Expert: Scott A Wilson - 9/3/2010

Question
1. Among the examinees in an examination 30%, 35% and   45% failed in Statistics, in Mathematics, and in at least one of the subjects respectively. An examinee is selected at random. Find the probabilities that :-

a.   He failed in Mathematics only
b.   He passed in Statistics, if it is known that he
       has failed in Mathematics

Q2.   From a set of 1000 observations known to be
       normally distributed, the mean is 534 cm and SD is
       13.5 cm.  How many observations are likely to
       exceed 561 cm? How many will be between 520.5 cm
       and 547.5 cm? Between what limits will the middle
       50% of the observations lie?

Q3.   Write a note on “standard error. Distinguish
       between Standard error and Sampling error.

Q4.   Compute the two regression equations on the basis
       of the following data:
                  X   Y
Mean                   40   45
Standard Deviation   10   9

Given that the coefficient of correlation between X & Y is 0.50. Also estimate the value of Y for X=48?  

Answer
Q1. Failure rate is
30% Statistics
35% Mathematics
45% in Statistics and/or Mathematics.

Since 30% + 35%  = 65%, and the total is only 45%, that means there is an overlap of 20%.  In this way, it can be seen that
30% + 35% - 20% = 65% - 20% = 45%, which is what was given.

a. So if, out of 35%, there were 20% that also failed in statistics, that means that there were 35% - 20% = 15% that failed in mathematics only.

b. If 45% of the students failed at one of two choices, that means that 55% passed at both.  Now it was said that 10% of those failed in statistics failed in statistics alone.  This means the total number who passed mathematics was 65%.  The answer would be 55/65.


Q2. Take the mean of 534 with a SD of 13.5.  Determine how many standard deviations by calculating (561-534)/13.5.  Look up on a normal distribution table and find the appropriate value.  For the other values given, subtract off the mean and divide by the standard deviation to get the normal value to look up.

www.mathsisfun.com/data/standard-normal-distribution-table.html
Using this table gives the chance of the number being between the average and that many standard deviations.  For one sided tests, addd 50% to find the chance of being below.  Subtract from 50% to get the chance of being above.  If the value is less than the mean, the opposite is done.


Q3. The more sample that are taken, the less the probability of there being much of a difference.  If all items are sampled, there is no chance of any difference.  In real-life, however, this would create costs that were out of bounds and the company that did this would quickly go bankruptcy.


Q4. Assuming that a 0 x would given a 0 y, a regression equation would be y = 45x/40, which reduces to y = 9x/8.

The E(Y given X) = E(Y) + r(Sy)(X-E(X))/Sx
where Sy is the standard deviation of y, Sx is the standard deviation of x, and r is the correlation coefficient.
The values to use are E(Y) = 45, E(X) = 40, Sy is 9, Sx is 10,
and r is 0.5