Question

The annual returns from a particular mutual fund are believed to be normally distributed. The following table lists the annual returns for this fund over a 20-year period.

Year	Return (%)	Year	Return (%)
1	6.2	11	5.4
2	11.3	12	26.0
3	16.1	13	13.5
4	16.9	14	24.2
5	9.8	15	–1.5
6	18.3	16	1.4
7	11.2	17	15.7
8	–12.1	18	11.8
9	19.2	19	1.9
10	17.6	20	17.8

a) Determine the mean and standard deviation of the annual returns for this fund.
b) What is the probability that an annual return will be
i) at least 9%?
ii) negative?
c) For how many of the next ten years would you expect the fund to have an annual return greater than 6%? What assumptions are necessary to answer this question?

Answer 1

a)

X	(X - X̄)²
6.2	28.46
11.3	0.06
16.1	20.84
16.9	28.78
9.8	3.01
18.3	45.77
11.2	0.112
-12.1	558.613
19.2	58.752
17.6	36.784
5.4	37.638
26	209.236
13.5	3.861
24.2	160.402
-1.5	169.911
1.4	102.718
15.7	17.347
11.8	0.070
1.9	92.833
17.8	39.250

	X	(X - X̄)²
total sum	230.7	1614.45
n	20	20

mean =    ΣX/n =    230.700   /   20   =   11.5350
                      
  
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (1614.4455/19)   =       9.2180

.........

b)

i)

µ =    11.535              
σ =    9.218              
                  
P ( X ≥   9.00   ) = P( (X-µ)/σ ≥ (9-11.535) / 9.218)        
= P(Z ≥   -0.275   ) = P( Z <   0.275   ) =    0.6083

ii)

µ =    11.535      
σ =    9.218      
          
P( X ≤    0   ) = P( (X-µ)/σ ≤ (0-11.535) /9.218)  
=P(Z ≤   -1.251   ) =   0.1054023

c)

µ =    11.535              
σ =    9.218              
                  
P ( X ≥   6.00   ) = P( (X-µ)/σ ≥ (6-11.535) / 9.218)          
= P(Z ≥   -0.600   ) = P( Z <   0.600   ) =    0.7259

out of ten years funds to have an annual return greatser tahn 6% = 0.7259 * 10

=7.259 = 7 years(round up)

please revert back for doubt