1. Correlation is the ratio of: a. 2 standard deviation b. two X^2 c. Covariance and 2 standard deviation d. Two regression coefficients
2.Which of the following are not the diagrams? a. Histogram, Frequency polygon b. Square and angular c. Cartogram and pictogram d. None
3. Two gives (less than and more than) bisects at a. Arithmetic mean b. Mode c. Geometric mean d. Median
4.The best measure of dispersion in which all observations participate: a. Mean deviation b. Standard deviation c. Quartile deviation d. All
5. In normal distribution, the area between +/- 2S.D. is a. 99.73% b. 95.45% c. 68.27% d. 100 %
6. The mean and variance are equal in a. Binomial Distribution b. Normal Distribution c. Poisson distribution d. None
7. Normal distribution was given by: a. Bernauli b. Fisher c. Student d. A. De. Moivre
8. Two samples' mean are tested by a. f-test b. t- test c. Chi- square test d. None
9. In throwing of two dice simultaneously, the probability of not getting two on the face is a. 1/36 b. 6/36 c. 35/36 d. 2/ 36
10. The sum of the deviation taken from the arithmetic mean is a. 1 b. 100 c. 0 d. infinity
11. Tabulation is the process of arranging the data in an orderly manner into ………and ……… capable of being read into proper directions: a. Row, Column b. Tables c. Class, Tables d. None
12.Which measure of dispersion, is calculated by only extreme values: a. Mean deviation b. Variance c. Range d. Standard deviation
13. If all the value are same, the S.D. is a. 1 b. 0 c. 100 d. 50
14. In simple bar diagram, ……. is kept constant but ………. varies. a. Length, width b. length, height c. width and length d. None
15. The range of f - test statistic value is a. 0 to 1 b. 0 to infinity c. negative infinity to positive infinity d. None
16. In testing of Hypothesis, we commit which types of error: a. 1 b. 2 c. 3 d. 4
17. Occurrence of one event excludes the occurrence of other, event is known as: a. Dependent b. independent c. Simple d. All above
18. If the correlation coefficient between two variable is -1, then the variables are: a. Uncorrelated b. positively correlated highly c. negatively correlated highly d. Just correlated
19. The limit of correlation coefficient is a. 0 to 1 b. 0 to -1 c. -1 to +1 d. -infinity to + infinity
20. The under root of which coefficient product is correlation coefficient: a. variation b. relation c. regression d. None
21. The degree of freedom we refer for a t-test to test correlation coefficient based on 15 pairs of observations shall be a. 15 b. 14 c. 13 d. 30
22. For construction the frequency polygon from histogram of each interval: a. Lower limits are joined b. Upper limits are joined c. Middle points are joined d. Cumulative frequencies are joined
23. Group of individuals under study is called: a. Statistic b. Parameter c. Population d. None
24. Which measures of dispersion is free from units: a. Range b. Standard deviation c. Coefficient of variation d. Variance
25. Following approach of defining probability relates set theory: a. A priori b. Axiomatic c. Empirical d. Classical
ANSWERS & EXPLANATIONS
1
c. (Covariance and 2 standard deviation)
-Correlation coefficient measures degree of linear association between variables.
-Correlation coefficient is the ratio of Covariance between variables to the product of standard deviation of the two variables. It is denoted by ‘r’.
-Correlation analysis is the statistical tool generally used to describe the degree to which one variable is related to another. The degree of relationship between variables is measured by finding the correlation coefficient. ( denoted by “r”).
Types of Correlations:
- positive and negative correlation
- linear and non- linear
- simple, partial and multiple
Method of measuring correlation
- Graphical method (By means of scatter diagram, useful to find out whether the relationship is linear or not but doesn’t give exact numerical value of correlation.)
- Karl’s Pearson's correlation coefficient: (ratio of covariance between variables to product of standard deviation of 2 variables, useful when data are quantitatively measured.)
- Spearman’s Rank Correlation: There are some variables like honesty, beauty, intelligence etc. cannot measured quantitatively. In such case, type of variable can be measured by assigning ranks. The degree of relationship between two sets of ranks is known as rank correlation.
Note:
Correlation coefficient is obtained from the square root of two regression coefficient. (Regression coefficient signifies the amount by which change in x must be multiplied to gives the corresponding average change in ‘y’).
2.
a. (Histogram, Frequency Polygon)
Histogram: It is a graph that show the frequency of numerical data using rectangles. The height of the rectangle ( the vertical axis) represents the distribution frequency of a variable.
A frequency Polygon: It is a type of line graph where the class frequency is plotted against the class midpoint and the points are joined by a line segment creating a curve.
3.
d. (Median)
Median of a distribution is the value of the variable which divides it into 2 equal parts. It is positional average. It is denoted by Md.
Arithmetic mean: A set of data is computed by dividing the sum of all observations by the number of observations . Also called average.
Geometric mean: Geometric mean of n non-zero and non - negative observations is the nth root of their products, denoted by G.M.
Mode: Mode is the variate value that is repeated maximum number of times in a distribution.
4.
b. (Standard deviation)
The scatterness of data from the central value - Dispersion. Measure of dispersion is the statistical measure used to measure the scatterness in the set of data. Less the value of dispersion, more the data will be representative.
Various measure of Dispersion is:
- Range: Simplest measure of dispersion, difference between largest and smallest observations in the set of observations. Denoted by R
- Quartile deviation: It is the half of the difference between the upper and lower quartile.
- Mean deviation/ Average deviation: Arithmetic mean of the absolute deviation taken from the central value, Median and, mode).
- Standard deviation: It is the positive square root of the arithmetic mean of the square of deviation taken from mean. Denoted by Sigma
- Variance: Square of Standard Deviation
- Lorenz’s curve: graphical method of determining dispersion, is the graph of one or more frequency distribution, which show the cumulative frequencies and cumulative total value both reduced to percentage.
5.
b. (95.45%)
Normal Distribution is a bell-shaped curve, where mean =median =mode.
Total area of normal distribution curve = 1
Area between +/- S.D. = 68.268%
Area between +/- 2S.D. = 95.45%
Area between +/- 3 S.D. = 99.73%
6.
c. (Poisson distribution)
- Binomial Distribution: It is a type of frequency distribution used when there are exactly two outcomes from each trial (either success or failure). Trial is random and independent, and probability of a success is the same for each trial. Mean of binomial distribution is calculated by multiplying the number of trials (n) by the probability of success (p) or np while variance of binomial distribution is calculated by. Var(x) = np(1-p)
- Poisson distribution: It is the limiting form of the binomial distribution in which number of trial (n) become very large and probability of success of event is very small. Mean of the poisson distribution is equal to Variance which is equal to “lambda”.
- Normal Distribution: Sometimes called bell - curve (De- Moivre distribution) is a distribution that occurs naturally in many situations. The bell curve is symmetrical i.e., half of the data will fall to the left of mean, half will fall to the right of the mean.
7.
d. (A. De. Moivre)
- Normal Distribution was given by: French Mathematician Abraham De Moivre
- Binomial distribution was given by: Swiss Mathematician Jacob Bernoulli
- Poisson distribution was given by: French mathematician Simeon Denis Poisson
8.
b. (t - test)
- f - test: Statistical test that is used in hypothesis testing to check whether the variance of two population or two sample are equal or not.
- t - test: Statistical test that compares means of two samples, suitable for unknown population variance. Used for limited sample size, not exceeding 30.
- z- test: statistical test that compares means of two samples, suitable for known population variance. Used for large sample size, (>30).
- Chi - square test: Chi- square test are used to test whether two categorical variables are related to each other or not.
9.
c. (35/36)
In throwing of two dice simultaneously, possible outcomes will be:
1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 3-1,3-2, 3-3, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 6-1, 6-2, 6-3, 6-4, 6-5, 6-6.
the probability of getting two on the faces = 1/36
the probability of not getting two on the faces = 35/36
10.
c. (0)
the sum of the deviation from arithmetic mean is always zero. It is due the property that the arithmetic mean is characterized as center of gravity i.e., the sum of positive deviation from mean is equal to the sum of negative deviation from the mean.
11.
a. (Rows and columns)
Tabulation is the process of arranging the data in an orderly manner into rows and columns capable of being read into proper directions.
Rows: horizontal lines of tables.
Columns: vertical lines of tables.
12.
c. (Range)
Range= (largest value - smallest value)
Mean deviation from mean = 1/n ∑ x - x
Standard deviation is simply square root of the arithmetic mean of the square of the deviation taken from arithmetic mean.
Variation is the square of Standard deviations.
13.
b. (0)
If all the values are same, the S.D. will be zero. Since there is no deviation from the mean value.
14.
c. (width, length)
A simple bar chart is used to represent data involving only one variable classified on a spatial, temporal or quantitative basis. In a simple bar chart, we make bars of equal width but variable length. i.e., magnitude of the quantity represents by the height or length of bar.
15.
b. (0 to infinity)
- The range of f test statistic value is 0 to infinity
- the range of variance is 0 to infinity
- the range of Chi -square test statistics is 0 to infinity
- the range of regression coefficient is negative infinity to positive infinity
- the range of normal distribution is negative infinity to positive infinity
- the range of heritability is 0 to 1
16.
b. (2)
In testing Hypothesis, we commit two types of error.
Type -1 Error: It’s occurring when true null hypothesis is rejected
Type -2 Error: It’s occurring when false null hypothesis is accepted
17.
b (Independent events)
When the occurrence of one event doesn't depend on the other events, called independence events
When the occurrence of one event depends on the other events, called Dependence events.
18.
c. (negatively correlated highly)
Interpretation of Calculated value of Karl’s Pearson’s correlation coefficient
- If r =+1, there is perfect positive linear correlation between two variables
- if r= -1, there is perfect negative linear correlation between two variables
- If r= 0, there is no correlation between two variables
- If r = (0.70-9), there is high degree positive correlation between two variables
- If r= (0.50-69), there is moderate degree positive correlation between two variables
- if r= (0.001- 0.499), there is low degree positive correlation between two variables
- If r= (- 0.99 -0.7), there is high degree negative correlation between two variables
- If r= (- 0.699 - 0.5), there is moderate degree negative correlation between two variables
- If r= ( -0.499-0.001), there is low degree negative correlation between two variables.
19.
c. ( -1 to +1)
the limit of correlation coefficient is -1 to +1.
20.
c. (regression)
Correlation coefficient is equal to under root of the regression coefficient product.
21.
b. (13)
The number of degrees of freedom is the value in the final calculation of the statistics that are free to vary.
| Test | Degree of Freedom |
| One sample t- test Paired t- test | N-1 |
| Independent t- test | N-2 |
| Chi - square test Goodness of fit Contigency table | N-1 (R-1) (C-1) |
| ANOVA df between groups df within groups | Number of group -1 Number of subjects - number of group - 1 |
| Correlation | N-2 |
22.
c. (Middle points are joined)
A frequency polygon is a line graph of class frequency plotted against class mid - point. It can be obtained by connecting the mid- points of the top of the rectangle in the histogram.
23.
c. (Population)
- Population: Groups of individuals under study
- Statistic: Statistic is the number describing the whole sample
- Parameter: Parameter is the number describing the whole population.
24.
c. (Coefficient of variation)
Coefficient of variation has no units.
Range is measured in the same units as the data.
Standard deviation is also measured in the same units as the data
Variance is expressed as the square of the original units of the data
25.
b. (Axiomatic)
Axiomatic approaches of probability relate to set theory.
There are following types of approaches of probability:
Classical/Priori approaches: According to this approach, outcomes of a random experiment are equally likely or mutually exclusive.
Relative frequency approach/ Empirical approach: It uses the relative frequencies of past occurrence as probabilities to determine the chances of occurrence of event in future.
Subjective Approach: Based on the belief and best judgement of the person making of probability assessments.
Axiomatic/modern approach: Based on three axioms:
probability of events ranges from 0 to 1
probability of entire sample space is 1
If A and B are mutually exclusive events, then the probability of occurrence of either A or B denoted by P(AUB)= P(A) +P(B).
Note
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