sleep-data x1 x2 condition y 3.3 7.72 sleep 8.42 2.23 12.88 sleep 9.83 6.01 11.95 sleep 8.78 2.21 15.48 sleep 10.8 6.16 5.68 sleep 7.62 6.44 6.99 sleep 13.57 3.86 11.74 wake 7.79 3.2 9.78 wake 7.56 6.35 13.99 wake 7.98 6.44 2.56 2.78 5.76 wake 5.2 wake 6.9 5.04 11.02 wake 7.81 4) Researchers design an experiment to study the impact of sleep on motor learning. The researchers hypothesize that sleep is required for memory consolidation to maintain the benefits of training. In this experiment, mice are trained to run on a rotating wheel with unevenly spaced bars. The speed of the rotating wheel gradually increases over time until the mouse can no longer perform the task. The speed of the wheel (in rotations per minute, rpm) when the mouse fails the task (falls off the wheel) is recorded as a measure of task performance. There are three measurements taken during this study: x1 is the baseline performance of a mouse before training, 22 is the performance of the mouse at the end of a training session, and y is the performance of the mouse after either 8 hours of sleep or wake. For each mouse, the dataframe indicates whether it was in the sleep or wake condition. a) [2] Create a scatter plot of the performance after sleep/wake (y) vs. the performance at the end of the training session (22). Use a different colour and symbol for the sleep and wake mice (you may want to use the points() function in R). b) Using only the mice in the sleep cond on, fit a multiple linear regression model for y with inputs 21 and 22. Compute a 95% confidence interval for the average value of y for a mouse who was able to run until 5.4 rpm at baseline and 9.8 rpm at the end of the training session. c) [2] Using only the mice in the wake condition, fit a multiple linear regression model for y with inputs x1 and 22. Compute a 95% confidence interval for the average value of y for a mouse who was able to run until 5.4 rpm at baseline and 9.8 rpm at the end of the training session. d) [1] Based on the results of (b) and (c), what conclusions would you draw about the research hypothesis? e) [1] Compute the p-value for testing the significance of the regression in (c). sleep-data x1 x2 condition y 3.3 7.72 sleep 8.42 2.23 12.88 sleep 9.83 6.01 11.95 sleep 8.78 2.21 15.48 sleep 10.8 6.16 5.68 sleep 7.62 6.44 6.99 sleep 13.57 3.86 11.74 wake 7.79 3.2 9.78 wake 7.56 6.35 13.99 wake 7.98 6.44 2.56 2.78 5.76 wake 5.2 wake 6.9 5.04 11.02 wake 7.81 4) Researchers design an experiment to study the impact of sleep on motor learning. The researchers hypothesize that sleep is required for memory consolidation to maintain the benefits of training. In this experiment, mice are trained to run on a rotating wheel with unevenly spaced bars. The speed of the rotating wheel gradually increases over time until the mouse can no longer perform the task. The speed of the wheel (in rotations per minute, rpm) when the mouse fails the task (falls off the wheel) is recorded as a measure of task performance. There are three measurements taken during this study: x1 is the baseline performance of a mouse before training, 22 is the performance of the mouse at the end of a training session, and y is the performance of the mouse after either 8 hours of sleep or wake. For each mouse, the dataframe indicates whether it was in the sleep or wake condition. a) [2] Create a scatter plot of the performance after sleep/wake (y) vs. the performance at the end of the training session (22). Use a different colour and symbol for the sleep and wake mice (you may want to use the points() function in R). b) Using only the mice in the sleep cond on, fit a multiple linear regression model for y with inputs 21 and 22. Compute a 95% confidence interval for the average value of y for a mouse who was able to run until 5.4 rpm at baseline and 9.8 rpm at the end of the training session. c) [2] Using only the mice in the wake condition, fit a multiple linear regression model for y with inputs x1 and 22. Compute a 95% confidence interval for the average value of y for a mouse who was able to run until 5.4 rpm at baseline and 9.8 rpm at the end of the training session. d) [1] Based on the results of (b) and (c), what conclusions would you draw about the research hypothesis? e) [1] Compute the p-value for testing the significance of the regression in (c)