A training director wishes to find out whether a particular training programme increases the efficiency of the employees. He plans to take a random sample of employees and record their efficiency rating before the training starts. After the completion of the programme, the efficiency of the same sample of employees is recorded as given in the table below. At 0.05 level of significance do you reject or accept the null hypothesis
H 0 : µd = 0 against H1 : µd > 0
where µd = 0 is the mean of the difference?______ ____ ______ ________ _____ _______ ___ ____ ____ _________ ___ _____.
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Suppose that a person selects a card at random from a deck of 52 cards and tells us that the selected card is red. Find the probability that the card is the kind of hearts given that it is red.
A continuous random variable X has the p.d.f.
Calculate the mean and standard deviation of X . Construct the distribution function F ( x) and hence evaluate P( X > 1/4) .
For what values of m will the equation
x 2 − 2x (1 + 3m) + 7 (3 + 2m) = 0
have equal roots?
If p is the length of the perpendicular from the origin to the line x/a + y/b = 1 prove that 1/p2 = 1/a2 + 1/b2.
Suppose that the temperature is normally distributed with expectation 50o C and variance 4o C . What is the probability that the temperature T will be between 48o C and 53o C ?
What is the probability that T ≥ 52o C ?
Evaluate the following integrals:
i) 
ii) ∫ x ln | x + 1| dx
Determine the value of the constant a if the function f (x) defined below is continuous at x = 2.
