Large Sample Tests - Part 3

 

V. Testing of sample standard deviation:

Let us consider a random sample x1, x2,....... xi,....xn of size n which is assumed to be drawn from a normal population.

Then the sample standard deviation as

Since,

We have,

We frame the null hypothesis:

OR

H_0: There is no significant difference sample standard deviation and population standard deviation.

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

To test H_0 we use the Z Statistic.

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

Confidence Interval of population standard deviation:

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

Combine (1) and (2) we get,

Problems:

1. Random sample of 200 students in a college has a standard deviation of IQ 26. Test whether the standard deviation of IQ of all the students is 28.

Given that,

    The sample size, sample standard deviation, population standard deviation are.

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. The standard deviation of IQ of all the students is 28.

2. A manufactured claimed that standard deviation of life of cells is 30 hours. To test this claim a random sample of 100 cells for tested and standard deviation of life of the sample is found to be 26 hours. Test the claim at 1% level of significance.

Given that,

    The sample size, sample standard deviation, population standard deviation as

We frame the null hypothesis:

OR

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

To test the H_0 we use the Z  Statistic.

Since, |Z cal| < 2.58, we accept H_0 at 1% level of significance. The manufactures claim is valid.

VI. Testing of difference between population standard deviation:

Let us consider two independent samples x1, x2, ....... xi, ...... xn1 and y1, y2,......yj,....yn2 of sizes n1 and n2 which are assumed to be drawn from normal population.

The standard deviation of the first sample as

The standard deviation of the second sample is

We frame the null hypothesis:

OR

H_0: There is no significant difference between the population standard deviation. ~ H_1 : There is significant difference between the population standard deviation.

To test H_0 we use the Z statistic.

If Sigma X and Sigma Y are unknown then we assume the,

Where, 

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

Problems:

1. From the following data test for the significant difference between the standard deviation.

	Size	Standard deviation
Boys	100	17
Girls	150	11

Given that, 

We frame the null hypothesis,

OR

H_0: There is no significant difference between the population standard deviation. ~ H_1: There is significant difference between the population standard deviation.

To test H_0 we use the Z - Statistic

Since, |Z cal| > 1.96 we reject H_0 at 5% level of significance. There is significant difference between the population standard deviation. Standard deviation of girls < Standard deviation of boys. The girls are performing better than boys.

2. The standard deviation of life of 2 brands of 100 and 200 electric bulbs are observed to be 200 and 180 hours respectively. Test whether there is only significant difference between standard deviation of life of two brands of electric bulbs.

Given that,

We frame the null hypothesis:

To test the Z Statistic.

Since |Z cal| < 1.96 we accept H_0 at 5% level of significance. There is no significant difference between the standard deviation of lives of two brands of electric bulbs.

Thank You!