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Thepvalueapproachfortestofhypothesis.doc

The p-value approach to hypothesis testing

Another approach to hypothesis testing is based on calculating p-value. The p-value is the probability of obtaining a test statistic equal to or more than the result obtained from the sample data, given that the null hypothesis is true. The p-value is often referred to as the observed level of significance, which is the smallest level at which the null hypothesis can be rejected for a given set of data. The decision rule for rejecting H0 in the p-value approach:

· If the p value is greater than or equal to (, the null hypothesis is not rejected.

· If the p value is less than (, the null hypothesis is rejected

Let’s look at the following example to discuss p-value further.

At a cereal plant, thousands of boxes of cereals are filled during each eight-hour shift. Boxes are supposed to contain an average, 368 grams of cereal as indicated in the label. The plant manager wants to determine whether or not the cereal-filling process is working properly. To study it, the plant manager plans to sample a random representative sample of 25 boxes, weigh each one, and then evaluate the difference between the sample statistic at a 95% confidence level.

First we could test the hypothesis as:

H0: ( = 368 vs. H1: ( ( 368

For ( = .05 (for two tail test we look at z for 0.25), we find z = ( 1.96. In testing suppose we obtained a z value (using the computational formula) of 1.50 and did not reject the null hypothesis as 1.50 is greater than –1.96 and less than 1.96.

Now we use the p=value approach. For our two tail test, we wish to find the probability of obtaining a test statistic Z that is equal to and extreme than 1.50 standard deviation units from the center of the standardized normal distribution. This means that we need to compute the probability of obtaining a z value greater than ( 1.65. From the z table we find out that the probability of obtaining z-value below ( 1.50 is 0.0668. So the probability of obtaining a value below +1.50 is 0.9332. Therefore, the probability of obtaining a value about +1.50 is

1 ( 0.9332 = 0.0668. Thus the p-value for this two-tailed test is 0.0668 + 0.0668 = 0.1336. This result is interpreted to mean that the probability of obtaining a result equal to more extreme than the one observed is 0.1336. Because this is greater than ( = .05, the null hypothesis is not rejected.

Unless we are dealing with a test statistic that follows the normal distribution, the computation of p-value can very difficult. Thus it is fortunate that software such as StatCrunch or, Microsoft EXCEL routinely present the p-value as part of the output for hypothesis-testing procedure.

In general, most of the time the decision to take an action is sometimes based on whether there is a sufficient evidence to reject a null hypothesis (H0: ( = ) by testing with a specified rejection value (example: .05). In such situation, it is often useful to know all of the information – called weight of evidence – that the hypothesis test provides against the null hypothesis and in favor of the alternative hypothesis.

The most informative way to measure the weight of evidence is to use p-value. For hypothesis test, we can interpret the p-value to be the probability, computed assuming that the null hypothesis H0 is true, of observing a value of the test statistic that is at least as extreme as the value actually computed from the sample data. The smaller the value is, the less likely are the sample results if the null hypothesis H0 is true. So the stronger is the evidence that H0 is false and the alternative hypothesis H1 is True. When considering using p-value, we could use the rule that “ We reject H0 at a level of significance α iff the p-value is less than α”.

We always use z test for the proportion.

The formula for computing the test statistic for proportion is:

Zo =(p1 hat ( p2 hat) ÷ (( (p hat(1( p hat) * ((1/n1 + 1/n2)}( where p1 hat = x1 bar / n1 , p2 hat = x2 bar / n2 and p hat=(x1+x2) ÷ (n1 + n2)

We will have all these values given in your problem.

You follow the following steps for conducting test of hypothesis utilizing p.

1. Ho: p1( p2 = 0 vs. H1: µ1( µ2 > or, < or ≠ 0, based on the description of the problem

2. α=

3. Zo =(p1 hat ( p2 hat) ÷ (( (p hat(1( p hat) * ((1/n1 + 1/n2)}( where p1 hat = x1 bar / n1 , p2 hat = x2 bar / n2 and p hat=(x1+x2) ÷ (n1 + n2) Compute

4. Determine the critical value from the z table based on the alternate hypothesis and value of α

5. Make decision