Is there a relationship between how much sleep someone gets and the amount of co
ID: 3224514 • Letter: I
Question
Is there a relationship between how much sleep someone gets and the amount of coffee they drink? To find out, ten graduate students were asked how many cups of coffee they consumed yesterday and how much they slept last night. Let x be the cups of coffee and y be the amount of sleep. The equation of the least squares regression line is y = 8.2742 - 0. 5323x, and r^2 = 0.4777. The data and scatterplot are below: a) What is the slope of the regression line and what does this value specifically tell you about sleep and coffee? b) Determine the correlation between hours of sleep and cups of coffee consumed. c) Describe the strength and direction of the relationship coffee consumed and amount of sleep. d) Predict the hours of sleep for someone drinking 4.5 cups of coffee and for someone drinking 7 cups of coffee. Are these valid predictions? Explain why or why not. e) Find the residual for a person drinking 2 cups of coffee. f) What percent in the variation of amount of sleep is NOT explained by the linear relationship with the cups of coffee consumed?Explanation / Answer
a. The slope of the regression line is -0.5323. This tells us of an inverse relationship between number of cups of coffee and the amount of sleep. This hints that drinking more cups of coffee can reduce hours of sleep.
b. The correlation between x and y is the square root of the regression R^2 value and here as we see an inverse relationship, the sign of the correlation coefficient would be negative. Hence, r = -sqrt(0.4777) = -0.69
c. The direction of association is inverse as the correlation coefficient is negative and the strength is high as the value is highly negative, indicating high negative collinearity
d. x=4.5, y = 8.2742-0.5323*4.5 = 5.88
x = 7, y = 8.2742-0.5323*7 = 4.55
This is only valid if the linear association of the variables in indeed true. There may even be a non-linear or no association between the variables
e. For 2 cups of coffee, prediction is 8.2742-0.5323*2 = 7.21 hours and actual is 7.5 hours. So, residual = 7.5-7.21 = 0.29 hours
f. 47.77% of variation of amount of sleep is explained by the linear relationship. So, percentage of variation not explained = 52.23%
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