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The z-test

Overview

Whenever the normal probability curve is set up as the statistical distribution for testing hypothesis about a population, the z-tests are used. There are 4 types of z test that will be taken up.

Lesson 1: z-test of Hypothesis about a Population Mean

Before the z-one population test of hypothesis about a population mean is applied, certain assumptions must be met: (1) The (population standard deviation) is known.

(2) The data are either interval or ratio.

(3) Only one group is specified.

(4) The distributions of the scores follow the normal distribution. A special table called the z-table is used to facilitate the work on hypothesis testing. These values can be obtained using the table under the normal curve:

The z table|

Type / | 0.025| 0.01| 0.05|

One-tailed| 1.96| 2.33| 1.65|

Two tailed| 2.33| 2.58| 1.96|

Application:

In the national level, the average score in the National Secondary Achievement Tests is 485 with a standard deviation of 95. A random sample of 135 freshmen entering the Philippine Normal University shows a mean score of 620. Can we say that the mean of this group comes from a population whose true mean () = 485? or is there a significant difference between the national average score and the mean score of the incoming freshman students of PNU? Research Question: Is there a significant difference between the average score in NSAT and the mean score of the incoming freshmen of PNU? or

Does the sample of 135 entering students of PNU come from a population with = 485? Ho: = 485 or

Ho: There is no significant difference between the national average score and the mean score of the incoming freshmen students of PNU. We are hypothesizing that the true mean of the population from where we drew our sample is 485. This is our null hypothesis. H1: 485 or

H1: There is a significant difference between the national average score and the mean score of the incoming freshmen students of PNU.

The alternative hypothesis will be accepted if there is a sufficient evidence to reject the null hypothesis (H0). The alternative hypothesis can also be regarded as the research hypothesis. Since we are making a hypothesis about the population the symbol for parameter mean is always used. level : 0.05; 2-tailed test

critical value: z = 1.96

This is the level of accuracy or level of significance that we want for our test, which means that we are willing to allow an error of 5% in our decision whether to reject or accept the null hypothesis. The accuracy of our decision is 95%. The critical value of z at 0.05 level of significance is 1.96. Decision Rule:

Reject Ho if z 1.96.

Accept Ho if z < 1.96

We have to reject Ho if the computed value of z is equal to or greater than 1.96 because in the z distribution or the normal probability curve, any value of z exceeding the value to the right or to the left of is associated with a probability of less than 0.05 (p < .05). This area, which is less than .05, is also known as the region of rejection. If we say for example, that the true mean () of the population is 485, the probability that this can happen by chance alone if the computed z is greater than 1.96 is less than .05. Since the true mean of 485 can occur less than 5 times in a hundred, which is quite rare, we have no reason to doubt that our null hypothesis is true.

Rejection

Rejection

Region of Acceptance

-196z

1.96zz

Accept Ho if z < 1.96. Again, as per normal probability curve any value of z, which is less than 1.96, has a probability that is greater than .05, i.e. the area towards the center of the distribution. This area between –1.96 and +1.96 values of z can be regarded as the region of acceptance.

By our 0.05 criterion, if the probability that the true mean of 485 can occur by chance alone is greater than 0.05, or more than 5 times in a hundred, we say...