Question

uestion (1) (30 Marks A) 5 Marks) The time required to build a computer is normally distributed with a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability thata computer is assembled in a time between 45 and 60 minutes? B) (5 Marks) t was noted that the amount of oil in each "32-ounce" bottle is actually a normally distributed random variable, with a mean of 32.2 ounces and a standard deviation of 0.3 ounce. If a customer buys a carton of twelve bottles, what is the probability that the mean amount of the twelve bottles will be greater than 32 ounces? C) (5 Marks) For each conjecture, state the null and alternative hypotheses. i) The average age of community college students is 24.6 years. ii) The average age of attorneys is at least 25.4 years. ii A researcher claims that the variation in the salaries of elementary school teachers is greater than the variation in the salaries of secondary school teachers. D) (5 Marks) A regional manager interested in studying the variation of the annual profit of different branches of his great stores. A sample of 20 stores has a mean profit of £40 million and a standard deviation of 5 million. Suppose that the sample is enlarged to 22 stores, by including two additional stores that are each has £40 million as annual profit. Will the standard deviation increase, decrease, or stay the same. Explain your answer E) (10 Marks) The "A" City Council has recently introduced a new transport system, which aims to give priority to buses. Before the introduction of the new system the journey from city "A" airport to city "B" railway station took an average of 35 minutes. A researcher has travelled on a random sample of 38 buses on this route from the airport and has found that the average time taken is now 32.5 minutes, with aa population standard deviation of 8.4 minutes, using a 1% level of significance determine whether there is a significant difference in the average time. [IMPORTANT NOTE: show the 5 steps of hypothesis clearly with the graphs)

Answer 1

Solution A:

X be the time required to build a computer

X »   N (µ=50 : σ = 10)      

P(45<X<60=F(60)-F(45) =

=ɸ (60-50/10) - ɸ(45-50/10) =

= ɸ(1)- ɸ(-0.5) =

=ɸ(1)-[1- ɸ(0.5)] =

= ɸ(1)+ ɸ(0.5)-1=

0.8413+0.6915-1=

=0.5328

Solution B

We want to find P(X>32) where X is normally distributed with (µ=32.2 : σ = .3)

Things we know,

1. X is normally distributed, therefore so will X
2. µ_x = µ=32.2 oz    
3. σ_{x =} σ/√n =.3/√12 = .3/3.46 = .087

P(X>32)= P(X- µ_x/ σ_x> 32-32.2/.087=P(Z>-2.29)= .9890

“There is about a 99% chance that the mean of the twelve bottles will exceed 32oz.”

Solution C:

Answer i :         H0: µ=24.6 and H1: µ≠24.6 b)

Answer ii:       H0: µ=25.4 and H1: µ > 25.4