Large Sample Tests - Part 1

 

Sampling:

They are two types of survey classified into Census survey and Sample survey. If the information is collected from all the individuals, then it is called a census survey. It is also known as "Population Survey's" or " Complete census" or " Complete enumeration". Usually, the census survey is conducted for every 10 years and hence known as "Decimal Census survey". 5 years as Quinquennial. If the information is collected only from limited member of individuals, then it is called a sample survey. We prefer sample survey because.

• They take less time.
• They are less expensive.
• They have greater scope.
• They are more accurate.
• If testing leads to destruction, then we have to use sample surveys.
• If the population is too large, then we use sample survey.

Population: 

The total number of individuals or units or characteristics is known as population. It is classified into two types.

1. Heterogenous Population.
2. Homogenous Population.

If the characteristics of the individuals differ then the population is called "Heterogenous Population". Otherwise, it is called a "Homogenous Population". A Heterogenous population is to be sub divided into Homogenous populations before the sample selection. The number of observations in a population is called population size and it is denoted by 'N'.

Sample:

A finite subset of population is called "Sample". The number of observations in a sample is called Sample Size. It is denoted by 'n'. If n > 30 then the sample is called a "Large Sample" and if n < 30 then the sample is called "Small Sample".

Parameter:

Function of population values are called parameters. The parameters are denoted by theta. 

• Population Size - N.
• Population Proportion - P.
• Population Mean - Mew or X bar.
• Population Standard Deviation - S or Sigma.
• Population Correlation Coefficient - Rho.

Statistic:

Function of sample values are known as statistics. Statistic are denoted by 't'. 

• Sample Size - n.
• Sample Proportion - p.
• Sample Mean - x bar.
• Sample Standard deviation - s
• Sample Correlation coefficient - r.

Sampling Distribution:

If the population is heterogenous then it is divided into homogenous sub-populations. These homogenous sub-populations are denoted by t1, t2, t3,........tn. Let the average of these n homogenous sub-populations be x1, x2, x3,........xn. The sampling distribution consists of sample values with the respective averages.

Standard Error:

Standard error is the standard deviation of sample values in a sampling distribution. Standard error is denoted by.

The uses of standard error are:

• To test the reliability of the sample.
• In tests of significance to test null hypothesis.
• To find confidence interval or fruducial limits.

Hypothesis:

Hypothesis is a statement which can be tested and cannot be proved. 

Null Hypothesis:

Null hypothesis is a statement of no significant difference. It is denoted by "H_0".

Alternative Hypothesis:

A possible alternative to null hypothesis is called an alternative hypothesis. It is denoted by "H_1".

Level of significance:

The level of significance denotes error usually level of significance is fixed as 1% to 5%.

Large Sample Test: 

If the sample size n > 30 then the sample is called a large sample. The large sample tests are.

1. Testing of single proportion.
2. Testing of two proportions.
3. Testing of single mean.
4. Testing of two means.
5. Testing of single standard deviation.
6. Testing of two standard deviations.
7. Testing of single correlation coefficient.
8. Testing of two correlation coefficient.

In the large sample test the table value Z alpha at 5% level of significance is 1.96 and 1% level of significance is 2.58.

Test procedure:

In large sample test

1. Frame the null hypothesis H_0.
2. Frame an alternative hypothesis H_1.
3. Fix the level of significance.
4. To test the null hypothesis, use the Z-Statistic.

        

       5. Compare |Z| with the table value Z alpha. If |Z| < Z alpha, then accept H_0. If |Z| > Z alpha, then reject H_0.

I. Test procedure of single proportion:

We know that sample proportion(p) = x/n. Where x is a number of successes, and n is the sample size.

We know that.

Let us consider.

We frame the null and alternative hypothesis:

Or

    H_0: There is no significant difference between sample proportion and the population proportion.

~ H_1: There is significant difference between sample proportion and population proportion.

To test the H_0 we use the Z - Statistic:

We compare |Z cal| with the table value Z alpha. If |Z cal| < Z alpha, then accept the H_0. And id |Zcal| > Z alpha, then reject the H_0.

Confidence Interval:

To find the confidence interval we use the condition for accepting the null hypothesis.

Problems:

1. In a large consignment of oranges, a random sample of 64 oranges revealed that 14 oranges where bad. Is it reasonable to assume that 20% oranges where bad?

    Given that sample size n = 64 

    The number of bad oranges x = 14

    Proportion of bad oranges in the sample p = x/n = 14/64 = 0.2187

    Proportion of bad oranges in the population P = 20%

    

    We frame the null hypothesis.

Or

    H_0: The claim is valid ~ H_1: The claim is invalid.

    To test the H_0 we use the Z statistic

Since, |Z cal| < 1.96, we accept H_0 at 5% level of significance. The claim is valid or 20% oranges are bad in the population.

2. A manufacturing claimed that 98% of steel pipes which he supplied to a factory conform to specification and examination of a sample of 500 pieces revealed that 30 were defective. Test the claim at 5% level of significance.

Given that. 

    The proportion of good items in the population 

    The sample size (n) = 500.

    The number of defective items = 300.

    The number of good items in the sample 500 - 30 = 470 = x.

    Sample proportion,

    

    We frame the null hypothesis.

Or

    H_0 : The claim is valid ~ H_1: The claim is invalid.

To test H_0 we use the Z Statistic.

Since, |Z cal| > 1.96 we reject H_0 at 5% level of significance.

II. Test procedure of two proportions:

Let us consider two independent samples of sizes n1 and n2. Let x1 and x2 be the number of successes in the first and second samples.

Sample proportion of the first sample

Sample proportion of the second sample

Let us assume that. 

Let us consider.

We frame the null hypothesis.

Or

H_0: There is no significant difference between the two population proportions. ~ H_1: There is significant difference between the two population proportions.

To test the null hypothesis, we use the Z - Statistics

If population proportion are unknown, then we assume that.

OR

We compare |Z cal| with the table value Z alpha, If |Z cal| Z alpha then we accept H_0 and If |Z cal| > Z alpha then we reject H_0.

Problems:

1. A machine produces 20 defectives in a random sample of 400 items. After the machine is overhauled, only 10 times are defective out of 300. Is there any significant difference in the proportion of defectives.

Given that.

    The size of the sample 1 n1 = 400.

    The size of the sample 2 n2 = 300.

The number of successes in sample 1 and the number of successes in sample 2 as

The sample proportion of the first and second sample as

We frame the null hypothesis.

OR

H_0: There is no significant difference between proportion of defectives before and after overhauling.

~ H_1: There is significant difference between the proportion of defectives before and after overhauling.

To test H_0: we use the Z - Statistics

Where,

Since |Z Cal| < 1.96 we accept H_0 at 5% level of significance. There is no significance difference between proportion of defectives before and after overhauling.

2. Before an increasing exercise duty on tea, 800 persons out of a sample of 1000 persons out of a sample of 1000 persons are tea drinkers. After an increase exercise duty, 800 people with tea drinker in a sample of 1200 people. Test whether there is significance difference in the consumption of tea before and after increasing exercise duty.

Given that.

The size of sample one and the size of sample two as.

The number of tea drinkers before increase and the number of tea drinkers after increase as

The sample proportion of the first sample and the second sample as 

We frame the null hypothesis.

OR

H_0: There is no significance difference between the proportions of tea drinkers. ~ H_1: There is significant difference between the proportion of tea drinkers.

To test H_0 we use the Z - Statistic

Where,

Since | Z Cal| > 1.96 we reject H_0 at 5% level of significance. There is significant difference in the consumption of tea before and after increase in exercise duty.

Thank You!