The mean income of a group of sample observations is $500 with standard deviatio

Question

The mean income of a group of sample observations is $500 with standard deviation of $40. (a) According to Chebyshev's theorem, at least what percentage of the incomes will lie between $400 and $600. (b) Suppose now we know the income distribution is symmetric and bell-shaped, find the proportion of incomes lying between $420 and $580; Suppose the new SAT scores for the 1^st time takers in the U.S. follow a bell-shaped distribution. The students scores have a mean of 1000 with a standard deviation of 150. Find: (a) Z-scores corresponding to the SAT scores of 850 and 1150 obtained from two randomly selected students; (b) Find the proportion (or %) of students who scored between these two test scores.

Explanation / Answer

a)as 400 and 600 lie 2.5 std deviation from mean

hence % of income b/w 400 and 600 =(1-1/k2)*100 =(1-1/2.52)*100=84%

b)P(420<X<580)=P(-2<Z<2) =0.97725-0.02275=0.9545

5)as z score =(X-mean)/std deviation

hence for 850 Z score =(850-1000)/150=-1

for 1150 z score =(1150-1000)/150=1

b)hence P(-1<Z<1) =0.8413-0.1587=0.6827 ~68.27%

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