## Introduction: Determining Statistical Significance Using a Z-test

__Overview:__

Purpose: In this instructable, you will learn how to determine if there is a statistical significance between two variables in regards to a social work problem. You will be using a Z-test to determine this significance.

Duration: 10-15 minutes, 10 steps

Supplies: Writing utensil, paper, and calculator

Level of Difficulty: Will need a basic understanding of algebra

__Terms (in alphabetical order):__

*Calculated mean* – The average of the values as determined by the tester

*Population size* – In statistics, all individuals, objects, or events that meet the criteria for study

*Null hypothesis* – The statement that there is no relationship between two variables of interest

*Rejection level* – Selected probability level at which the null hypothesis is rejected

*Two-tailed* - the relationship between the variables goes in either direction, meaning that the test is determining if there is one variable that has an overall effect on the other variable. Ex. Among medical social workers, females and males will differ in their job-satisfaction levels

*One-tailed* - the relationship between the variable is in one specific direction. Ex. Female medical social workers will have higher levels of job satisfaction than male medical social workers

*Statistical significance* – Judged too unlikely to have occurred because of sampling error

*True/Expected mean* – The original average of the values

*True standard deviation* – How much a set of values varies; allows for us to find how likely it is for a specific value to be obtained by doing a Z-test

*Z-score* - A measure of how many standard deviations below or above the population mean a score is

*Z–test* – A hypothesis-testing procedure used to decide if variables have statistical significance

*Z-table* – A table used in calculating the statistical significance

## Step 1: Read the Following Problem

I am interested in studying anxiety among students studying for midterms. I know the true mean on the anxiety scale of all students is 4 with a true standard deviation of 1. I’m studying a group of 100 students who are studying for midterms. I calculate a mean for these students on this scale of 4.2. (Note: higher scores = higher anxiety). The rejection level is 0.05. Is there a statistically significant difference between the general student population and students who are studying for midterms on this scale?

## Step 2: Identify

a. The true mean (expected mean)

b. The true standard deviation of the population

c. The calculated mean (observed mean)

d. The population size

e. The rejection level

## Step 3: Use the Following Formula to Find the "z-score"

z = __(observed mean-expected mean)__

(standard deviation/√population size)

## Step 4: Subtract the Rejection Level From "1"

Write down this value

## Step 5: Two-tailed or One-tailed Test?

For definitions and examples of two-tailed and one-tailed test, refer to the beginning of the instructable to the section titled: “Terms”

Write down if the test is two-tailed or one-tailed.

## Step 6: Additional Step for Two-tailed Test

If the test is one-tailed, leave the number calculated in step 3 as is. If it’s two-tailed, divide the value you calculated from step 3 in half.

Write down this number.

## Step 7: Use the Z-table

Access the Z-table, which is the first table under this step. Using the number you wrote down in step 6, find it in the center of the table. Once you find the number in the center, use the far left column and the top row to determine the value.

Write the value. For further instructions to find this value, the following is an example of how to use the z-table:

If your number was “0.0438” calculated in step 6, as found in the cross-section of column 3 and row 3 in the z-table excerpt, your value would be 0.11. The far left column of the table has the value of the first place decimal. The top row has the value for the second place decimal. See the second picture of an excerpt of the z-table for an example.

## Step 8: Reject the Null Hypothesis or Fail to Reject the Null Hypothesis

Compare the number you found in step 7 with the number you calculated in question 3 to determine if you are to reject the null hypothesis or if you are to fail to reject the null hypothesis.

Write down the number from step 3 Write down the number from step 7

If the number you calculated from step 7 is less than the number you calculated in step 3, you are to reject the null hypothesis. If the number you calculated from step 7 is greater than the number you calculated in step 3, you fail to reject the null hypothesis

Reject the null hypothesis or fail to reject the null hypothesis?

## Step 9: Determine Statistical Significance

If you reject the null hypothesis, then there is a statistical significance between the variables. If you fail to reject the null hypothesis, there is not a statistical significance between the variables.

Write down if there is or if there isn’t a statistical significance

## Step 10: Check Your Answers

- Step 3: 2
- Step 5: Two-tailed
- Step 6: 0.475
- Step 7: 1.96
- Step 8: Since 1.96 < 2, you are to reject the null hypothesis
- Step 9: There is a statistical significance