Problem 5 (20points). The figure shows the average global temperature (relative
ID: 3324929 • Letter: P
Question
Problem 5 (20points). The figure shows the average global temperature (relative to the year 1921) from 1880 to 2005. Although it may seem 0.6 obvious from the figure that the 04 temperature is increasing at a higher 0.2 rate the last 35 years, many people0 believe that such an increase can be explained by random fluctuations, and that it is statistically insignificant. In order to prove or disprove these allegations you will estimate the slope of the regression line for a few samples and test its significance at the 95% confidence level. In the table below x represents the number of years elapsed from 1879 (so x=1 in 1880) and y is the average global temperature (relative to the year 1921) Giobal Temperature 02+-- .0.6 0.8 1880 900 920 1940 1960 190 2000 Year Period ST 1971-2005 3,815 1880-1970 4,186 1880-2005 8,001 9.87 21.7 11.83 1,180 693 486 419,405 255,346 674,751 6.78 9.35 16.13 35 91 126 a) test the hypotheses that global warming can be explained by statistical fluctuations b) Test the hypotheses that the global warming rate has increased in 1971-2005 (compared c) Provide a particular prediction (not the mean response) for the global temperature for the around a constant mean that does not grow in time. to the rate in 1880-1970), and that this increase is statistically significant. year 2010 (x-131), together with a 95% confidence interval for that prediction.Explanation / Answer
(a). Now b1 which is the slope of the linear regression line is given as under
b1 = Sum(XiYi) - Sum(XiYi)/n / Sum(Xi^2) - (Sum(Xi))^2/n
=1180 - 1180/35 / 419405 - 3815^2/35 = 0.008 for the period 1971 -2005
=-693 + 693/91 / 255346 - 4186^2/91 = -0.01 for period 1880-1970
=486-486/126 / 674751 - 8001^2/126 = 0.002 for period 1880 to 2005
The hypothesis that global warming can be explained by statistical fluctuations will formulated as
H0: b1 =0
Ha: b1 !=0
the t statistic t = b1/sb1 where sb1 =S/Sum(Xbar-Xi)^2 where is the std dev. of the sample
If the sum of area to the right and left of this test statistic under the std. normal curve = pvalue = < sig level chosen which is generally 0.05 then we can conclude the increase is not due to statistical fluctuations and vice versa
(b) The hypothesis that global warming has increased in 1971-2005 compared to 1880-1970 can forrmulated by comapring their slopes B1 and B1*
H0 : B1 > = B1*
Ha: B1<B1*
The rejection or acceptance of this null hypothesis can be done in a similar fashion to the steps outlined in the first part and whether it is statistically significant
(c) The regression eqn states Y = B0 + B1*X, Now we know B1 so we can calculate B0 and then we can find a Y for a particular X= 131 by sustituting X in the equation above.
At the 95% C.I. the interval can computed as b1 +- t(alpha/2)*Sb1 where alpha = 1 - 0.95 -0.05