We Have Learnt How to Use

T-distribution

We have learnt how to use :

(i) Z- Normal Distribution Table

(ii)Poisson Distribution Table

(iii)Binomial Distribution Table

A small question? What are the differences between (i), (ii) and (iii) above? When are they used?

Now we are going to learn how to use t-distribution table. After this we are going to visit F-distribution and ( CHI SQUARE TEST ).

When to use ‘t-distribution’? Answer:

When we estimated confidence intervals , we learnt that the difference in size between large and small samples is important when the population standard deviation Ϭ is unknown and must be estimated from the sample standard deviation. If the sample size n is 30 or less and Ϭ is not known , we use the t-distribution. The appropriate t distribution has (n-1) degrees of freedom. These rules apply to hypothesis testing , too.

Example:

A personnel doctor of a major corporation is recruiting a lot of Banglabest workers for a special job in southern tip of Malaysian peninsular. During the testing process, management asks how things are going, and she replies ,’fine. I think the average score on the aptitude test will be 90.’ When the management reviews 20 of the test results compiled, it finds that the mean score is 84, and the standard deviation is 11.

Data: (hypothesized value of the population mean)

N=20; // 84 is sample mean; and 11 is sample standard deviation.

If management wants to test the hypothesis at the 0.10 level of significance , what is the procedure?

H0: μ = 90 - null hypothesis

H1:  alternative hypothesis

And α = 0.10  level of significance for testing this hypothesis

Calculating the standard error of the mean = 11

Now we can compute the standard error of the mean  = 11/ = 11/4.47 = 2.46

So 2.46 is the estimated standard error of the mean.

Now try to draw.

I want you to draw.

A normal distribution curve with mean 90 and the end of each upper and lower part of the graph as 0.05 of the area.

Since management is interested in knowing whether the true mean score is larger or smaller than the hypothesized score a two tailed test is now used.

Take out the t-table from your book RM18 or RM70 Bluman’s book. You should get this at the back of the books respectively.

Since n=20; the appropriate number of degrees of freedom is 19 that is 20-1. Therefore we look in the t-distribution table. Under the 0.10 column until we reach the 19 degrees of freedom row. We got t value of 1 . 729

So to compute the limits of the acceptance region is:

Upper Limit = 94.25 upper limit

Lower Limit = 90 – 1.729(2.46) = 85.75 lower limit

If you have drawn the picture the limits of acceptance region 94.25 and 85.75 , and the sample mean , 84. From the drawing , we can see that the sample mean lies outside the acceptance region.

Therefore management reject the null hypothesis ( the doctors claim that the true mean score of the Banglabest workers being tested is 90).

ONE TAILED TEST under t-distribution

It is the same procedure like what we did with z Normal Distribution Table. See my drawings in the notes last week and this week. You should know how to draw 95% Confidence Interval, right on the graph itself. Easy?

Exercise 600:

Q1. Given a sample mean of 83, a sample standard deviation of 12.5 and sample size 22. Test the hypothesis that the value of the population mean is 70. Against the alternative that it is more than 70. Use 0.025 significance level.

Q2. Data given n = 25; sample mean 52 and sample variance 4.2; test the hypothesis that the population mean is 65, against the alternative that it is some other value. Use 0.01 significance level.

F-distribution/

Week after PRU13 we jump into

CHI SQUARED TEST

Then

2 Weeks after PRU 13 , we jump into

Reference: UIA Levin R.I and Rubin, D.S.(1980). Statistics for management. 5th edition. New York: prentice hall . pg 370 - 372