Prelim II Solutions

# Prelim II Solutions - ENGRD 2700, Spring `09 Prelim 2...

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ENGRD 2700, Spring ‘09 Prelim 2 Prelim 2 Solutions Apr 9, 2009 Problem 1 (a) A right-skewed density function looks like this: The right tail is longer, and the mass of the distribution is concentrated on the left of the ﬁgure. If an iid sample comes from a right-skewed distribu- tion, the smallest and the largest sample quantiles will be larger than those expected from a normal sample, so in the normal probability plot, points on the two ends of the plot will fall above a straight line through the middle part, yielding a curved pattern. We observe this in plot (d). (b) A thick-tailed density curve has “thicker” probability tails than a normal density curve, so it has more probability mass on the tails compared to a normal distribution. In the ﬁgure below, the blue curve is a normal density and the red curve is a thick-tailed density. If an iid sample comes from a thick-tailed distribution, the largest sample quantiles will be larger than those expected from a normal sample and the smallest sample quantiles will be smaller than those expected from a nor- mal sample. On a normal probability plot, points on the far right will lie 1

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ENGRD 2700, Spring ‘09 Prelim 2 above the straight line, and points on the far left will lie below it, yielding an S-shaped pattern. We observe this in plot (c). (c) Plots (b) and (e) do not exhibit any signiﬁcant deviations from a straight line, so the samples depicted in those plots arguably came from normal dis- tributions. Plots (c) and (d) were discussed before. Plot (a) depicts a light-tailed sample, and plot (f) depicts a left-skewed sample. Problem 2 (a) Denote by X the total amount of time (in minutes) the professor has to spend with the students. Clearly, X = X 1 + X 2 + X 3 . Since X 1 ,X 2 ,X 3 are normally distributed and independent, X is also nor- mally distributed, with expected value μ = μ 1 + μ 2 + μ 3 = 28 and standard deviation σ = q σ 2 1 + σ 2 2 + σ 2 3 = 21 . (b)
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## This note was uploaded on 10/29/2011 for the course ENGRD 270 taught by Professor Resnick during the Spring '07 term at Cornell.

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Prelim II Solutions - ENGRD 2700, Spring `09 Prelim 2...

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