Statistics and Probability Answers

I. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is  a) replaced,  b) not replaced. II. A ball is drawn at random from a box containing 6 red balls, 4 white balls, and 5 blue balls. Determine the probability that it is (a) red, (b) white, (c) blue, (d) not red, (e) red or white. III. Prove that the mean and variance of a binomially distributed random variable are, respectively, µ = np and σ2 = npq. 

The probability of a customer arrival at a grocery service counter in any 1 second is equal to 0,1. Assume that customers arrive in a random stream and hence that the arrival any 1 second is independent of any other. Calculate the probability that the first arrival will occur during the third   1-second interval.

If X is a random variable with Poisson distribution satisfying P(X=0) = P(X=1), calculate E(X). 

For the past few years, the number of customers of a drive-in bottle shop has averaged 20 per hour. This year, another bottle shop one kilometre away opened a drive-in window. The manager of the first shop believes that this will result in a decrease in customers. A random sample of 50 hours showed an average of 18.7 customers per hour with a standard deviation of 3.0. Can we conclude at the 5% level of significance that the manager’s belief is correct?

    A sample of 60 Grade 11 student’s ages was obtained to estimate the mean age of all Grade 11 students. X=16.7 years and the population variance is 16.

1. Find the 90% confidence interval for ?

1. Find the 95% confidence interval for ?

1. What conclusions can you make based on each estimate

(b) Global Insurance Company has four salespeople working in Hill town. The number of policies sold during the last month is given in matrix A, as 

o1 Malley Coponi Steomberg Deutsch

8 7 6 8 Automobile

6 9 11 5 Life

A= 4 3 2 0 Health

0 2 1 3 Homeowners

i) Let S= [1 1 1 1]. Find SA and interpret its elements.

ii) Find 𝐴𝑆௧ and interpret its elements. 

The probability that a vehicle entering the Luray Caverns has Canadian 

license plates is 0.12; the probability that it is a camper is 0.28; and the 

probability that it is a camper with Canadian license plates is 0.09. What is 

the probability that 

(i) A camper entering the Luray Caverns has Canadian license Plates.

(ii) A vehicle with Canadian license plates entering the Luray Caverns is a 

camper. 

(iii) A vehicle entering the Luray Caverns does not have Canadian plates or 

is not a camper.

c) Determine the location and values of the absolute maximum and absolute minimum for the given function: 𝑓(𝑥) = (−𝑥 + 2) ସ , 𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑥 ≤ 3

A study was conducted to determine the marrying age of teachers. It was found out

that the mean marrying ager of teachers is 30 years old. Fifteen teachers were surveyed

randomly and found that their mean marrying age was 33 years old with a standard

deviation of 5 years. Use 10% level of significance to test the hypothesis and assume that

the population is normally distributed.

A Z score of 3 is approximately how many standard deviations above the mean?