4. An investment company claims that only 10% of their portfolios give a negativ

Question

4. An investment company claims that only 10% of their portfolios give a negative return to investment in a 1 year span of time. You take a random sample of 150 of their portfolios and nd that 20 had a negative 1 year return (a) Show that the number of portfolios that have a negative 1 year return follows a binomial distribution (b) Find the probability that at least 20 portfolios have a negative 1 year return using a normal approximation (c) Your team has decided that if P(X 20) is lower than 0.01, then the investment companys claim that only 10% of their portfolios give a negative return to investment in a 1 year span of time should be rejected. What should your team decide based on the probability found in (b)? Explain 4. An investment company claims that only 10% of their portfolios give a negative return to investment in a 1 year span of time. You take a random sample of 150 of their portfolios and nd that 20 had a negative 1 year return (a) Show that the number of portfolios that have a negative 1 year return follows a binomial distribution (b) Find the probability that at least 20 portfolios have a negative 1 year return using a normal approximation (c) Your team has decided that if P(X 20) is lower than 0.01, then the investment companys claim that only 10% of their portfolios give a negative return to investment in a 1 year span of time should be rejected. What should your team decide based on the probability found in (b)? Explain 4. An investment company claims that only 10% of their portfolios give a negative return to investment in a 1 year span of time. You take a random sample of 150 of their portfolios and nd that 20 had a negative 1 year return (a) Show that the number of portfolios that have a negative 1 year return follows a binomial distribution (b) Find the probability that at least 20 portfolios have a negative 1 year return using a normal approximation (c) Your team has decided that if P(X 20) is lower than 0.01, then the investment companys claim that only 10% of their portfolios give a negative return to investment in a 1 year span of time should be rejected. What should your team decide based on the probability found in (b)? Explain

Explanation / Answer

a) Let the number of portfolios be X

p= success of any trial = number of portfolios that have negative return

q = failure of any trail = number of portfolios that have positive return

P(SSFSSS..FSF) = P(S) X P(S) X P(F) X ..... X P(F) X P(S) X P(F) where "S" represents success and "F" represents failure.

= p.p.q.......q.p.q

= p^20 x q^(150-20)

Hence the probability of 20 success in 150 trails in any order is given by addition theorem of probability by the expression (150 C 20) x p^20 x q^130 which is similar to the pdf of binomial distribution.

Hence, the number of portfolios that have negative 1 year return follow binomail distribution.

b) X ~ Bin (150,0.1)

By normal approximation

np = 150 X 0.1 = 15

npq = 150 X 0.1 X 0.9 = 13.5

X ~ N (np , sqrt (npq) )

X ~ N (15 , 3.674)

Z = [(X-15)/3.674] ~ N (0,1) (Converted into standard normal variate)

P (Z <= 20) = P [ Z<= (20-15/3.674) ] = P [Z <= 1.36] (value to be taken from Normal tables )

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