6 4 254.4 4044.96 16 65.5 8.1 530.55 4290.25 65.61 66.8 8.7 581.16 4462.24 75.69 96.4 9.9 954.36 9292.96 98.01 85.7 9.3 797.01 7344.49 86.49 79 5.3 418.7 6241 28.09 614.6 56.1 4368.14 48150.96 434.69 𝑟 = 𝑛 ∑ ?? − ∑ ? ∑ ? √(𝑛 ∑ ? 2 − (∑ ?) 2 )(𝑛 ∑ ? 2 − (∑ ?) 2 ) 𝑟 = (8 ∗ 4368.14) − (614.6 ∗ 56.1) √((8 ∗ 48150.96) − (614.6) 2 )((8 ∗ 434.69) − (56.1) 2 ) 𝑟 = 0.297

b. Describe the type of correlation coefficient and interpret the correlation in the context of the data. The correlation coefficient ( r ) of 0.297 reveals a positive correlation between the number of wildland fires and the wildland acres burned in the context of the provided data. This implies that, on average, as the number of wildland fires increases, there is a tendency for a rise in the wildland acres burned. 2. The numbers of pass attempts and passing yards for seven professional quarterback for a recent year are listed in the table below. Round all answers to the nearest 1000 th . Pass attempts (x) 449 565 528 197 670 351 218 Passing Yards (y) 3265 4018 3669 1141 5177 2362 1737 a. Calculate the sample correlation coefficient, r. Pass attempts (X) Passing Yards (Y) XY X 2 Y 2 449 3265 1465985 201601 10660225 565 4018 2270170 319225 16144324 528 3669 1937232 278784 13461561 197 1141 224777 38809 1301881 670 5177 3468590 448900 26801329 351 2362 829062 123201 5579044 218 1737 378666 47524 3017169

2978 21369 10574482 1458044 76965533 𝑟 = 𝑛 ∑ ?? − ∑ ? ∑ ? √(𝑛 ∑ ? 2 − (∑ ?) 2 )(𝑛 ∑ ? 2 − (∑ ?) 2 ) 𝑟 = (7 ∗ 10574482) − (2978 ∗ 21369) √((7 ∗ 1458044) − (2978) 2 )((7 ∗ 76965533) − (21369) 2 ) 𝑟 = 0.991 b. Describe the type of correlation coefficient and interpret the correlation in the context of the data. The correlation coefficient of 0.991 between pass attempts and passing yards for the seven professional quarterbacks indicates an exceptionally strong positive correlation. This implies that as the number of pass attempts increases, there is a robust tendency for passing yards to also increase. c. Test the significance of the correlation coefficient, r, that you found in part a. Use α=.05. Is there enough evidence to conclude that there is a significant linear relationship between the data? 𝐻 𝑜 : 𝜌 = 0 𝐻 𝑎 : 𝜌 ≠ 0 Test statistics:

𝑡 = 𝑟 ∗ √ 𝑛 − 2 1 − 𝑟 2 𝑡 = 0.991 ∗ √ 7 − 2 1 − 0.991 2 𝑡 = 16.28 Degrees of freedom =n-2= 7-2 =5 The P-value for 2-tailed test using Excel =T.DIST.2T(16.28,5)=0 Since the P-value (0) is less than alpha(0.05) we reject the null hypothesis. Since the null hypothesis is rejected, the correlation is significant d. Find the equation of the regression line for the data. ? = 𝑛 ∑ ?? − ∑ ? ∑ ? (𝑛 ∑ ? 2 − (∑ ?) 2 ? = (7 ∗ 10574482) − (2978 ∗ 21369) (7 ∗ 1458044) − (2978) 2 ? = 7.762 a = ? ̅ − ??̅ ? = ( 21369 7 ) − (7.762 ∗ 2978 7 ) a= -249.46 Equation of the regression: y= 7.762x – 249.46 e. Use the regression equation to predict the average number passing yards if the pass attempts are 250.

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