Brittany Oliva UID 003933164 Stats 101a HW3 Problem 1 a) The 95% CI for the slope of the regression model is (9.514971e-1, 1.01266). 1 is plausible for the slope because it is included in our interval and more so because we would expect the current week’s gross income to be about the same or more as the previous week’s and thus having a slope of 1 would allow this to happen. b) When we test H0: B0 = 10000 against a two sided alternative, we get a p-value of 0.7517731 which tells us we must fail to reject the null hypothesis. c) CurrentWeek = B0 + B1*($400000) = (6.805e+03) + (9.821e+-1)*($400000) = $399645. From our 95% prediction interval, we can see that $450000 is not feasible in that it is way out of our prediction interval range of (359832.8, 439442.2) which is already wide enough as is since a prediction interval gives us a prediction for a single value. d) The promoter’s prediction rule can be correct if the slope is 1 and the intercept is 0, and looking at our confidence intervals for both numbers, we see that the intercept does include 0 and the slope does include 1. Given this information, their prediction rule is plausible. Problem 2 a) 95% CI for the slope of the regression model is (-4.163454, -0.3335853). There is evidence of a significant negative linear association because the entire interval for the slope of the regression model contains negative numbers. But we should be cautious of the linearity of this model. b) E ( Y | X =4) => PriceChange = B0 + B1 * (LoanPaymentsOverdue) = 4.5145 + (-2.2485)*(4) = -4.4795. The 95% CI for E ( Y | X =4) is (-6.648849, -2.310322) which shows that 0% is not a feasible value because it is not in the interval.Problem 3 a) The 95% CI for the startup time, B0, is: (calculations done in R the rest done by hand) b) When we test H0: B1 = 0.01 against a two sided alternative, we get a p-value of 1.87 which makes absolutely no sense in that there can never be a probability over 1. But if we take the –t value, since we already had a positive t value, then we get a better p-value of 0.1257517, which means we fail to reject the null that B1 = 0.01. c) Point estimate for 130 invoices is Time = B0 + B1*(Invoices) = .6417099 + .0112916 * (130) = 2.109618 hours. And the 95% prediction interval for the time taken to process 130 invoices is (1.422947, 2.7963) Problem 5 a) 95% Confidence interval for x=60 : (55.72647, 58.39702) 95% Prediction interval for x=60 : (41.9662, 72.1573) The confidence interval tells us we are 95% confident that if a group of students have a mean reading score of 60, the mean writing score for this group will fall in the confidence interval we calculated. The prediction interval tells us that 95% of the time when the reading score is 60, if an individual has a reading score of 60, his/her will have a writing score falling in our prediction interval. b) Equations for CI and PI:c) Standard error for 95% CI = 10.25294 and Standard error for 95% PI = 52.23 d) Standard error for the prediction is larger because predicting an outcome for a person is more uncertain than predicting an outcome for a group. e) Comparing the two confidence intervals we see that the prediction interval is wider because it tells you about the distribution in individual values and must account for more uncertainty. Problem 6 a) Slope = 0.6892 b) SE of the slope = 0.12 c) When testing the null that B1 = 0 we find ourselves in a position to reject the null of B1 = 0 once we compare the t value and t critical. We find t = 5.787 and t* = 2.02, so our t-value is larger than t* thus confirming we must reject the null because the true slope must be different from 0. d) Showing that F = t^2Problem 7 It is possible that 95% of the observations fall outside of the 95% CI, as depicted in the figures, if we happen to be looking at single points. The boundaries of the CI represent estimations which means they represent averages of groups. As N (the sample size) increases, the variance of the distribution narrows the distribution shape and has less capacity to hold the amount of observations in the sampling