Question

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 find that 20 had a negative 1 year return

1. Show that the number of portfolios that have a negative 1 year return follows a binomial distribution

2. Find the probability that at least 20 portfolios have a negative 1 year return using a normal approximation

3. 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

Choose: (a) Claim should be rejected.

(b) Claim should not be rejected.

(c) Not possible to tell

(d) 0.15

Answer 1

1. In any random experiment,if there is two outcomes in any trial,which are mutually exclusive and exhaustive then the trials are called Bernoulli trial.When more than one Bernoulli trial occur then they generate Binomial distribution.

Here,for the investment comapany there is two type of returns from their portfolios i.e. negative return and positive return(including 'zero' return) and we are taking random sample of 150(>1),so each trial is Bernoulli trial and the distribution is Binomial.

2. X:No. of portfolios that have a negative 1 year return

$X\sim Binomial(n=150,p=\frac{20}{150}\simeq 0.13)$

To find, $P[X\geqslant 20]=1-P[X<20]=1-\sum_{i=1}^{19}P[X=i]$

$=0.53617\simeq 0.54$

3. Since $P[X\geqslant 20]=0.54 > 0.01$ , we can't say that the investment company's claim of 10% portfolios' negative return in 1 year is rejected so we have to decide

(b) Claim should not be rejected