# T Distribution-Statistics-Solved Quizes, Exams for Statistics

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Statistics study consist on topics like estimates, F distribution, sampling, multiplication theorems, probability, random variable, T distribution, geometric probability distribution, marginal probability, skewness, symm...
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STA 301 FINAL TERM SOLVED ALL PAPERS Dated 16-12-2012 to 19-07-2012

STA301 BY AWAIS REHMAN PTNS01 TAUNSA SHARIF CAMPUS

Q. No 1

Ans T distribution

The Student‟s t-Distribution:

The mathematical equation of the t-distribution is as follows:

This distribution has only one parameter ν, which is known as the degrees of freedom of the t-

distribution. PROPERTIES OF STUDENT‟S t-DISTRIBUTION

The t-distribution has the following properties:

i) The t-distribution is bell-shaped and symmetric about the value t = 0, ranging from – ∞ to ∞.

ii) The number of degrees of freedom determines the shape of the t-distribution.

Thus there is a different t-distribution for each number of degrees of freedom.

As such, it is a whole family of distributions.

The t-distribution, for small values of ν, is flatter than the standard normal distribution which

means that the t distribution

is more spread out in the tails than is the standard normal distribution.

As the degrees of freedom increase, the t-

distribution becomes narrower and narrower, until, as n tends to infinity, it tends to coincide with

the standard normal distribution.

(The t-distribution can never become narrower than the standard normal distribution.)

iii) The t-distribution has a mean of zero, when ν ≥ 2. (The mean does not exist when ν = 1.) iv)

The median of the t-distribution is also equal to zero. docsity.com

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v) The t-distribution is unimodal. The density of the distribution reaches its maximum at t = 0

and thus the mode of the t- distribution is t = 0.

(The students will recall that, for any hump-shaped symmetric distribution, the mean, median

and mode are equal.)

Q. No 2

Husband and wife apply for same job and husband probability for job is like this 9/3 and wife job

probability was something 3/5 like, what is the probability that husband could get job and wife

not?

Q. No 3

a boy sell umbrellas on rainy days and earn 30\$ and in other days he can earn 24\$ , if the

probability of rain is 0.3% what will he expectation ?

Q. No 4

CENTRAL LIMIT THEOREM

The theorem states that: “If a variable X from a population has mean μ and finite variance σ2,

then the sampling distribution of the sample mean ⎯ X approaches a normal distribution with

mean μ and variance σ2/n as the sample size n approaches infinity.” As n → ∞, the sampling

distribution of ⎯ X approaches normality.

Due to the Central Limit Theorem, the normal distribution has found a central place in the theory

of statistical inference.(Since, in many situations, the sample is large enough for our sampling

distribution to be approximately normal, therefore we can utilize the mathematical properties of

the normal distribution to draw inferences about the variable of interest). The rule of thumb in

this regard is that if the sample size, n, is greater than or equal to 30, then we can assume that the docsity.com

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sampling distribution of ⎯ X is approximately normally distributed. On the other hand, If the

POPULATION sampled is normally distributed, then the sampling distribution of ⎯ X will also

be normal regardless of sample size. In other words, ⎯ X will be normally distributed with mean

μ and variance σ2/n.

Q. No 5

What is difference between constant and random variable?

A very important point to note here is that, from the MATHEMATICAL standpoint, simple

linear regression requires that X is a NON-RANDOM variable, whereas Y is a RANDOM

variable. For example, consider the case of agricultural experiments. If we conduct an

experiment to determine the optimal amount of a particular fertilizer to obtain the maximum

yield of a certain crop, then the amount of fertilizer is a non-random variable whereas the yield is

a random variable. This is so because the amount of fertilizer is in our OWN control. But, the

yield is a random variable because it is NOT in our control. In connection with determining the

line of BEST fit, the first point is that.Portion probability

Q. No 6

LEVEL OF SIGNIFICANCE

The probability of committing Type-I error can also be called the level of significance of a test.

Now, what do we mean by Type-I error? In order to obtain an answer to this question, consider

the fact that, as far as the actual reality is concerned, H0 is either actually true, or it is false. Also,

as far as our decision regarding H0 is concerned, there are two possibilities --- either we will

accept H0, or we will reject H0. The above facts lead to the following table:

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Q. No 7

Q. No 8

Degrees of freedom

In statistics, the number of degrees of freedom is the number of values in the final calculation of

a statistic that are free to vary. Mathematically, degrees of freedom is the dimension of the

domain of a random vector, or essentially the number of 'free' components: how many

components need to be known before the vector is fully determined.. In equations, the typical

symbol for degrees of freedom is (lowercase Greek letter nu). In text and tables, the

abbreviation "d.f." is commonly used.

. No 9

Statistical hypothesis testing

A statistical hypothesis test is a method of making decisions using data, whether from a

controlled experiment or an observational study (not controlled). In statistics, a result is called

statistically significant if it is unlikely to have occurred by chance alone, according to a pre-

determined threshold probability, the significance level. The phrase "test of significance" was

coined by Ronald Fisher: "Critical tests of this kind may be called tests of significance, and when

such tests are available we may discover whether a second sample is or is not significantly

different from the first. Hypothesis testing is sometimes called confirmatory data analysis, in

contrast to exploratory data analysis. In frequency probability, these decisions are almost always

made using null-hypothesis tests. These are tests that answer the question assuming that the null

hypothesis is true, what is the probability of observing a value for the test statistic that is at least

as extreme as the value that was actually observed?) More formally, they represent answers to

the question, posed before undertaking an experiment, of what outcomes of the experiment

would lead to rejection of the null hypothesis for a pre-specified probability of an incorrect

rejection.

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Q. No 10

DISTRIBUTION FUNCTION

The distribution function of a random variable X, denoted by F(x), is defined by F(x) = P(X < x).

The function F(x) gives the probability of the event that X takes a value LESS THAN OR

EQUAL TO a specified value x. The distribution function is abbreviated to d.f. and is also called

the cumulative distribution function (cdf) as it is the cumulative probability function of the

random variable X from the smallest value up to a specific value x.

Q. No 11

Q. No 12

EXPERIMENTAL DESIGN

By an experimental design, we mean a plan used to collect the data relevant to the problem under

study in such a way as to provide a basis for valid and objective inference about the stated

problem. The plan usually includes:

• The selection of treatments, whose effects are to be studied,

• The specification of the experimental layout, and

• The assignment of treatments to the experimental units.

All these steps are accomplished before any experiment is performed. Experimental Design is a

very vast area. In this course, we will be presenting only a very basic introduction of this area.

There are two types of designs:

Q. No 13

Define an independent and dependent variable in regression? 2 Marks

INDEPENDENT EVENTS

Two events A and B in the same sample space S, are defined to be independent (or statistically

independent) if the probability that one event occurs, is not affected by whether the other event

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has or has not occurred, that is P (A/B) = P (A) and P (B/A) = P (B). It then follows that two

events A and B are independent if and only if

P (A ∩ B) = P (A) P (B) and this is known as the special case of the Multiplication Theorem of

Probability.

OR

Independent Variable Definition

What is an independent variable? The independent variable is the variable that is

manipulated by the researcher. The independent variable is something that is hypothesized to

influence the dependent variable. The researcher determines for the participant what level or

condition of the independent variable that the participant in the experiment receives. For

example, each participant in the experiment may be randomly assigned to either an experimental

condition or the control condition.

Dependent Variable Definition

What is a dependent variable? The dependent variable is the variable that is simply

measured by the researcher. It is the variable that reflects the influence of the independent

variable. For example, the dependent variable would be the variable that is influenced by being

randomly assigned to either an experimental condition or a control condition.

Q. No 14

For the data set, we find out five number summary result: Xo=200, Xm=500 and Q2=350, (3

Marks)

Q. No 15

Discuss three properties of normal distribution? 3 Marks

NORMAL DISTRIBUTION

A continuous random variable is said to be normally distributed with mean μ and standard

deviation σ if its probability density function is given by

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Q. No 16

The 90% confidence interval for the population mean is 11 to 20, interpret this result? 3 Marks

Q. No 17

Define LSD test? 3 Marks

THE LEAST SIGNIFICANT DIFFERENCE (LSD) TEST

According to this procedure, we compute the smallest difference that would be judged

significant, and compare the absolute values of all differences of means with it. This smallest

difference is called the least significant difference or LSD, and is given by:

LEAST SIGNIFICANT DIFFERENCE (LSD):

where MSE is the Mean Square for Error, r is the size of equal samples, and tα/2 (ν) is the value

of t at α/2 level taken against the error degrees of freedom (ν).

The test-criterion that uses the least significant difference is called the LSD test. docsity.com

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Q. No 18

CORRELATION CO-EFFICIENT OF TWO RANDOM VARIABLES

Let X and Y be two r.v.‟s with non-zero variances σ2X and σ2Y. Then the correlation coefficient

which is a measure of linear relationship between X and Y, denoted by ρXY (the Greek letter

rho) or Corr(X, Y), is defined as:

If X and Y are independent r.v.‟s, then ρXY will be zero but zero correlation does not necessarily

imply independence.

Q. No 19

Confidence Interval?5 Marks

A confidence interval is an interval computed from the sample observations x1, x2….xn, with a

statement of how confident we are that the interval does contain the population parameter.

Confidence interval

In statistics, a confidence interval (CI) is a kind of interval estimate of a population parameter

and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated

from the observations), in principle different from sample to sample, that frequently includes the

parameter of interest, if the experiment is repeated. How frequently the observed interval

contains the parameter is determined by the confidence level or confidence coefficient

Confidence intervals consist of a range of values (interval) that act as good estimates of the

unknown population parameter.

Q. No 20

Calculate Harmonic Mean of given data?5 Marks

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Harmonic Mean Definition:

Harmonic mean is used to calculate the average of a set of numbers. Here the number of

elements will be averaged and divided by the sum of the reciprocals of the elements. The

Harmonic mean is always the lowest mean.

Harmonic Mean Formula :

Harmonic Mean = N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)

where

X = Individual score

N = Sample size (Number of scores)

Harmonic Mean Example: To find the Harmonic Mean of 1,2,3,4,5.

Step 1: Calculate the total number of values.

N = 5

Step 2: Now find Harmonic Mean using the above formula.

N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)

= 5/(1/1+1/2+1/3+1/4+1/5)

= 5/(1+0.5+0.33+0.25+0.2)

= 5/2.28

So, Harmonic Mean = 2.19

Q. No 21

What is the difference between an outcome and an event? (2)

Event

An event is any collection of outcomes of an experiment.

Formally, any subset of the sample space is an event.

Any event which consists of a single outcome in the sample space is called an elementary or

simple event. Events which consist of more than one outcome are called compound events.

Outcome

An outcome is the result of an experiment or other situation involving uncertainty.

The set of all possible outcomes of a probability experiment is called a sample space.

Q. No 22 docsity.com

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The mean of a Poisson distribution is 5 while its standard deviation is 4. Comment on it

POISSON DISTRIBUTION

The Poisson distribution is named after the French mathematician Sime‟on Denis Poisson (1781-

1840) who published its derivation in the year 1837.THE POISSON DISTRIBUTION ARISES

IN THE FOLLOWING TWO

SITUATIONS:

• It is a limiting approximation to the binomial distribution, when p, the probability of success is

very small but n, the number of trials is so large that the product np = μ is of a moderate size;

• a distribution in its own right by considering a POISSON PROCESS where events occur

randomly over a specified interval of time or space or length.

EXAMPLE

Two hundred passengers have made reservations for an airplane flight. If the probability that a

passenger who has a reservation will not show up is 0.01, what is the probability that exactly

three will not show up?

SOLUTION

Let us regard a “no show” as success. Then this is essentially a binomial experiment with n =

200 and p = 0.01. Since p is very small and n is considerably large, we shall apply the Poisson

distribution, using

μ= np = (200) (0.01) = 2.

Therefore, if X represents the number of successes (not showing up), we have

Q. No 22

If an automobile is driven on the average no more than 16000 Km per year then formulate the

null and alternative hypothesis (2)

NULL AND ALTERNATIVE HYPOTHESES

NULL HYPOTHESIS docsity.com

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A null hypothesis, generally denoted by the symbol H0, is any hypothesis which is to be tested

for possible rejection or nullification under the assumption that it is true.

A null hypothesis should always be precise such as „the given coin is unbiased‟ or „a drug is

ineffective in curing a particular disease‟ or „there is no difference between the two teaching

methods‟. The hypothesis is usually assigned a numerical value. For example, suppose we think

that the average height of students in all colleges is 62″. This statement

is taken as a hypothesis and is written symbolically as H0 : μ = 62″. In other words, we

hypothesize that μ = 62″.

ALTERNATIVE HYPOTHESIS

An alternative hypothesis is any other hypothesis which we are willing to accept when the null

hypothesis H0 is rejected. It is customarily denoted by H1 or HA. A null hypothesis H0 is thus

tested against an alternative hypothesis

H1. For example, if our null hypothesis is H0 : μ = 62″, then our alternative hypothesis may be

H1 : μ ≠ 62″ or H1 : μ < 62″.

Q. No 23

If the population proportions are gives as: P1 = 0.4, P2 = 0.20 find sigma^2 P-hat 1 - P-hat 2 ,

where n = 12.

Q. No 24

How many parameters are associated with F- distribution and what is the range of the

distribution? (3)

This distribution has two parameters ν1 and ν2, which are known as the degrees of freedom of

the F-distribution.The Fdistribution having the above equation have ν1 degrees of freedom in

the numerator and ν2 degrees of freedom in the denominator. It is usually abbreviated as F

(ν1, ν2).

PROPERTIES OF F-DISTRIBUTION

1. The F-distribution is a continuous distribution ranging from zero to plus infinity.

2. The curve of the F-distribution is positively skewed.

Q. No 25

Which of the following statement represents continuous data and discrete data? (5) docsity.com

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Continuous Data

A set of data is said to be continuous if the values / observations belonging to it may take on any

value within a finite or infinite interval. You can count, order and measure continuous data. For

example height, weight, temperature, the amount of sugar in an orange, the time required to run a

mile.

Discrete Data

A set of data is said to be discrete if the values / observations belonging to it are distinct and

separate, i.e. they can be counted (1,2,3,....). Examples might include the number of kittens in a

litter; the number of patients in a doctors surgery; the number of flaws in one meter of cloth;

gender (male, female); blood group (O, A, B, AB).

i) Number of shops in a plaza.

ii) Hourly temperature recorded by whether bureau.

iii) Inches of rainfall in a city.

iv) Number of passengers carried by rail every year.

v) Height measurements of boys studying in a college.

parameters of hyper geometric distribution

HYPERGEOMETRIC PROBABILITY DISTRIBUTION

There are many experiments in which the condition of independence is violated and the

probability of success does not remain constant for all trials. Such experiments are called hyper

geometric experiments. In other words, a hyper geometric experiment has the following

properties:

PROPERTIES OF HYPERGEOMETRIC EXPERIMENT

• The outcomes of each trial may be classified into one of two categories, success and failure.

• The probability of success changes on each trial.

• The successive trials are not independent.

• The experiment is repeated a fixed number of times. docsity.com

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The number of success, X in a hyper geometric experiment is called a hyper geometric random

variable and its probability distribution is called the hyper geometric distribution. When the

hyper geometric random variable X assumes a value x, the hyper geometric probability

distribution is given by the formula range of chi-square distribution. ans(0 to infinity) given x-

x(bar)=3,-3,4,-4,1,-1 find out mead deviation. ans is o what are steps involved in statistical

research find out upper quartile of standard normal distribution

ANS

Calculate sampling error if sample is 102 and population mean is 100

SAMPLING & NON-SAMPLING ERRORS

1. SAMPLING ERROR

The difference between the estimate derived from the sample (i.e. the statistic) and the true

population value (i.e. the parameter) is technically called the sampling error. For example,

ANS

Quartile deviation

QUARTILE DEVIATION

The quartile deviation is defined as half of the difference between the third and first quartiles i.e.

ANS

Pearson co-efficient of skewness:

A candidate for mayor in a large city hires the services of a poll-taking organization, and they

found that 62 of 100 educated voters interviewed support the candidate, and 69 of 150

uneducated voters support him. docsity.com

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