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Commuting Time Probability Analysis: A Statistical Inquiry

## Question 1: Commuting Time Probability

A professor commutes daily from Clementi Station to his office. The avenge time for a one-way trip is 24 minutes. with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.

(a) Compute the probability that a trip will take at least 0.5 hour. If the professor has to be in his office at 8.30 am and he leaves Clementi Station at 8.15 am daily. compute the percentage of the time that he is late for work.

(b) If the professor leaves Clementi Station at 8.05 am daily and coffee is served at staff lounge from 8.20 am until 8.30 am, compute the probability that he misses coffee. Compute the probability that 2 out of the next 3 trips will take at least 0.5 hour.

(a) f(x)= 3.x 2.4.6. 0.elsewhere. We draw a random sample of size 54. selected with replacement, from the discrete uniform distribution. Calculate the mean ;IT and the variance cri of the sample mean

(b)  Find the probability that sample mean 7 is greater than 4.1 but less than 4.4. (6 marks)
A process yields 10 % defective items. If 100 items are randomly selected from the process. use Normal approximation to find the following probabilities.

(i) Compute the probability that the number of defectives exceeds 13.

(ii) Calculate the probability that the number of defectives is less than S.

(a) The heights of a random sample of 50 male college students showed a mean of 174.5 centimetres and a standard deviation of 6.9 centimetre'.
(b) Construct a 90 %. 95 % 99 % confidence interval for the mean height of all male college students. (12 marks)
Compare the confidence intervals obtained above. comment on which has the largest interval. and which has the smallest interval.

(A) A rocket motor is manufactured by bonding inwether two tunes of nronellank. an igniter.

(b) A process yields 10 % defective items. If 100 items are randomly selected from the process. use Normal approximation to find the following probabilities.

(i) Compute the probability that the number of defectives exceeds 13.

(ii) Calculate the probability that the number of defectives is less than 8.

(a) The heights of a random sample of 50 male college students showed a mean of 174.5 centimetres and a standard deviation of 6.9 centimetres.
(b) Construct a 90 %. 95 %. 99 % confidence interval for the mean height of all male college students. Compare the confidence intervals obtained above. comment on which has the largest interval. and which has the smallest interval.

A rocket motor is manufactured by bonding together two types of propellants. an igniter and a sustainer. The shear strength of the bond y is thought to be a linear function of the age of the propellant x when the motor is cast. Test data arc shown in the Table Q4.