Statistics and Probability Answers

An unbiased coin is tossed twice. The four possible outcomes are equiprobable. If A is the event “both head or tail have occurred” and B is the event: “at most one tail is observed”, then find P(A), P(B), P(A/B) and P(B/A)?

Michael, has been asked by “Custom Calculators” to address an issue
the company is having with their warranty system. The company’s email to
Michael. "Dear Michael,
we produce and sell calculators. So, we know from market research that a calculator made by our factory will stop working on average after 3000 days, according to a bell-curve (that’s what the market research person told us, but we don’t understand what he means). Anyway, we decided to set our warranty at 2000 days (so that anyone who bought a calculator from us could have it replaced provided it was bought no later than 2000 days ago), but we’ve found that we’re replacing 10% of calculators we sell. We can’t really afford this, and have only budgeted to replace 3% of calculators we sell.
Can you tell us what we should change our warranty time to so that we only have to replace 3%

What should Michael’s advice be?

In a case of 144 boxes of light bulbs, it is found that exactly 4 boxes contain at least


one broken light bulb. If 5 boxes are selected from the case at random, what is the

probability (rounded to 4 decimal places) that none of the boxes will contain a broken

light bulb?

A civil engineer is analyzing the compressive strength of concrete. A random sample of 12 specimens has a mean compression strength of 3250 psi with variance 1000 (psi)2.
i) Construct a 95% confidence interval for the population mean compressive strength.
ii) Construct a 95% confidence interval for the population standard deviation of compressive strength.

In measuring user reaction time to the mouse movement, a psychologist estimates that the standard deviation is 0.05 second. How large a sample measurements must he take in order to be 95% confident that the error in his estimate of mean reaction time will not exceed 0.01 second?

Probability of a coin

(1) Suppose that in a group of people you find that X percent of people in the group have
heights that are greater than the average (that is, the mean) height in that group. Which of
the following is a true statement about X?
a) X can be any percentage but less than 100%.
b) X can be any percentage as long as it is less than 50%.
c) X cannot be bigger than 25%.
d) X can be bigger than 25% but cannot be bigger than 50%.
e) X can be bigger than 50% but cannot be as high as 99.9%.

(2) Suppose that in a group of people you find that X percent of people in the group have
heights that are greater than the median height in that group. Which of the following is a
true statement about X?
a) X can be any percentage but cannot be equal to 100%.
b) X can be any percentage.
c) X cannot be bigger than 25%.
d) X is approx. 50%
e) X is approx. 75%

The lifetime X of a bulb is a random variable with the probability density function:
f(x) = 6[0.25 - (x-1.5)^2] when 1≤x≤2
0 otherwise
X is measured in multiples of 1000 hrs. What is the probability that none of the three
bulbs in a traffic signal have to be replaced in the first 1500 hrs of their operation.

If 500 people each select a number a random between 1 and 100, what is the probability
that 4 people select the number 25?

Two file servers are compared according to their response time for retrieving a small file. The mean response time of 50 such requests submitted to server 1 was measured to be 682ms with a known standard deviation of 25ms. A similar measurement in server 2 resulted in a sample mean of 676ms with a standard deviation of 28ms. Do these samples provide sufficient evident to conclude that server 1 provides better response than server 2. Perform this test at 0.05 level of significance.