Step 4: Make a Decision and Report the Results
II. Thought Questions About Error
A. The probability of a Type II error decreases as the alpha level increases from .01 to .05. This is because a .05 alpha level makes the H 0 easier to reject and we would be failing to reject it less often.
B. Given the same alpha level, the probability of a Type I error decreases as we move from a one-tailed test to a two-tailed test. This is because the rejection region in a two-tailed test is split in two and falls farther into the extremes, making it more dif- fi cult to reject H 0 . Thus, we would be rejecting a H 0 that is actually true less often.
C. α = .001. We should minimize the probability of a Type I error because of the potentially dangerous side effects of the drug. (We want to make it more likely that any differences found are due to treatment and not chance, hence the lower α .)
Answers to “Your Turn!” Problems (continued)
Using Microsoft Excel for Data Analysis
if you are using excel for the fi rst time for statistical analysis, you may need to load the add-in tool that allows these functions. The information for loading the Data Analysis Toolpak as well as general instructions for using excel for data analysis are at the beginning of the excel section in Chapter 4.
One-Sample z -Test in Excel
To use excel for a one-sample z -test, we will be providing formula instructions for excel to use for the necessary calculations. To demonstrate the procedures, we’ll work with the sample research question from this chapter. First, type in the information shown below in column A of the spreadsheet that will be needed for the calculations. Also type in the summary information in column B that was provided for this problem.
Rather than using the table at the back on your text, excel will determine your zcrit values. We are using a nondirectional test and an alpha level of .05. since it is a two-tailed test, remember that you have to divide that .05 in half to obtain the zcrit
values associated with the extreme 2.5% of the normal distribution. With your cursor in cell B5, type the following exactly as written: =ABs(noRMsinV(.025))
After you hit the enter key, zcrit (1.959964) will appear in the cell. if you were using a one-tailed test, you would use .05 instead of .025. For an alpha level of .01, you would use .005 for a two-tailed test and .01 for a one-tailed test.
next, we want to calculate the standard error of the mean using the formula M / n. in essence, we will be instructing excel to divide the population stan- dard deviation by the square root of n by grabbing those values from the cells where they are located. For this example, with your cursor in cell B6, type the following exactly as written: =B3/sQRT(B4)
After you hit the enter key, the standard error (2) will appear in the cell.
Lastly, we need to calculate our obtained z-value using the formula zobt M / M. With your cursor in cell B7, type the following exactly as written: = (B1−B2)/B6
obt
Additional Practice Problems
Answers to odd numbered problems are in Appendix C at the end of the book.
1. Discuss the difference between the null hypothesis and the alternative hypothesis.
2. Discuss the difference between directional and nondirectional alternative hypotheses.
3. Explain what is meant by the alpha level and the critical region of a sampling distribu- tion and how they are related.
4. A new activity program is being developed to determine its effectiveness in influenc- ing the amount of time that children spend watching television, which is currently μ = 30 hours per week. In symbolic form, write the notation for both the null hypothesis and the alternative hypothesis for:
a. a nondirectional hypothesis test.
b. a directional test that specifies that the program would reduce the number of hours spent in front of the television.
5. If our obtained sample mean is close in value to the population mean, is H0 or H1 more likely to be supported? Explain.
6. What are two conditions that researchers control that influence the likelihood of reject- ing the null hypothesis?
7. For the values listed below, indicate whether to reject or fail to reject the null hypothesis, and whether the obtained probability is greater than or less than alpha.
zcrit zobt α a. −2.33 −2.07 .01 b. ±1.96 −2.38 .05 c. +1.65 +1.83 .05 d. ±2.58 +2.54 .01
8. For the values listed below, indicate whether the obtained z-value is significant (S) or nonsignificant (NS).
zcrit ssobt
a. ±2.58 −2.47
b. −1.65 −2.13
c. +2.33 +2.40
d. +3.10 +3.00
9. What is a Type I error and how is it related to the alpha level?
10. What is a Type II error? What can researchers do to make a Type II error less likely to occur?
11. A psychology professor has developed a workshop designed to reduce the apprehension of students taking university courses for the first time. All incoming freshmen take a standardized apprehension assessment which has resulted in a normal distribution with μ = 80 and σ = 12. A random sample of n = 44 new freshmen who attended the profes- sor’s workshop obtained a M = 76.
a. Using the four-step hypothesis testing procedure, test at α = .05 using a one-tailed test.
b. What was the size of the effect?
c. If an alpha level of α = .01 was used, would the statistical decision remain the same?
Explain.
d. Based on your answers to “c” above, how is the power of a statistical test related to the alpha level that is set by the researcher?
12. A cognitive psychologist has developed a new technique which he believes will influence the ability to concentrate while performing difficult cognitive tasks. A standardized test of logical reasoning is normally distributed and has a μ = 36 and a σ = 8. A random sample of n = 18 adults is taught the concentration technique and then administered the logic test. A sample mean of M = 38 was obtained. Use the hypothesis testing procedure with α = .05 to determine the effectiveness of the concentration technique.
13. A standardized mood assessment has a μ = 65 and a σ = 15. Higher scores reflect a more positive mood state. A pop music producer, interested in how music affects peoples’
mood, hires a research company to assess this relationship. As a part of the study, the research company obtains a random sample of n = 40 adults and administers the mood assessment while they are listening to soft piano music. The mean for the sample was M = 69.
a. Using a two-tailed test and α = .05, use the four-step hypothesis testing procedure to test the music producer’s hypothesis that music affects mood.
b. Given the same α = .05, but using a one-tailed test hypothesizing that the music would improve the mood of the listeners, would the outcome have been the same?
Explain.
c. Based on your answer to “b” above, how is the power of a statistical test related to the directionality of the research hypothesis?
Notes
1 American Psychological Association. (2001). Publication manual of the American Psychological Association (5th ed.). Washington, DC: American Psychological Association.
2 Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Law- rence Erlbaum Associates.
133
THE t-STATISTIC
Now that we have established the foundational concepts for hypothesis testing, we are ready to build further on these ideas. Previously, we used the z-statistic in testing hypotheses about the population mean (μ). However, the z-test requires knowing the value of the population standard deviation (σ) – information that is, in fact, not usually known. A standard devia- tion is needed to calculate the standard error, which is needed in turn for the z-score formula.
Even though we cannot calculate the population standard deviation directly, remember that we can estimate it by using n − 1 in the standard deviation formula presented in Chapter 4:
s SS n
1
We can then use this estimate in the formula for the standard error. Instead of using
M n , we use s s
M = n ∙
The statistic used to test hypotheses about population means when σ is not known is called the t-statistic, which is given by the formula:
t M
SM
obt
Notice that the only difference between this formula and the z-score formula is the use of the estimated standard error in the denominator in place of the actual standard error.
THE t-DISTRIBUTION
Whenever we engage in hypothesis testing, we will use a sampling distribution of some kind. The t-distribution, like the z-distribution, is theoretical, symmetrical, and bell- shaped, but the appearance of the curve changes according to the size of the sample. The z-distribution is based on an infinite number of cases resulting in what we refer to as the normal distribution curve. However, with small samples the curve will be shaped
9
One-Sample t TestOne-Sample t TestOne-Sample t Test
differently. The t-distribution is actually a family of curves, one for each sample size. The particular t-distribution that we use will be based on the degrees of freedom associated with the sample.
We encountered the term degrees of freedom (df) when we used n − 1 in the denomina- tor of the formula for estimating the population standard deviation from sample data. This corrected for the underestimation of the sample standard deviation. We also use df when estimating other population parameters to increase the accuracy of our estimates. The par- ticular df to use will vary from one statistical procedure to another depending on the sample size and the number of data sets used to estimate a population parameter. When using the t-distribution for a single sample, df will also be n − 1, the size of the sample minus 1. For instance, if your sample size is 16, df will be 15.
Shape of the t-Distribution
The t-distribution is used in the same way that we used the normal distribution, except that the curves of the t-distribution depart from normality as df decrease. The illustration below shows what the curves look like for three different df → 5, 15, and infinity.
The t-scores are shown along the baseline, just like z-scores. Notice that the curves become flatter with smaller df and that the t-values extend farther into the tails. As df increase, the t-distribution looks more and more like the normal distribution. When df are infinite, there is no difference in shape between the t-curve and the normal distribution curve.
Tip!
Because smaller samples result in a distribution that is flatter and more spread out, more extreme t-values would be required to reject H0 (i.e., we would have to go farther into the tails to reach significance with smaller samples).
Critical Values for the t -Distribution
To determine the critical values for the t-distribution, we will use Table 2 near the end of this book. This table shows the critical values of t that would be necessary for rejecting the null hypothesis. Some important features of the table are as follows:
• In the left column, various degrees of freedom (df) are listed.
• The first row across the top lists various levels of significance for a one-tailed test.
When using the table for the normal distribution curve, you were required to divide the proportion associated with the alpha level in half for two-tailed tests. This is not necessary for the table used to determine critical values for the t-distribution.
• The remaining columns in the body of the table list the critical values necessary for rejecting H0 at the significance levels identified at the top of each column.
In order to reject H0, your obtained t-value will have to exceed the value listed in the chart.
However, the chart is abbreviated, and the critical values are not shown for all df. If the df for your particular research problem are not shown, then use the critical values associated with the next-lowest df. For instance, if your sample size is 54, actual df would be 53 (n – 1), which is not shown. In this case, you would use 40 to ensure that your obtained t-value falls inside the critical region.
The “Your Turn!” exercise below will give you some practice working with this table and will reinforce some ideas that were covered in the last chapter.
Your Turn!