Unit 7

Sampling

7.1 Sampling, Sampling distribution, Standard error

Population-

The population is the collection or group of observations under study.

The total number of observations in a population is known as population size and it is denoted by N.

Types of population-

1. finite population – the population contains finite numbers of observations is known as finite population
2. Infinite population- it contains infinite number of observations.
3. Real population- the population which comprises the items which are all present physically is known as real population.
4. Hypothetical population- if the population consists the items which are not physically present but their existence can be imagined, is known as hypothetical population.

Sample –

To get the information from all the elements of a large population may be time consuming and difficult. And also, if the elements of population are destroyed under investigation then getting the information from all the units is not make a sense. For example, to test the blood, doctors take very small amount of blood. Or to test the quality of certain sweet we take a small piece of sweet in such situations, a small part of population is selected from the population which is called a sample

Complete survey-

When each and every element of the population is investigated or studied for the characteristics under study then we call it complete survey or census.

Sample Survey-

When only a part or a small number of elements of population are investigated or studied for the characteristics under study then we call it sample survey or sample enumeration Simple Random Sampling or Random Sampling

The simplest and most common method of sampling is simple random sampling. In simple random sampling, the sample is drawn in such a way that each element or unit of the population has an equal and independent chance of being included in the sample. If a sample is drawn by this method then it is known as a simple random sample or random sample

Simple Random Sampling without Replacement (SRSWOR)

In simple random sampling, if the elements or units are selected or drawn one by one in such a way that an element or unit drawn at a time is not replaced back to the population before the subsequent draws is called SRSWOR.

Suppose we draw a sample from a population, the size of sample is n and the size of population is N, then total number of possible samples is

Simple Random Sampling with Replacement (SRSWR)

In simple random sampling, if the elements or units are selected or drawn one by one in such a way that a unit drawn at a time is replaced back to the population before the subsequent draw is called SRSWR.

Suppose we draw a sample from a population, the size of sample is n and the size of population is N, then total number of possible sample is .

Parameter-

A parameter is a function of population values which is used to represent the certain characteristic of the population. For example, population mean, population variance, population coefficient of variation, population correlation coefficient, etc. are all parameters. Population parameter mean usually denoted by μ and population variance denoted by

Sample mean and sample variance-

Let be a random sample of size n taken from a population whose pmf or pdf function f(x,

Then the sample mean is defined by-

Statistic-

Any quantity which does not contain any unknown parameter and calculated from sample values is known as statistic.

Suppose is a random sample of size n taken from a population with mean μ and variance then the sample mean-

Is a statistic.

Estimator and estimate-

if a statistic is used to estimate an unknown population parameter then it is known as estimator and the value of the estimator based on observed value of the sample is known as estimate of parameter.

Standard error-

The standard deviation of the sampling distribution is known as standard error.

When we increase the sample size then standard increases.

It plays an important role in large sample theory.

Note- The reciprocal of the standard error is called ‘precision’.

If the size of the sample is less than 30, it is considered as small sample otherwise it is called large sample.

The sampling distribution of large samples is assumed to be normal.

Key takeaways

1. Population-

The population is the collection or group of observations under study.

2.     Simple Random Sampling without Replacement (SRSWOR)

If the elements or units are selected or drawn one by one in such a way that an element or unit drawn at a time is not replaced back to the population before the subsequent draws is called SRSWOR.

3.     Simple Random Sampling with Replacement (SRSWR)

If the elements or units are selected or drawn one by one in such a way that a unit drawn at a time is replaced back to the population before the subsequent draw is called SRSWR.

4.     Statistic-

Any quantity which does not contain any unknown parameter and calculated from sample values is known as statistic.

7.2 Testing of hypothesis for means, Confidence limits for means

Hypothesis-

A hypothesis is a statement or a claim or an assumption about the value of a population parameter.

Similarly, in case of two or more populations a hypothesis is comparative statement or a claim or an assumption about the values of population parameters.

For example-

If a customer of a car wants to test whether the claim of car of a certain brand gives the average of 30km/hr is true or false.

Simple and composite hypotheses-

If a hypothesis specifies only one value or exact value of the population parameter then it is known as simple hypothesis. And if a hypothesis specifies not just one value but a range of values that the population parameter may assume is called a composite hypothesis.

Null and alternative hypothesis

The hypothesis which is to be tested as called the null hypothesis.

The hypothesis which complements to the null hypothesis is called alternative hypothesis.

In the example of car, the claim is and its complement is .

The null and alternative hypothesis can be formulated as-

Critical region-

Let be a random sample drawn from a population having unknown population parameter .

The collection of all possible values of is called sample space and a particular value represent a point in that space.

In order to test a hypothesis, the entire sample space is partitioned into two disjoint sub-spaces, say, and S – . If calculated value of the test statistic lies in , then we reject the null hypothesis and if it lies in then we do not reject the null hypothesis. The region is called a “rejection region or critical region” and the region is called a “non-rejection region”.

Therefore, we can say that

“A region in the sample space in which if the calculated value of the test statistic lies, we reject the null hypothesis then it is called critical region or rejection region.”

The region of rejection is called critical region.

The critical region lies in one or two tails on the probability curve of sampling distribution of the test statistic it depends on the alternative hypothesis.

Therefore, there are three cases-

One tailed and two tailed tests-

Procedure for testing a hypothesis-

Step-1: first we set up null hypothesis and alternative hypothesis .

Step-2: after setting the null and alternative hypothesis, we establish a

criteria for rejection or non-rejection of null hypothesis, that is,

decide the level of significance (), at which we want to test our

hypothesis. Generally, it is taken as 5% or 1% (α = 0.05 or 0.01).

step-3: The third step is to choose an appropriate test statistic under H0 for

testing the null hypothesis as given below

Now after doing this, specify the sampling distribution of the test statistic preferably in the standard form like Z (standard normal), , t, F or any other well-known in literature

Step-4: Calculate the value of the test statistic described in Step III on the basis of observed sample observations.

Step-5: Obtain the critical (or cut-off) value(s) in the sampling distribution of the test statistic and construct rejection (critical) region of size .

Generally, critical values for various levels of significance are putted in the form of a table for various standard sampling distributions of test statistic such as Z-table, -table, t-table, etc

Step-6: After that, compare the calculated value of test statistic obtained from Step IV, with the critical value(s) obtained in Step V and locates the position of the calculated test statistic, that is, it lies in rejection region or non-rejection region.

Step-7: in testing the hypothesis we have to reach at a conclusion, it is performed as below-

First- If calculated value of test statistic lies in rejection region at level of significance then we reject null hypothesis. It means that the sample data provide us sufficient evidence against the null hypothesis and there is a significant difference between hypothesized value and observed value of the parameter

Second- If calculated value of test statistic lies in non-rejection region at level of significance then we do not reject null hypothesis. Its means that the sample data fails to provide us sufficient evidence against the null hypothesis and the difference between hypothesized value and observed value of the parameter due to fluctuation of sample

Procedure of testing of hypothesis for large samples-

The sample size more than 30 is considered as large sample size. So that for large samples, we follow the following procedure to test the hypothesis.

Step-1:  first we set up the null and alternative hypothesis.

Step-2: After setting the null and alternative hypotheses, we have to choose level of significance. Generally, it is taken as 5% or 1% (α = 0.05 or 0.01). And accordingly, rejection and non-rejection regions will be decided.

Step-3: Third step is to determine an appropriate test statistic, say, Z in case of large samples. Suppose Tn is the sample statistic such as sample

mean, sample proportion, sample variance, etc. for the parameter

then for testing the null hypothesis, test statistic is given by

Step-4: the test statistic Z will be assumed to be approximately normally distributed with mean 0 and variance 1 as

By putting the values in above formula, we calculate test statistic Z.

Suppose z be the calculated value of Z statistic

Step-5: After that, we obtain the critical (cut-off or tabulated) value(s) in the sampling distribution of the test statistic Z corresponding to assumed in Step II. we construct rejection (critical) region of size α in the probability curve of the sampling distribution of test statistic Z.

Step-6: Take the decision about the null hypothesis based on the calculated and critical values of test statistic obtained in Step IV and Step V.

Since critical value depends upon the nature of the test that it is one tailed test or two-tailed test so following cases arise-

Case-1 one-tailed test- when

(right-tailed test)

In this case, the rejection (critical) region falls under the right tail of the probability curve of the sampling distribution of test statistic Z.

Suppose is the critical value at level of significance so entire region greater than or equal to is the rejection region and less than

is the non-rejection region

If z (calculated value ) ≥   (tabulated value), that means the calculated value of test statistic Z lies in the rejection region, then we reject the null hypothesis H0 at level of significance. Therefore, we conclude that sample data provides us sufficient evidence against the null hypothesis and there is a significant difference between hypothesized or specified value and observed value of the parameter.

If z <    that means the calculated value of test statistic Z lies in non-rejection region, then we do not reject the null hypothesis H0 at level of significance. Therefore, we conclude that the sample data fails to provide us sufficient evidence against the null hypothesis and the difference between hypothesized value and observed value of the parameter due to fluctuation of sample.

So, the population parameter

Case-2:  when

(left-tailed test)

the rejection (critical) region falls under the left tail of the probability curve of the sampling distribution of test statistic Z.

Suppose -   is the critical value at level of significance then entire region less than or equal to -   is the rejection region and greater than -is the non-rejection region

If z ≤-, that means the calculated value of test statistic Z lies in the rejection region, then we reject the null hypothesis H0 at level of significance.

If z >-, that means the calculated value of test statistic Z lies in the non-rejection region, then we do not reject the null hypothesis H0 at   level of significance.

In case of two tailed test-

In this case, the rejection region falls under both tails of the probability curve of sampling distribution of the test statistic Z. Half the area (α) i.e. α/2 will lies under left tail and other half under the right tail. Suppose and are the two critical values at the left-tailed and right-tailed respectively. Therefore, entire region less than or equal to and greater than or equal to are the rejection regions and between -is the non-rejection region.

If Z that means the calculated value of test statistic Z lies in the rejection region, then we reject the null hypothesis H0 at level of significance.

If that means the calculated value of test statistic Z lies in the non-rejection region, then we do not reject the null hypothesis H0 at level of significance.

Testing of hypothesis for population mean using Z-Tesnt

For testing the null hypothesis, the test statistic Z is given as-

The sampling distribution of the test statistics depends upon variance

So that there are two cases-

Case-1: when is known -

The test statistic follows the normal distribution with mean 0 and variance unity when the sample size is the large as the population under study is normal or non-normal. If the sample size is small then test statistic Z follows the normal distribution only when population under study is normal. Thus,

Case-1: when is unknown –

We estimate the value of by using the value of sample variance

After that, we calculate the value of test statistic as may be the case ( is known or unknown) and compare it with the critical value at prefixed level of significance α.

Example: A company of pens claims that a certain pen manufactured by him has a mean writing-life at least 460 A-4 size pages. A purchasing agent selects a sample of 100 pens and put them on the test. The mean writing-life of the sample found 453 A-4 size pages with standard deviation 25 A-4 size pages. Should the purchasing agent reject the manufacturer’s claim at 1% level of significance?

Sol.

It is given that-

The null and alternative hypothesis will be-

Also the alternative hypothesis left-tailed so that the test is left tailed test.

Here, we want to test the hypothesis regarding population mean when population SD is unknown. So we should used t-test for if writing-life of pen follows normal distribution. But it is not the case. Since sample size is n = 100 (n > 30) large so we go for Z-test. The test statistic of Z-test is given by

We get the critical value of left tailed Z test at 1% level of significance is

Since calculated value of test statistic Z (= ‒2.8) is less than the critical value

(= −2.33), that means calculated value of test statistic Z lies in rejection region so we reject the null hypothesis. Since the null hypothesis is the claim so we reject the manufacturer’s claim at 1% level of significance.

Example: A big company uses thousands of CFL lights every year. The brand that the company has been using in the past has average life of 1200 hours. A new brand is offered to the company at a price lower than they are paying for the old brand. Consequently, a sample of 100 CFL light of new brand is tested which yields an average life of 1220 hours with standard deviation 90 hours. Should the company accept the new brand at 5% level of significance?

Sol.

Here we have-

The company may accept the new CFL light when average life of

CFL light is greater than 1200 hours. So the company wants to test that the new brand CFL light has an average life greater than 1200 hours. So our claim is > 1200 and its complement is ≤ 1200. Since complement contains the equality sign so we can take the complement as the null hypothesis and the claim as the alternative hypothesis. Thus,

Since the alternative hypothesis is right-tailed so the test is right-tailed test.

Here, we want to test the hypothesis regarding population mean when population SD is unknown, so we should use t-test if the distribution of life of bulbs known to be normal. But it is not the case. Since the sample size is large (n > 30) so we can go for Z-test instead of t-test.

Therefore, test statistic is given by

The critical values for right-tailed test at 5% level of significance is

 1.645

Since calculated value of test statistic Z (= 2.22) is greater than critical value (= 1.645), that means it lies in rejection region so we reject the null hypothesis and support the alternative hypothesis i.e. we support our claim at 5% level of significance

Thus, we conclude that sample does not provide us sufficient evidence against the claim so we may assume that the company accepts the new brand of bulbs

Significance test of difference between sample means

Given two independent examples and with means standard derivations from a normal population with the same variance, we have to test the hypothesis that the population means   are same For this, we calculate

It can be shown that the variate t defined by (1) follows the t distribution with degrees of freedom.

Cor. If the two samples are of same size and the data are paired, then t is defined by

Example

Eleven students were given a test in statistics. They were given a month’s further tuition and the second test of equal difficulty was held at the end of this. Do the marks give evidence that the students have benefitted by extra coaching?

Sol. We compute the mean and the S.D. of the difference between the marks of the two tests as under:

 we find that (for v=10) =2.228. As the calculated value of , the value of t is not significant at 5% level of significance i.e. the test provides no evidence that the students have benefitted by extra coaching.

Example:

From a random sample of 10 pigs fed on diet A, the increase in weight in certain period were 10,6,16,17,13,12,8,14,15,9 lbs. For another random sample of 12 pigs fed on diet B, the increase in the same period were 7,13,22,15,12,14,18,8,21,23,10,17 lbs. Test whether diets A and B differ significantly as regards their effect on increases in weight?

Sol. We calculate the means and standard derivations of the samples as follows

Testing of hypothesis for difference of two population means using Z-Test-

Let there be two populations, say, population-I and population-II under study.

Also let denote the means and variances of population-I and population-II respectively where both are unknown but may be known or unknown. We will consider all possible cases here. For testing the hypothesis about the difference of two population means, we draw a random sample of large size n1 from population-I and a random sample of large size n2 from population-II. Let be the means of the samples selected from population-I and II respectively.

These two populations may or may not be normal but according to the central limit theorem, the sampling distribution of difference of two large sample means asymptotically normally distributed with mean and variance

Example: A college conducts both face to face and distance mode classes for a particular course indented both to be identical. A sample of 50 students of face-to-face mode yields examination results mean and SD respectively as-

and other sample of 100 distance-mode students yields mean and SD of their examination results in the same course respectively as:

Are both educational methods statistically equal at 5% level?

Level of significance-

The probability of type-1 error is called level of significance of a test. It is also called the size of the test or size of the critical region. Denoted by .

Basically, it is prefixed as 5% or 1% level of significance.

If the calculated value of the test statistics lies in the critical region then we reject the null hypothesis.

The level of significance relates to the trueness of the conclusion. If null hypothesis do not reject at level 5% then a person will be sure “concluding about the null hypothesis” is true with 95% assurance but even it may false with 5% chance.

Confidence limits-

Key takeaways

7.3 Student’s t distribution

General procedure of t-test for testing hypothesis-

Let X1, X2,…, Xn be a random sample of small size n (< 30) selected from a normal population, having parameter of interest, say,

which is actually unknown but its hypothetical value- then

Step-1: First of all, we setup null and alternative hypotheses

Step-2: After setting the null and alternative hypotheses our next step is to decide a criteria for rejection or non-rejection of null hypothesis i.e. decide the level of significance at which we want to test our null hypothesis. We generally take = 5 % or 1%.

Step-3: The third step is to determine an appropriate test statistic, say, t for testing the null hypothesis. Suppose Tn is the sample statistic (may be sample mean, sample correlation coefficient, etc. depending upon ) for the parameter then test-statistic t is given by

Step-4: As we know, t-test is based on t-distribution and t-distribution is described with the help of its degrees of freedom, therefore, test statistic t follows t-distribution with specified degrees of freedom as the case may be.

By putting the values of Tn, E(Tn) and SE(Tn) in above formula, we calculate the value of test statistic t. Let t-cal be the calculated value of test statistic t after putting these values.

Step-5: After that, we obtain the critical (cut-off or tabulated) value(s) in the sampling distribution of the test statistic t corresponding to assumed in Step II. The critical values for t-test are corresponding to different level of significance (α). After that, we construct rejection (critical) region of size in the probability curve of the sampling distribution of test statistic t.

Step-6: Take the decision about the null hypothesis based on calculated and critical value(s) of test statistic obtained in Step IV and Step V respectively.

Critical values depend upon the nature of test.

The following cases arises-

In case of one tailed test-

Case-1:

[Right-tailed test]

In this case, the rejection (critical) region falls under the right tail of the probability curve of the sampling distribution of test statistic t.

Suppose is the critical value at level of significance then entire region greater than or equal to is the rejection region and less than is the non-rejection region.

If ≥  that means calculated value of test statistic t lies in the rejection (critical) region, then we reject the null hypothesis at level of significance.

If < that means calculated value of test statistic t lies in non rejection region, then we do not reject the null hypothesis at level of significance.

Case-2:

[Left-tailed test]

In this case, the rejection (critical) region falls under the left tail of the probability curve of the sampling distribution of test statistic t.

Suppose -   is the critical value at level of significance then entire region less than or equal to - is the rejection region and greater than - is the non-rejection region.

If  ≤ −  that means calculated value of test statistic t lies in the rejection (critical) region, then we reject the null hypothesis   at level of significance.

If > −, that means calculated value of test statistic t lies in the non-rejection region, then we do not reject the null hypothesis at level of significance.

In case of two tailed test-

In this case, the rejection region falls under both tails of the probability curve of sampling distribution of the test statistic t. half the area (α) i.e. α/2 will lies under left tail and other half under the right tail. Suppose -, and are the two critical values at the left- tailed and right-tailed respectively. Therefore, entire region less than or equal to -and greater than or equal to are the rejection regions and between -and is the non rejection region.

If   ≥ or ≤ -, that means calculated value of test statistic t lies in the rejection(critical) region, then we reject the null hypothesis   at level of significance.

And if - <    < , that means calculated value of test statistic t lies in the non-rejection region, then we do not reject the null hypothesis at level of significance.

Testing of hypothesis for population mean using t-Test

There are the following assumptions of the t-test-

1. Sample observations are random and independent.
2. Population variance is unknown
3. The characteristic under study follows normal distribution.

For testing the null hypothesis, the test statistic t is given by-

Example: A tube manufacturer claims that the average life of a particular category

of his tube is 18000 km when used under normal driving conditions. A random sample of 16 tube was tested. The mean and SD of life of the tube in the sample were 20000 km and 6000 km respectively.

Assuming that the life of the tube is normally distributed, test the claim of the manufacturer at 1% level of significance using appropriate test.

Sol.

7.4 Chi-square distribution as a test of goodness of fit.

When a fair coin is tossed 80 times, we expect from the theoretical considerations that heads will appear 40 times and tail 40 times. But this never happens in practice that is the results obtained in an experiment do not agree exactly with the theoretical results. The magnitude of discrepancy between observations and theory is given by the quantity (pronounced as chi squares). If the observed and theoretical frequencies completely agree. As the value of increases, the discrepancy between the observed and theoretical frequencies increases.

Definition.

Chi – square distribution

If be n independent normal variates with mean zero and s.d. unity, then it can be shown that is a random variate having distribution with ndf.

The equation of the curve is

(1)  Properties of distribution

Goodness of fit

The values of is used to test whether the deviations of the observed frequencies from the expected frequencies are significant or not. It is also used to test how will a set of observations fit given distribution therefore provides a test of goodness of fit and may be used to examine the validity of some hypothesis about an observed frequency distribution. As a test of goodness of fit, it can be used to study the correspondence between the theory and fact.

This is a nonparametric distribution free test since in this we make no assumptions about the distribution of the parent population.

Procedure to test significance and goodness of fit

Example. A set of five similar coins is tossed 320 times and the result is

Solution.

Example. Fit a Poisson distribution to the following data and test for its goodness of fit at level of significance 0.05.

Solution.

Example. In experiments of pea breeding, the following frequencies of seeds were obtained

Theory predicts that the frequencies should be in proportions 9:3:3:1. Examine the correspondence between theory and experiment.

Solution.

Key takeaways

References

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3. S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
4. W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
5. N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
6. B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.
7. T. Veerarajan, “Engineering Mathematics”, Tata Mcgraw-Hill, New Delhi, 2010
8. Higher engineering mathematics, HK Dass