Statistics

(a) Explain what is meant by the skewness of the distribution, and define a suitable measure of Skewness. For given p.d.f.

$$f(x) = k\,(x - x^2), \qquad 0 \le x \le 1$$

Find the Skewness and discuss.

(b) Define the normal distribution and obtain its mean and variance. Show that for the normal distribution, the mean, mode and median are the same.

(a) If X has a binomial distribution, then show that $E(X) = np$, $Var(X) = npq$. Derive the m.g.f of the binomial distribution and explain its uses?

(b) What are the main characteristics of the Poisson distribution? Explain with the help of examples. Also give its properties, applications and relationship with other distributions.

(a) From the following set of values:

1) Compute the residuals and verify that they add to zero and draw the conclusion about the results.

2) Compute the standard error of estimate, $s_{y.x}$.

(b) Under what conditions the correlation among the random variables exists. Also describe the properties of correlation coefficient. Calculate the correlation co-efficient for a sample of 20 pairs of observations, given that

Mean of X = 2, Mean of Y = 8, $\sum X^2 = 180$, $\sum Y^2 = 1424$ and $\sum XY = 404$

Also interpret its results.

(a) What is finite-correction factor? When is it appropriately used in sampling applications and when can it, without the great undesirable consequences, be ignored?

(b) Given the population 2, 4, 8, 8, 10, 10.

1) How many samples of size $n = 2$ can be drawn without replacement from this population?

2) Compute and tabulate the sampling distribution of the mean for samples of size $n = 2$.

(a) Explain what is meant by:

(i) a statistical hypothesis,

(ii) test-statistic,

(iii) test of significance,

(iv) level of significance,

(v) type-I error and type-II error

(b) The weights of 4 persons before they stopped smoking and 5 weeks after they stopped smoking are as follows:

Person | 1 | 2 | 3 | 4

Before | 148 | 176 | 153 | 116

After | 154 | 176 | 151 | 121

Use the t-test for paired observations to test the hypothesis at the 0.05 level of significance, that giving up smoking has no effect on a person’s weight.

(a) Explain the procedure of randomization in a completely randomized design where we have 3 varieties of wheat and 18 experimental plots available. Also explain what is explained by significant F value in an experiment.

(b) Four varieties of wheat were tried in a randomized complete block design in four replications. Yield in kilogram per plot is shown in the table given below. Test the hypothesis that there is no difference in the means of four varieties. $\alpha = 0.05$.