Statistics and Probability Answers

A bottled water company has found in the past that 2% of their bottled water does not meet the company’s high standards. As such periodic samples are taken and tested for their quality. If from the last batch a sample of 12 bottles are taken and tested, determine the probability:

i. Carefully defining the random variable of interest [1]

ii. What is the probability distribution? State the values of the parameters [2]

iii. Justifying the suitability of the probability distribution identified in part (ii) [4]

iv. Calculate the probability that at most 2 bottles do not meet the company’s Standards [4]

v. The expected number of bottles that do not meet the company’s standards. [2]

vi. The amount of sodium sulfate, in mg, found in the bottled water follows a normal distribution with a mean 13mg and a standard deviation of 2.5mg. It is known that 76% of the bottles have a mean that is more than x mg. Find the value of x

If P (A) = 1/3

, P (B^c) = 1/4

, then P (AB) = 0.

True or False?

A population consist of 2,4,5,9 and 10, with sample size of 3. Compute the mean and variance of the

sampling distribution of the sample mean.

The total number of hours, measured in units of

100 hours, that a family runs a vacuum cleaner over a

period of one year is a continuous random variable X

that has the density function

f(x) =

⎧

⎨

⎩

x, 0 2 − x, 1 ≤ x < 2,

0, elsewhere.

Find the variance of X.

Find the average number of hours per year that families run their vacuum cleaners.

Toss a fair coin twice. You win Ghc 1 if at least one of the two tosses comes out heads.

(a) Assume that you play this game 300 times. What is, approximately, the probability that you win at least Ghc 250?

(b) Approximately how many times do you need to play so that you win at least Ghc 250 with probability at least 0.99?

 How many times do you need to toss a fair coin to get 100 heads with probability 90%?

 Suppose it is known that the weight of a certain population of individuals are approximately normally distributed with a mean 80 kg and a standard deviation of 16 kg. What is the probability that a person picked at random from this group will weight between 60 and 110 kg?

 Suppose that all athletes run 200 metres and the time they take to run is normally distributed with mean 12 seconds and a standard deviation of 3 seconds. The coach has decided that 40 percent of the athletes who can run the distance in the least time will be sent to participate in the Olympics. What is the cutoff score that will be decide which members of the time will qualify.

 The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 50 minutes and 1.5 hours to see a doctor?

According to the Insurance Institute of America, a family of four spends between Ghc 400 and Ghc 3,800 per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts.

(a) What is the mean amount spent on insurance?

(b) What is the standard deviation of the amount spent?

(c) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year?

(d) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year? (e) What is the probability a family spends more than Ghc 3,000 per year?