Running head: z-Scores, Type I and Type II Error, Null Hypothesis Testing ANSWER TEMPLATE 1

z-Scores, Type I and Type II Error, Null Hypothesis Testing ANSWER TEMPLATE 2

## z-Scores, Type I and Type II Error, Null Hypothesis Testing Answer Template

## Ashley Moore

## Capella University

# z-Scores, Type I and Type II Error, Null Hypothesis Testing Answer Template

The following assignment includes three sections consisting of:

1. *z *scores in SPSS.

2. Case studies of Type I and Type II errors.

3. Case studies of null hypothesis testing.

Additional notes:

· Answer in complete sentences.

· Follow APA rules for scholarly writing.

· Include a reference list if necessary.

· Save your answers and upload this template to the assignment area for grading.

## Section 1: *z* Scores in SPSS

A *z *score is typically analyzed when population mean (µ) and population standard deviation (σ) are known. However, in SPSS, we can still calculate *z *scores with the **grades.sav **data using the sample mean (*M*) and sample standard deviation (*s*). To do this, open **grades.sav **in SPSS. On the **Analyze** menu, point to **Descriptive Statistics**, and then click **Descriptives…**

You will be calculating and interpreting *z *scores for the **total **variable. In the **Descriptives** dialog box, move the **total **variable into the **Variable(s)** box. Select the **Save standardized values as variables **option and click **OK**.

SPSS provides descriptive statistics for **total** in the **Output** window. SPSS also creates a new variable in the far right column, labeled **Ztotal**, in the **Data Editor** area**. Ztotal** provides a *z *score for each case on the **total **variable. You are now prepared to answer the following Section 1 questions.

### Question 1

[What is the sample mean (*M*) and sample standard deviation (*s*) for **total**? You will use these values in Question 2 below.

Mean = 100.06

Std. Deviation = 14.19]

### Question 2

A z-score for this sample is calculated as [(X– M) ÷ s]. Locate Case #53’s unstandardized total score (X)in the Data Editor. In the formula below, replace X, M, s, to show how the z score in Ztotal is derived for Case #53.

[ (75-100.06) + 14.190] = -10.87

(*X *– *M*) ÷ *s* = -10.87

### Question 3

Run Descriptives on Ztotal. What are the mean and standard deviation of Ztotal? (Hint: “0E7” in SPSS is scientific notation for 0). Are the mean and standard deviation what you would expect? Justify your answer.

### [Its standard deviation is equal to 1.0 and its mean is equal to 0.0.

### Yes, that is what I expected because it is standardized normal variate; therefore, it should have a mean equal to zero and a standard deviation equal to 1.]

### Question 4

Case number 6 has a **Ztotal **score of 1.19. What does a *z* value of 1.19 represent?

[It means that when we standardize the normal variate then the probability is 0.4192.The observed z value is 1.19. A z value greater than zero shows the total deviations that some element is from the mean. In the given case it is 0.004192%.]

### Question 5

Identify the case with the lowest *z* score. Refer to Appendix A in the Warner (2013) text. Interpret the percentile rank of this *z* score rounded to whole numbers.

[The lowest z score is of case # 66 which is -3.24007. So according to attached appendix it shows a probability of 0.5006].

### Question 6

Identify the case with the highest *z* score. Refer to Appendix A in the Warner (2013) text. Interpret the percentile rank of this *z* score rounded to whole numbers.

[The highest z score is of case # 10 which is -1.53133. So according to attached appendix it shows a probability of 0.4370].

## Section 2: Cases Studies of Type I and Type II Errors

### Question 7

A jury must determine the guilt of a criminal defendant (not guilty, guilty). Identify how the jury would make a correct decision. Analyze how the jury would commit a Type I error versus a Type II error.

[In the present justice system an individual is innocent until a proven guilty protocol (the null hypothesis). If the jury commit a Type I error when finding if a defendant is innocent or guilty, they are not accepting (rejecting) the fact that the defendant is innocent hence they are not accepting the null hypothesis and an innocent person will be sentenced for a crime they did not commit. The same is right in Type II errors if the jury accepts the null hypothesis (innocence) then there are chances that a criminal will be set free.]

### Question 8

An I/O psychologist asks employees to complete surveys measuring job satisfaction and organizational citizenship behavior. She intends to measure the strength of association between these two variables. The researcher is concerned that she will commit a Type I error. What research decision influences the magnitude of risk of a Type I error in her study?

[In the given study, the psychologist measures the strength of association between the two variables, measuring job satisfaction and organizational citizenship behavior. Moreover, the researcher is concerned that she would make a Type I error. That is, rejecting the null hypothesis when it is true.

The critical region of the test is denoted by alpha which, represents the level of significance. Also, for any research study, it is difficult to avoid Type I error. Hence, the significance level is fixed for any test that is performed. This Type I error can be reduced when the sample size increases.

In other words, as the sample size increases the extent of Type I error can be reduced. This is because when the sample size is increased, there are more subjects to represent the population.

Hence, it can be concluded that the sample size that is selected for the study influences the magnitude of risk for a Type I error.]

### Question 9

A clinical psychologist is studying the efficacy of a new drug medication for depression. The study includes a placebo group (no medication) versus a treatment group (new medication). He then measures the differences in depressive symptoms across the two groups.

What would a Type I error represent within the context of his study? How can he reduce the risk of committing a Type I error? How does this decision affect the risk of committing a Type II error?

[A clinical psychologist is examining the viability of another medication prescription for depression. The investigation incorporates a placebo/fake treatment gathering versus a treatment gathering. The psychologist estimates the distinctions in depressive indications over the two gatherings and or groups.

What might a Type I error speak to inside the setting of his investigation? How might he diminish the danger of submitting a Type I error? How does this choice influence the danger of conferring a Type II error? A Type I error would speak to the dismissal that the new medication has turned out to be effective for depression treatment, while a Type II error would propose that the new medication has demonstrated ineffective for depression. The danger of submitting a Type I error can be decreased by expanding the example measure. The use of a substantial example or gathering will set a low alpha level and reduction odds of making a Type II error.]

## Section 3: Case Studies of Null Hypothesis Testing

### Question 10

You are running a series of statistical tests in SPSS using the standard criterion for rejecting a null hypothesis. You obtain the following *p* values.

Test 1 calculates group differences with a *p* value = .07.

Test 2 calculates the strength of association between two variables with a *p* value = .50.

Test 3 calculates group differences with a *p* value = .001.

For each test below, state whether or not you reject the null hypothesis. For each test, also explain what your decision implies in terms of group differences (Test 1 and Test 3) and in terms of the strength of association between two variables (Test 2).

Test 1 (group differences) = you do not reject the null hypothesis. This does not mean the null is true since .07 >.05. The results are 7% of the time.

Test 2 (strength of association) = Failed to reject the null hypothesis.

Test 3 (group differences) = Reject the null hypothesis.

### Question 11

A researcher calculates a statistical test and obtains a *p* value of .86. He decides to reject the null hypothesis. Is this decision correct, or has he committed a Type I or Type II error? Explain your answer.

[It will rely upon the level of significance of the test. On the off chance that p-value is ≥ level of significance α, at that point the choice ought to be not to reject the null hypothesis. On the off chance that null is rejected at such circumstances, type I error is submitted. In the event that p-value is < level of significance α, the choice ought to be to reject the null hypothesis. If the null is not rejected at such circumstances Type II error is submitted.]

### Question 12

You are proposing a research study that you would like to conduct while attending Capella University. During the proposal, a committee member asks you to explain in your own words what you meant by saying “*p* less than .05.” Provide an explanation.

[The p-value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event.

So, if a P-value is too low, meaning it is less than a particular critical point, which is .05 here, we reject null hypothesis. Because the probability of that certain event occurring is very low.]

# References

Warner, R. M. (2013). *Applied statistics: From bivariate through multivariate techniques* (2nd ed.). T