Statistical Review Solutions: Normal Distribution, Confidence Intervals, Hypothesis Testin, Study notes of Statistics

Download Statistical Review Solutions: Normal Distribution, Confidence Intervals, Hypothesis Testin and more Study notes Statistics in PDF only on Docsity! 1 Solutions to review problems #1 Here are solutions to the …rst set of review problems: 4.77: The sampling distribution of y is normal with mean y = 60 and standard deviation y = = p n = 5= p 16 = 1:25: Since the distribution of y is normal, approximately 95% of the values of y should fall within y  1:96 y which is 60  1.96(1.25), giving the interval (57.55, 62.45). 5.41 a) A 99% con…dence interval for  is given by y t:005;14 s= p n = 31.47  2:977 (5:04)= p 15 = 31.47  3:87, which gives the interval (27.6, 35.34). b) The question is a bit unclear on exactly what a null hypothesis might be, but if we choose H0 :  = 35, then at the = :01 signi…cance level we will fail to reject H0, since the 99% con…dence interval includes  = 35: 11.65 a) Yes, the plotted points seem to follow a line. b) From the printout, byi = 12:51 + 35:83xi . 11.66 a) b2" = 1n2P(yi byi)2 =MSE = 1:069: b) From the printout, s:e:(b 1) = 6:96: c) For this research hypothesis, H0 : 1 = 0 and Ha : 1 > 0 since they are interested in detecting a positive relationship. The p value in the printout for H0 : 1 = 0 is p = :0004, but the printout is for the two-sided alternative hypothesis Ha : 1 6= 0: Thus, to get the p value for our one-sided alternative hypothesis we divide the printed p value by 2, yielding p = :0004=2 = :0002: 11.30 a) The plot looks good, there could be one or more in‡uential points. b) The estimated regression equation is byi = 99:78 + 51:92xi, and the residual standard deviation isb" = pMSE = p148:999 = 12:21: c) A 95% con…dence interval for 1 is given by b 1  t:025;28 s:e:(b 1); yielding 51.92  2:048 (:586) or 51.9  1:2, giving an interval of (50.7, 53.1). 11.31 a and b) From the printout, t = 88:53; with a p value of p < :0001: 11.32 a) F = 7837:26; and p < :0001: b) They are equal, because both tests are testing the same null hypothesis when we do simple linear regression. The test statistics are related by t2 = F: 1