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## Karnataka 1st PUC Statistics Model Question Paper 4 with Answers

Time: 3 Hrs 15 Min

Max. Marks: 100

Note :

- Graph sheets and statistical tables will be supplied on request.
- Scientific calculators may be used.
- All working steps should be clearly shown.

Section-A

I. Answer any TEN of the following questions. (10 × 1 = 10)

Question 1.

State ‘Croxton and Cowdon’s’ definition of statistics.

Answer:

Statistics is a science of collection, presentation, analysis and interpolation of numerical data.

Question 2.

Who is an investigator?

Answer:

The person who conducts statistical investigation is a investigator.

Question 3.

What is tabulation of the data?

Answer:

Tabulation is the process of systematic arrangement of statistical data in vertical columns and horizontal rows, in the form of a table.

Question 4.

Write the formula of mid-points of a class.

Answer:

Question 5.

Which graph is used to locate a median?

Answer:

Ogives (less than Ogive) are used to locate the value of median.

Question 6.

What is class frequency?

Answer:

The number of observations corresponding to a particular class is known as the class frequency.

Question 7.

Find geometric mean of 2 and 8.

Answer:

We know that GM = \(\sqrt{a \times b}=\sqrt{2 \times 8}=\sqrt{16}\) = 4

Question 8.

For a data if D_{5} = 50, then what is the value of P_{50}.

Answer:

Here D_{5} = 50, which divides into two equal parts. ∴ P_{50} also divides into two equal parts i.e. P_{5o} = 50.

Question 9.

Mention the type of correlation between ‘speed of a vehicle and distance covered by it.

Answer:

It is a positive correlation.

Question 10.

What is interpolation?

Answer:

Interpolation is the technique of estimating the unknown value of dependent variable (y) for a given value of independant variable (x) which is within the limits or range of the independent variable.

Question 11.

If P(A) = \(\frac { 2 }{ 5 }\) , then find P(A’)

2 2 3

Answer:

Given P(A) = \(\frac { 2 }{ 5 }\), then P(A’) = 1 – P(A) = 1 – \(\frac { 2 }{ 5 }\) = \(\frac { 3 }{ 5 }\)

Question 12.

Define a random variable.

Answer:

Random variable is a function which assigns a real number-to every sample point in the sample space.

Section-B

II. Answer any TEN of the following questions. (10 × 2 = 20)

Question 13.

What is a continuous variable? Give an example.

Answer:

A random variable which assumes all the possible values in its range is a continuous random variable.

Ex: Height or weight of children.

Question 14.

Mention two methods of sampling.

Answer:

- Simple random sampling.
- Systematic sampling
- Stratified sampling.

Question 15.

What do you mean by inclusive class interval? Give an example.

Answer:

In a class if lower limit as well as upper limits are included in the same class, such a class is called inclusive class interval.

Ex.C.I. = (0 – 9), (10 – 19)…. (40 – 49).

Question 16.

What are stubs and captions of a table?

Answer:

Row headings are called stubs and column headings are called captions of a table.

Question 17.

Mention two objectives of diagrams and graphs.

Answer:

- They can be remembered for longer period of time.
- They facilitates comparison.

Question 18.

What is histogram?

Answer:

Histogram is a pictorial representation of graphs of frequency distribution by means of adjacent rectangles, whose areas are proportional to the frequencies represented.

Question 19.

Find the harmonic mean of 1, \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\)

Answer:

Question 20.

For a data, if median is 50 and mean deviation from median is 12, then find the coefficient.

Answer:

Question 21.

What are regression lines? Where they intersect?

Answer:

The graphs of regression equations are called regression lines.

They intersect at (x̄, ȳ)

Question 22.

In case of two attributes, if N = 250, (AB) = 30, (A) = 100 and (B) = 50, then find the remaining classes and their frequencies.

Answer:

In a 2 × 2 contingency table of two attributes :

Question 23.

Two cards are drawn from a pack of 52 playing cards, what is the probability that they are king?

Answer:

Question 24.

If E(X) = 3 and E(X^{2}) = 25, then find SD (X).

Answer:

Section-C

III. Answer any EIGHT of the following questions. 8 × 5 = 40

Question 25.

Write the functions of statistics.

Answer:

The functions of statistics are :

- It simplifies complexity of the data.
- It reduces the bulk of the data.
- It adds precision to thinking.
- It helps in comparing different sets of figures.
- It guides in the formulation of policies and helps in planning. \
- It indicates trends and tendencies.

Question 26.

What are the guidelines for the construction of a questionnaire?

Answer:

The following points are considered while framing a schedule/questionnaire.

- Questions should be simple and easy to understand to get spontaneous answers.
- Questions should not confuse the reader and they should give one and only one meaning
- The number of questions should be kept minimum
- If possible the questions should be capable of getting a definite answers, with either yes or no, a number , a place or a date etc.,
- Questions should be capable of tabulating
- The questions should be such that the least intelligent and a educated can answer them with least trouble
- The questions should be arranged in logical order
- The questions which hurt the feelings of the informant and which are of personal and confidential nature should be avoided
- A far as possible the questions put should be corroborator at least on the point of importance 0 The questions should pre-tested in a small group of individuals before it is used
- Questionnaire should look attractive
- If possible a covering letter is attached along with the questionnaire.

Question 27.

Prepare a blank table showing the particulars relating to the students of a college classified according to :

(a) Faculty : Arts, Commerce, Science.

(b) Caste : SC/ST, Group I to III, others

(c) Sex : Male and female

Answer:

Students strength of a college according to faculty, caste and sex.

Question 28.

Following is the data regarding strength of a college. Draw percentage bar diagram for this data.

Answer:

Similarly others can be calculated.

Percentage bar diagram shows the students strength of a college according to sex for the year 2009-10.

Question 29.

For the following observations, Find mean, median and mode : 12, 42, 25, 35, 67, 25, 56, 5, 75

Answer:

Arrange in ascending order : 5, 12, 25, 25, 35,42, 56, 67, 75

n = 9

mode (z) = size of most repeated item = 25

Question 30.

Explain types of correlation with examples.

Answer:

The correlation may be

1. positive

2. negative

3. zero

1. Positive correlation : The two variables are said to be positively correlated if they vary • in the same direction.

Example

- Demand and supply
- Income and expenditure

2. Negative correlation : The two variables are said to be negatively correlated, if they vary in opposite direction.

Example

- Price and Demand/Sales
- Production and price of vegetables

3. Zero/non-correlation : The two variables does not show any related variation, they are said to be zero/non-correlated.

Example 1. Sales of pig iron and sale of pig iron.

Question 31.

Marks obtained by five students in two subjects are as follows : Find Spearman’s rank correlation coefficient for the data.

Answer:

Here the data is non-repeated so,

R_{1}, R_{2} be the ranks in accountancy. (X) and Statistics (Y)

There exists a high degree positive correlation between marks in Accountancy and statistics

Question 32.

200 candidates appeared for II PUC examination in a college and 60 of them succeeded. 35 received a special coaching or tutorial class and out of them 20 candidates succeeded. Using Yule’s coefficient, discuss whether the special coaching is effective or not.

Answer:

Let A and B be ‘Attending coaching class’ and success in examination.

Let α not attending coaching class and β – failure in the examination.

then contingency table can be prepared as below.

The Yule’s coefficient of association :

Conclusion : Here Q > 0.5, there exists a positive association i.e. special coaching is effective in success in examination.

Question 33.

Interpolate the index for 2008 from the following data.

Answer:

Let X and Y be the year and index no.

From the data there are 4 known values of y are given,

so expand the Binomial (y -1)^{4} = 0

i.e. y_{4} – 4y_{3} + 6y_{2} – 4y_{1} + y_{0} =0

322 – 4 × 313 + 6y_{2} = 4 × 281 +278 = 0

322 – 1252 + 6y_{2} – 1124 + 278 = 0

600 – 2376 + 6y_{2} = 0

-1776 + 6y_{2} = 0

∴ y_{2} = \(\frac { 1776 }{ 6 }\) = 296

Question 34.

State and prove addition theorem of probability for two non-mutually exclusive events.

Answer:

Statement: Let A and B be two events with respective probabilities P(A) and P(B). Then the probability of occurrence of at least one of these two events is

P(A ∪B) = P(A) + P(B) – P(A ∩B)

Proof: A random experiment results ‘n’ exhaustive events m and m, events are favourable to events A and B respectively. And P out comes are common to A × B.

The probabilities are

Event (A ∪B) i.e. occurrence of at least one of the events is (m_{1}, + m_{2} -D) favourable events.

P(A ∪B)= P(A) + P(B) – P(A ∩B): from result (1).

Hence the proof.

Question 35.

A bag contains 6 red balls and 4 white balls. What is the probability that two balls drawn are

(i) of the same colour

(ii) of different colours.

Answer:

(i) P (2 balls drawn are of same colour) = P(2 red balls) OR P(2 white balls)

(ii) P(drawing different coloured balls) = 1 – P(drawing 2 same coloured balls)

= 1 – \(\frac { 21 }{ 45 }\) = 1 – 0.47 = 0.53

∵ P(A’) = 1 – P(A)

Question 36.

If X is a random variable and ‘a’ is any constant, then prove that E(ax) = aE(X) and var(ax) = a^{2}var(x)

Answer:

Proof : (i) E(ax) = Σax(px)

By definition E(X) = ΣxP(X) = aΣxP(X)

∴ E(ax) = aE(x)

(ii) var(ax) = E[ax – aE(x)]^{2}

by definition of var (x) = E [x – E(x)]^{2} = E [ax – aE(x)]^{2} = a^{2}E [x – E(x)]^{2}

var(ax)=a^{2} var(x)

Section-D

IV. Answer any TWO of the following questions. (2 × 10 = 20)

Question 37.

Which series is better? and which is more consistent?

Answer:

Betterness will be decided on the basis of means of X and Y.

Consistency will be decided on the basis of C.V’s. (coefficient variations)

Here X̄ > Ȳ ; X series is better and

CV(Y) < (V(X); Y series is more consistent.

Question 38.

Find Karl Pearson’s coefficient of skewness for the following data.

Answer:

X = Midpoints or C.l.

Modal class = size of highest frequency

After converting into exclusive CI = 19.5 – 24.5

The distribution is negatively skewed

Question 39.

For the following bivariate data, find Karl Pearson’s coefficient of correlation

Answer:

There exists a high degree positive correlation between x and y.

Section-E

IV. Answer any TWO of the following questions. (2 × 5=10)

Question 40.

(a) The probability that a boy will pass an examination is \(\frac { 3 }{ 5 }\) and that of a girl is \(\frac { 2 }{ 5 }\). Find the probability that

(i) both of them passes the examination

(ii) at least one of them passes the examination.

Answer:

(i) P(Boy and Girl passes examination)

= P(B ∩G) = P(B).P(G) = \(\frac { 3 }{ 5 }\) × \(\frac { 2 }{ 5 }\) = \(\frac { 6 }{ 25 }\)

(ii)

P (at lest one of them passes the examination)

= P(A ∪B) = P(A) + P(B) – P(A ∩B)

P(B ∪G) = P(B) + P(G) – P(B ∩G)

\(=\frac{3}{5}+\frac{2}{5}-\frac{6}{25}=\frac{3 \times 5+2 \times 5-6}{25}=\frac{19}{25}=0.76\)

Question 40.

(b) Find the mathematical expectations of number of beads obtained when two fair coins are tossed.

Answer:

Let’X denote the number of heads obtained which take the values of X : 0 (No heads), 1 (one head) and 2 (2 heads) with respective probabilities \(\frac { 1 }{ 4 }\), \(\frac { 2 }{ 4 }\), \(\frac { 1 }{ 4 }\) .

Question 41.

Following are the weights in (kgs) of 40 students of a college. Prepare a continuous frequency distribution table with suitable class interval.

Weight in (kgs) :

45 56 50 41 55 51 46 50 45 57 64 48 53 43 63 45 57 44 54 59 49 52 42 61 51 63 48 56 45 50 55 50

Answer:

Highest value = 64, Lowest value = 41

Range = HV – LV = 64 – 41 = 23

Question 42.

Draw a histogram. Hence find the value of mode(Z) through the graph.

Answer:

Question 43.

For the following data find the second Quartile. (Q_{2}).

Answer:

Convert the less than cumulative frequency distribution in to simple (i.e. upper limits and L.C.F.are given)

Question 44.

There are 10 tickets in a bag which are numbered 1, 2, 3…10. Two tickets are drawn randomly one after the other with replacement. Find the expectation of the sum of the numbers drawn.

Answer:

Let X and Y be the numbers on the first and second tickets drawn.

Then the numbers on the 1st ticket drawn is a discrete random variable which takes the values : X – 1, 2, 3 …….. 10 with each probability \(\frac { 1 }{ 10 }\) each.

To

E(X) = ΣX.P(X)

= 1 × \(\frac { 1 }{ 10 }\) + 2 × \(\frac { 1 }{ 10 }\) + 3 × \(\frac { 1 }{ 10 }\) + …. +10 × \(\frac { 1 }{ 10 }\)

= \(\frac { 1 }{ 10 }\) [1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10] = \(\frac { 55 }{ 10 }\) = 5.5 10L J 10

Since X and Y are independent takes same values with respective probabilities.

∴ E(X + Y) = E(X).E(Y) = 5.5 + 5.5 = 11