Methods2023 Exam 2 Section A · Q15CAS active1 mark

Comparing two normal density curves

Let $X$ be a normal random variable with mean of 100 and standard deviation of 20. Let $Y$ be a normal random variable with mean of 80 and standard deviation of 10.

Which of the diagrams below best represents the probability density functions for $X$ and $Y$ , plotted on the same set of axes?

Five grids labelled A to E, each showing two bell-shaped curves: a dashed curve for X and a solid curve for Y, as given in a key. The options differ in which curve is further left, which is taller, and which is wider.

Which of the diagrams below best represents the probability density functions for $X$ and $Y$ , plotted on the same set of axes?

1 mark

ADiagram A: the solid curve (

Y

) is centred to the left and is tall and narrow; the dashed curve (

X

) is centred to the right and is lower and wider, with peak about half the height of the solid one.BDiagram B: the dashed curve (

X

) is centred to the left and is lower and wider; the solid curve (

Y

) is centred to the right and is tall and narrow.CDiagram C: the solid curve (

Y

) is centred to the left and is low and wide; the dashed curve (

X

) is centred to the right and is taller and narrower.DDiagram D: the solid curve (

Y

) is centred to the left and the dashed curve (

X

) to the right, both with about the same height and spread.EDiagram E: the dashed curve (

X

) is centred to the left and is taller and narrower; the solid curve (

Y

) is centred to the right and is lower and wider.

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