Question

6.5---11 and 12

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 475 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean μ = 1.6% and standard deviation σ = 1.2%.

(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 475 stocks in the fund) has a distribution that is approximately normal? Explain.

----- Yes , x is a mean of a sample of n = 475 stocks. By the  --central limit theory--- central limit theorem law of large numbers , the x distribution  --is not approximately normal.

(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.)________________


(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the standard deviation decreases.

Yes, probability increases as the mean increases.   

Yes, probability increases as the standard deviation increases.

No, the probability stayed the same.

(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 = 1.6%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)
P(x > 2%) = _________________

Explain.

This is very unlikely if μ = 1.6%. One would suspect that the European stock market may be heating up.

This is very likely if μ = 1.6%. One would not suspect that the European stock market may be heating up.    

This is very likely if μ = 1.6%. One would suspect that the European stock market may be heating up.

This is very unlikely if μ = 1.6%. One would not suspect that the European stock market may be heating up.

12) The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.5 minutes and a standard deviation of 2.5 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway.

(a) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)_________


(b) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)_________


(c) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)______________

Answer 1

(a)

Since the sample size is greater than 30,

Yes , x is a mean of a sample of n = 475 stocks. By the central limit theorem, the x distribution is approximately normal.

(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.)_______________

Standard error of mean = 1.2/ = 0.4

probability that the average monthly percentage return x will be between 1% and 2% = P(1 < < 2)

= P( < 2) - P( < 1)

= P[z < (2 - 1.6)/0.4] - P[z < (1 - 1.6)/0.4]

= P[z < 1] - P[z < -1.5]

= 0.8413 - 0.0668

= 0.7745

(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.)________________

Standard error of mean = 1.2/ = 0.2828427

probability that the average monthly percentage return x will be between 1% and 2% = P(1 < < 2)

= P( < 2) - P( < 1)

= P[z < (2 - 1.6)/0.2828427] - P[z < (1 - 1.6)/0.2828427]

= P[z < 1.41] - P[z < -2.12]

= 0.9207 - 0.0170

= 0.9037

(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the standard deviation decreases.

(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 = 1.6%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)
P(x > 2%) = _________________

P(x > 2%) = P[z > (2 - 1.6)/0.2828427] = P[z > 1.41] = 0.0793

Since the probability is still above 0.05,

This is very likely if μ = 1.6%. One would not suspect that the European stock market may be heating up.    

12)

(a) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)_________

Mean of total taxi takeoff time = 33 * 8.5 = 280.5

Standard deviation of  total taxi takeoff time = * 2.5 = 14.36141

P(x < 320) = P[z < (320 - 280.5)/14.36141] = P[z < 2.75] = 0.9970

(b) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)_________

P(x > 275) = P[z > (275 - 280.5)/14.36141] = P[z < -0.383] = 0.6491

(c) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)______________

P(275 < x < 320) =  P(x < 320) - P(x < 275)

= 0.9970 - (1 - P(x > 275))

= 0.9970 - (1 - 0.6491)

= 0.6461