Question

Question 3: The following data set contains the winning times (in seconds) of the 22 men's 200 meter Olympic sprints held between 1900 and 1996. (Notice that the Olympics were not held during the World War I and II years.) Year Men200m 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 A statistician is attempting to model the relationship in the provided scatter plot using 21.6 22.6 21.7 21.6 E 21.8 21.2 S 20.7 21.1 20.7 20.6 20.5 20.3 1900 1920 1940 1960 1980 mens200m$Year a) Provide the least-squares estimate of the model given above 19.83 20.19 19.8 19.75 20.01 19.32 b) Find the coefficient of determination and interpret its meaning in the context of the data. [2] c) Is there statistical evidence to the Men's 200m Olympic winning time can expressed as a negative linear function of the year? Why or why not? State the appropriate null and alternative hypothesis, find the P-value, and (i) HO state a decision/conclusion 12] HA (ii) Test Statistic- iii) P-value- (iv) Decision and conclusion Construct a 95% confidence interval estimate for β-In addition, interpret its meaning in the context of this data d)

Answer 1

a) The step-by-step instructions to obtain a simple linear regression output using Excel2010 * Store the variable Year(X) and Men200m (Y) in the columns of the Exce12010 worksheet Choose the Stat-Data ->Data Analysis >Regression Select the range of the variable Y and then select the range of variable X * Choose the confidence level is 95% * Tick the labels. xClick ok. The simple linear regression output using Excel2010

SUMMARY OUTPUT Rearession Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.948075218 0.89884662 0.893788951 0.298134436 ANOVA Ms Significance F Regression Residual Total 1 15.79644898 15.79644898 177.71952.07378E-11 20 1.7776828380.088884142 21 17.57413182 Coefficients P-value Lower 95% Uper 95% Lower 95.0% Upper 95.0% 76.153369314.15222611718.34037145 5.61E-14 67.49197741 84.81476122 67.49197741 84.81476122 0.028383313 0.002129097 13.33114931 2.07E-11 -0.032824532 -0.023942094 -0.03282453 -0.023942094 Standard Error t Stat Intercept Year

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, The simple linear regressi on least-squares equation of the Exce12010 output is Men200m-A+AYear Men200m 76.15336931-0.028383313Year. Men200m 76.1534-0.0283Year (b) The values of SSR- 15.79644898, and SST 17.57413182 of the Excel2010 outputis The formula for calculating the coefficient of determination is Sum of squares due to regres sion Total sum of squares SSR SST 15.79644898 17.57413182 0.89884662 R2 0 8988 The interpretation of the coefficient of determination,R The value of the coefficient of determination is R2 29,88% of the variation in the dependent variable Men200m (Y) can be explained by the independent variable Year (X)

(c) Step1: To, test whether the populati on slope (B) is significant or not. The null and alternative hypotheses are as follows Ho : A = 0 (There is no linear relationship) H, -0 (There is linear relationship) Step2: The level of significance is c 0.05 Step3: The values of hi ands are: 0028383313s 0.002129097 The formula for calculating the t- test statistic is 6-A 0.028383313-0 0.002129097 -0.028383313 0.002129097 13.33114931 t13.3311 Step4: If t 2.085963447. Reject the null hypothesis Ho Otherwise, we do not reject Ho Step5. Conclusion : Since, the P-value = 0.000000000021 < α= 0.05 We reject the null hypothesis Ho Since, the t-value is t 13.3311>t0.05, 2.085963447. We reject the null hypothesis Ho Hence, there is sufficient provide sufficient evidence to conclude that the independent vari able Year(X) is a significant predictor of the dependent variable Men's 200m(Y)

(d) The 95% confidence interval for the slope of the model. The coefficients of the param eters of the simple linear regression output of the Exce12010 is given below The values of b,S^ are 0.028383313 and S, -0.002129097 of the Excel2010 -0.028383313tps-2 (0.002129097) -0.028383313t0 0.002129097) -0.0283833132.085963447 0.002129097) -0.0283833132.085963447 0.002129097) -0.028383313t 0.004441219 → (-0023942094,-0032824532) A tW2.-2SA-1-0. 02394S A s-0.03282| The interpretation of the confi dence interval of the slope coefficient, Fear: we can be 99% confident that the population slope coefficient Year (x), Ais in the interval [-0.03282,-0.02394]