U4A1 – z Scores, Type I and II Errors, Hypothesis Testing

This is your second IBM SPSS assignment. It includes three sections in which you will:

1. Generate z scores for a variable in grades.sav and report and interpret them.

2. Analyze cases of Type I and Type II errors.

3. Analyze cases to either reject or not reject a null hypothesis.

Download the Unit 4 Assignment 1 Answer Template from the Resources area and use the template to complete the following sections:

• Section 1: z Scores in SPSS.

• Section 2: Case Studies of Type I and Type II Errors.

• Section 3: Case Studies of Null Hypothesis Testing.

Format your answers in narrative style, integrating supporting statistical output (table and graphs) into the narrative in the appropriate places (not all at the end of the document). See the Copy/Export Output Instructions in the Resources area for assistance.  Submit your answer template as an attached Word document in the assignment area.

Resources

z Scores, Type I and II Errors, Hypothesis Testing Scoring Guide.

Copy/Export Output Instructions.

Unit 4 Assignment 1 Answer Template.

APA Style and Format.

Guideline

Includes all relevant output; no irrelevant output is included. No errors in SPSS output.

Evaluates the computation, interpretation, and application of z scores.

Evaluates a real-world application of Type I and Type II errors and the research decisions that influence the relative risk of each.

Applies the logic of null hypothesis testing in an exemplary manner.

Without exception, communicates in a manner that is scholarly, professional, and consistent with the expectations for members in the identified field of study instructions

SPSS output can be selectively copied and pasted into Word by using the Copy command:

1. Click on the SPSS output in the viewer window.

2. Right-click for options.

3. Click on the COPY command.

4. Paste the output into a Microsoft Word document.

The Copy command will preserve the formatting of the SPSS tables and charts when pasting into Microsoft Word. An alternative method is to use the Export command:

1. Click on the SPSS output in the Viewer window.

2. Right-click for options.

3. Click on the Export command.

4. Save the file as Word/RTF (.doc) to your computer.

5. Open the .doc file.

INTRODUCTION – U4A1 ASSIGNMENT

In Unit 4, we begin the transition from descriptive statistics to inferential statistics that are studied in the

remainder of the course, including correlation, t tests, and analysis of variance. The interpretation of inferential

statistics requires an understanding of probability and the logic of null hypothesis testing. The logic and

interpretative skills addressed in this unit are vital to your success in interpreting data in this course as well as to

the success of learners who take advanced courses in statistics.

The Standard Normal Distribution and z Scores

A student receives an intelligence quotient (IQ) score of 115 on a standardized intelligence test. What is his or

her percentile rank? To calculate the percentile rank, you must understand the logic and application of the

standard normal distribution. The standard normal distribution is introduced in Chapter 1 of the Warner text

(Figure 1.4, p. 28). The advantage of the standard normal distribution is that the proportional area under the

curve is constant. This constancy allows for the calculation of a percentile rank of an individual score X for a

given distribution of scores.

When a population mean (μ) and population standard deviation (σ) are known, such as a distribution of scores

for a standardized intelligence test, the standard normal distribution determines the percentile rank of a given X

score (for example, 50th percentile on IQ). Note in Figure 1.4 (Warner, 2013) that around two-thirds of standard

scores (68.26%) fall within ±1 population standard deviation from the population mean. Approximately 95% of

standard scores fall within ±2 population standard deviations. Approximately 99% of standard scores fall within

±3 population standard deviations. Knowing this, we can begin the process of determining a percentile rank

from an individual score.

Consider a standardized IQ test where μ = 100 and σ = 15. A standard score, or z score, is calculated with

individual X scores rescaled to μ = 0 and σ = 1. The formula for a z score is [( X − μ) ÷ σ]. Refer to Figure 2.13 in

the Warner text (p. 61). For example, an IQ score of X = 100 would be rescaled to z = 0.00 [(100 − 100) ÷ 15 =

0]. A z score of 0 means that the X score is 0 standard deviations above or below the mean. An IQ score of X =

85 would be rescaled to z = −1.00 [(85 − 100) ÷ 15 = −1.00]. A z score of −1 represents an IQ score 1 standard

deviation below the mean. An IQ score of 130 would be rescaled to z = 2.00 [(130 − 100) ÷ 15 = 2.00]. A z score

of 2 represents an IQ score 2 standard deviations above the mean. In short, a negative z score falls to the left of

the population mean, whereas a positive z score falls to the right of the population mean.

Once a given z score is calculated for a given X score, its percentile rank can be determined. Refer to Appendix

A of the Warner text (pp. 1051–1055). For example, an IQ score of 115 is z = 1.00 (Column A). The area under

the curve between z = 0.00 and z = 1.00 is .3413 (Column B), and the area under the curve to the right of z =

1.00 is .1587 (Column C). If we add the area under the curve of .3413 to the other half of the distribution (.50),

we know that an IQ score of 115 is at about the 84th percentile (.50 + .3413). We can say that a student who

receives an IQ score of 115 outperforms about 84% of students on the IQ test, whereas about 16% of students

outperform him or her on the IQ test. Conversely, an IQ score of 85 is z = −1.00 (Column A at z = 1.00; only

positive z scores are listed). In this case, we know that the area under the curve between z = 0.00 and z = −1.00

is also .3413 (Column B). The area under the curve to the left of z = −1.00 is .1587 (Column C). An IQ score of

85 is therefore at about the 16th percentile. A student who scores an 85 on an IQ test outperforms about 16

percent of other students on the IQ test, whereas about 84% of students outperform him or her on the IQ test.

Unit 4 – Probability and Hypothesis Testing

INTRODUCTION

An important z score is ±1.96, where 95% of scores fall under the area of the curve, whereas 2.5% fall to the left

and 2.5% fall to the right (2.5% + 2.5% = 5%). A z score beyond these cutoffs is typically considered to be

“extreme” (Warner, 2013). In addition, we will see below that most inferential statistics set “statistical

significance” to obtained probability values of less than 5%.

Hypothesis Testing

Probability is crucial for hypothesis testing. In hypothesis testing, you want to know the likelihood that your

results occurred by chance. No matter how unlikely, there is always the possibility that your results have

occurred by chance, even if that probability is less than 1 in 20 (5%). However, you are likely to feel more

confident in your inferences if the probability that your results occurred by chance is less than 5% compared to,

say, 50%. Most researchers in the social sciences find it reasonable to designate less than a 5% chance as a

cutoff point for determining statistical significance. This cutoff point is referred to as the alpha level. An alpha

level is set to determine when a researcher will reject or fail to reject a null hypothesis (discussed next). The

alpha level is set before data are analyzed to avoid “fishing” for statistical significance. In high-stakes research

(such as testing a new cancer drug), researchers may want to be even more conservative in designating an alpha

level, such as less than 1 in 100 (1%) that the results are due to chance.

Null and Alternative Hypotheses

The null hypothesis ( H0) refers to a given population parameter, such as a population mean of 100 on IQ.

Imagine that we ask two groups of students to complete a standardized IQ test, and then we calculate the

mean IQ score for each group. We observe that the mean IQ for Group A is 100 ( MA = 100), whereas the mean

IQ for Group B is 115 ( MB = 115). Is a mean difference of 15 IQ points statistically significant or just due to

chance? The null hypothesis predicts that H0: MA = MB. That is, the null hypothesis predicts no difference

between groups. Remember that “null” also means “zero,” so we could also state the null hypothesis as H0: MA

− MB = 0. When comparing groups, in general, the null hypothesis predicts that group means will not differ.

When testing the strength of a relationship between two variables, such as the correlation between IQ scores

and grade point average (GPA), in general, the null hypothesis is that the relationship between variable A and

variable B is zero.

By contrast, the alternative hypothesis ( H1) does predict a difference between two groups, or in the case of

relationships, that two variables are significantly related. An alternative hypothesis can be directional ( H1:

Group X has a higher mean score than Group Y) or nondirectional ( H1: Group X and Group Y will differ).

In hypothesis testing, you either reject or fail to reject the null hypothesis. Note that this is not stating, “accept

the null hypothesis as true.” By default, if you reject the null hypothesis, you accept the alternative hypothesis as

true. However, if you do not reject the null hypothesis, you cannot accept the alternative hypothesis as true.

You have simply failed to find statistical justification to reject the alternative hypothesis.

Type I and Type II Errors

If you commit a Type I error, this means that you have incorrectly rejected a true null hypothesis. You have

incorrectly concluded that there is a significant difference between groups, or a significant relationship, where

no such difference or relationship actually exists. Type I errors have real-world significance, such as concluding

that an expensive new cancer drug works when actually it does not work, costing money and potentially

endangering lives. Keep in mind that you will probably never know whether the null hypothesis is “true” or not,

as we can only determine that our data fail to reject it.

If you commit a Type II error, this means that you have not rejected a false null hypothesis when you should

have rejected it. You have incorrectly concluded that no differences or no relationships exist when they actually

do exist. Type II errors also have real-world significance, such as concluding that a new cancer drug does not

work when it actually does work and could save lives.

Your alpha level will affect the likelihood of making a Type I or a Type II error. If your alpha level is small (such

as .01, less than 1 in 100 chance), you are less likely to reject the null hypothesis, so you are less likely to commit

a Type I error. However, you are more likely to commit a Type II error.

You can decrease the chances of committing a Type II error by increasing the alpha level (such as .10, less than

1 in 10 chance). However, you are now more likely to commit a Type I error. Since the chances of committing

Type I and Type II errors are inversely proportional, you will have to decide which type of error is more grievous.

You need to assess the risk associated with each type of error. Your research questions will help in this decision.

In standard social sciences research, the alpha level is set to .05 (that is, a 1 in 20 chance of committing a Type I

error). An alpha level of .05 is used throughout the remainder of this course.

Probability Values and the Null Hypothesis

The statistic used to determine whether or not to reject a null hypothesis is referred to as the calculated

probability value or p value, denoted p. When you run an inferential statistic in SPSS, it will provide you with a

p value for that statistic. If the test statistic has a probability value of less than 1 in 20 (.05), we can say “p < .05,

the null hypothesis is rejected.” Keep in mind in the coming weeks that we are looking for values less than .05

to reject the null hypothesis. This may seem counterintuitive at first, because usually we assume that bigger is

better. In the case of null hypothesis testing, the opposite is the case—if we expect to reject a null hypothesis,

remember that, for p values, smaller is better. Any p value less than .05 (such as .02, .01, or .001) means that we

reject the null hypothesis. Any p value greater than .05 (such as .15, .33, or .78) means that we do not reject the

null hypothesis. Make sure you understand this point, as it is a common area of confusion among statistics

learners.

Based on your understanding of the null hypothesis, the alternative hypothesis, the alpha level, and the p value,

you can begin to make statements about your research results. If your results fall within the rejection region, you

can claim that they are “statistically significant,” and you reject the null hypothesis. In other words, you will

conclude that groups do differ in some way or that two variables are significantly related. If the results do not

fall within the rejection region, you cannot make this claim. Your data fail to reject the null hypothesis. In other

words, you will conclude that groups do not differ in some way or that two variables are unrelated.

This unit covers the terminology and concepts behind hypothesis testing, which prepares you for the remaining

units in this course. The statistical tests include:

• Correlation (Units 5–6).

• t tests (Units 7–8).

• Analysis of variance (Units 9–10).

Reference

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

Oaks, CA: Sage.

OBJECTIVES

To successfully complete this learning unit, you will be expected to:

1. Analyze case studies of Type I and Type II errors.

2. Analyze case studies of null hypothesis testing.

Unit 4 Study 1- Readings

Use your Warner text, Applied Statistics: From Bivariate Through Multivariate Techniques , to complete

the following:

• Read Chapter 3, “Statistical Significance Testing,” pages 81–124. This reading addresses the

following topics:

◦ The logic of null hypothesis testing.

◦ Type I and Type II errors.

◦ The z test.

◦ Null hypothesis and alternative hypothesis.

◦ Limiting Type I error.

◦ Effect size.

◦ Statistical power.

SOE Learners – Suggested Readings

Some programs have opted to provide program-specific content designed to help you better understand

how the subject matter in this study is incorporated into your particular field of study. The following readings

are suggested for SOE learners.

Harrison, J., Thompson, B., & Vannest, K. J. (2009). Interpreting the evidence for effective interventions to

increase the academic performance of students with ADHD: Relevance of the statistical significance

controversy. Review of Educational Research, 79(2), 740–775.