Questions: A study of adults aged 20 to 40 investigated the relationship between the average number of hours of exercise per week x and average number of hours of sleep per day y. The regression line was found to be y=8.5-0.2x. Which of the following is a true statement? a) For every additional hour of weekly exercise, there is a corresponding 0.2 decrease in daily hours of sleep. b) We can correctly predict that a 60-year-old adult who exercises 6 hours per week sleeps on average 7.3 hours per day. c) For every additional hour of weekly exercise, there is a corresponding 8.5 increase in daily hours of sleep. d) Increasing amount of weekly exercise causes a decrease in daily hours of sleep.

A study of adults aged 20 to 40 investigated the relationship between the average number of hours of exercise per week x and average number of hours of sleep per day y. The regression line was found to be y=8.5-0.2x. Which of the following is a true statement?
a) For every additional hour of weekly exercise, there is a corresponding 0.2 decrease in daily hours of sleep.
b) We can correctly predict that a 60-year-old adult who exercises 6 hours per week sleeps on average 7.3 hours per day.
c) For every additional hour of weekly exercise, there is a corresponding 8.5 increase in daily hours of sleep.
d) Increasing amount of weekly exercise causes a decrease in daily hours of sleep.

Transcript text: A study of adults aged 20 to 40 investigated the relationship between the average number of hours of exercise per week $x$ and average number of hours of sleep per day $y$. The regression line was found to be $y=8.5-0.2 x$. Which of the following is a true statement? a) For every additional hour of weekly exercise, there is a corresponding 0.2 decrease in daily hours of sleep. b) We can correctly predict that a 60-year-old adult who exercises 6 hours per week sleeps on average 7.3 hours per day. c) For every additional hour of weekly exercise, there is a corresponding 8.5 increase in daily hours of sleep. d) Increasing amount of weekly exercise causes a decrease in daily hours of sleep.

Solution

To determine the true statement, we need to interpret the given regression line equation $ y = 8.5 - 0.2x $. The equation suggests that for each additional hour of exercise per week ($x$), the average number of hours of sleep per day ($y$) decreases by 0.2 hours. This is because the coefficient of $x$ is -0.2, indicating a negative relationship between exercise and sleep. Therefore, statement (a) is true. Statement (b) is incorrect because the regression model is based on adults aged 20 to 40, not 60-year-olds. Statement (c) is incorrect because it misinterprets the regression coefficient. Statement (d) is misleading because correlation does not imply causation.

Schritt 1: Interpretation der Regressionsgleichung

Die gegebene Regressionsgleichung lautet $ y = 8.5 - 0.2x $. Diese Gleichung zeigt, dass für jede zusätzliche Stunde wöchentlicher Bewegung ($x$) die durchschnittliche Anzahl der Schlafstunden pro Tag ($y$) um $0.2$ Stunden abnimmt. Dies wird durch den negativen Koeffizienten $-0.2$ angezeigt.

Schritt 2: Überprüfung der Aussage (a)

Um die Aussage (a) zu überprüfen, berechnen wir die Änderung der Schlafstunden, wenn $x$ um 1 erhöht wird. Die Änderung ist $y(1) - y(0) = (8.5 - 0.2 \times 1) - (8.5 - 0.2 \times 0) = -0.2$. Aufgrund von Rundungsfehlern in der Berechnung kann das Ergebnis leicht abweichen, aber die theoretische Änderung beträgt $-0.2$.

Schritt 3: Schlussfolgerung

Die Aussage (a) ist korrekt, da sie die richtige Interpretation der Regressionsgleichung widerspiegelt. Die anderen Aussagen sind entweder falsch oder irreführend.

Endgültige Antwort

Die Antwort ist $\boxed{\text{a}}$.

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