Question

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 250 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean μ = 1.4% and standard deviation σ = 1%.

(b) After 9 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)


(c) After 18 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that μ = 1.4%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)
P(x > 2%) =

Answer 1

Part b)

X ~ N ( µ = 1.4 , σ = 1 )
P ( 1 < X < 2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 1 - 1.4 ) / ( 1 / √(9))
Z = -1.2
Z = ( 2 - 1.4 ) / ( 1 / √(9))
Z = 1.8
P ( -1.2 < Z < 1.8 )
P ( 1 < X̅ < 2 ) = P ( Z < 1.8 ) - P ( Z < -1.2 )
P ( 1 < X̅ < 2 ) = 0.9641 - 0.1151
P ( 1 < X̅ < 2 ) = 0.849

Part c)

X ~ N ( µ = 1.4 , σ = 1 )
P ( 1 < X < 2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 1 - 1.4 ) / ( 1 / √(18))
Z = -1.6971
Z = ( 2 - 1.4 ) / ( 1 / √(18))
Z = 2.5456
P ( -1.7 < Z < 2.55 )
P ( 1 < X̅ < 2 ) = P ( Z < 2.55 ) - P ( Z < -1.7 )
P ( 1 < X̅ < 2 ) = 0.9945 - 0.0448
P ( 1 < X̅ < 2 ) = 0.9497

Part e)

X ~ N ( µ = 1.4 , σ = 1 )
P ( X > 2 ) = 1 - P ( X < 2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 2 - 1.4 ) / ( 1 / √ ( 18 ) )
Z = 2.5456
P ( ( X - µ ) / ( σ / √ (n)) > ( 2 - 1.4 ) / ( 1 / √(18) )
P ( Z > 2.55 )
P ( X̅ > 2 ) = 1 - P ( Z < 2.55 )
P ( X̅ > 2 ) = 1 - 0.9945
P ( X̅ > 2 ) = 0.0055

Since the probability is less than 5% i.e < 0.05, that tend to shake your confidence in the statement that μ = 1.4% and it would be possible that the European stock market might be heating up.