Section 43-45 Computation of t for Independent Data; Reporting the Results of t Tests

Question 1

t-Test (Comparing SAMPLE Means)

Answer 1

_*t* - Test_ is used to compare SAMPLE MEANS in order to determine whether or not the difference between the means is due ONLY to random SAMPLING ERROR (i.e. to determine if you can correctly REJECT The NULL HYPOTHESIS).

• IN this instance you CANNOT use the z-score because there is NO AVAILABLE POPULATION MEAN or STANDARD DEVIATION, which is required for the z-score calculation.

SURVEYS and EXPERIMENTS are frequently conducted, and they often yield two SAMPLE means each (because you’re usually comparing two groups).

• The t-Test looks at these two SAMPLE means and standard deviations and gives a probability that a given null hypothesis is correct.
• Consider what makes the t-Test work.

1. The larger the samples, the less likely that the difference between two means was created by sampling errors. Thus, when large samples are used, the t-test is more likely to yield a probability (that the Null Hypothesis is correct) low enough to allow us to reject the Null Hypothesis.
2. The larger the difference between the two means, the less likely it is that the difference was created by sampling errors. Random sampling tends to create many small differences and few large ones. Thus, when large differences between means are obtained, the t-test is more likely to yield a probability low enough to allow us to reject the null hypothesis.
3. The smaller the variance among the subjects, the less likely it is that the difference between two means was created by sampling errors. To understand this, consider a population in which all individuals are identical: They all look alike, think alike, and speak and act in unison. How many do you have to sample to get a good sample? Only one because they are all the same. Thus, when there is no variation among subjects, it is not possible to have sampling errors. As the variation increases, sampling errors are increasingly more likely to occur.

There are 2 Types of t-Tests:

_*t*-Test for INDEPENDENT DATA_ – Sometimes called the “Uncorrelated Data” – Refers to data that has NO factor present that might mitigate the differences between the groups being compared.

• Ex: An investigator wanted to determine whether there were differences between male and female voters in their attitudes toward welfare. Samples of men and women were drawn at random and administered an attitude scale so that a score for each subject could be obtained. Means for the two samples were computed. Women had a mean of 40.00 (on a scale from 0 to 50, where 50 was the most favorable). Men had a mean of 35.00. What accounts for the 5-point difference? One possible explanation is the null hypothesis, which states that there is no true difference between men and women-that the observed difference is due to sampling errors created by random sampling.

_*t*-Test for DEPENDENT DATA_ – Sometimes called “Correlated Data” – Refers to data that HAS one or more factors present that might mitigate the differences between the groups being compared.

• Ex: In a study of visual acuity, same-sex siblings (two brothers or two sisters) were identified. For each pair of siblings, a coin was tossed to determine which one received a vitamin supplement and which one received a placebo. Thus, in the control group, there is a subject who is the same-sex sibling of each subject in the experimental group.

FACTORS that might MITIGATE the DIFFERENCES – The means obtained in the DEPENDENT DATA Example above are subject to LESS ERROR than the means from the INDEPENDENT DATA Example above. Why?

• In the INDEPENDENT DATA, there was no matching or pairing of subjects before assignment to conditions.
• In the DEPENDENT DATA, the matching of subjects (in both GENETICS and GENDER) assures us that the two groups are more similar than if just two independent samples were used.
  
  • To the extent that genetics and gender are associated with visual acuity, the TWO GROUPS WILL BE MORE SIMILAR at the onset of the experiment than the two groups in the INDEPENDENT DATA.
  • The t-test for DEPENDENT data takes this possible reduction of error into account. Thus, it is important to select the right t-test.

Question 2

t-Test for Independent Data (Computation)

Answer 2

INDEPENDENT DATA are obtained when there is no matching or pairing of subjects across groups. In this section, we will first examine how to compute t (called the OBSERVED VALUE of *t* ) for independent data and how to interpret it using the t table. The formula for t is simple (See below).

• The numerator of the formula is easy to understand. It is the difference between the two means. As you can see, the larger the difference, the larger the value of t.
• The denominator starts with the familiar symbol S (for standard deviation). The subscripts (_{<strong>D</strong>} for difference and _m for means) indicate that it is the standard deviation of the Difference Between Means. This standard deviation is called the STANDARD ERROR of the DIFFERENCE BETWEEN MEANS.
  
  • In this case, we want to know whether the difference between two means is an unlikely event. If it is unlikely (e.g., likely to occur fewer than 5 times in 100 due to chance alone), the difference will be declared statistically significant (i.e., unlikely to be the result of random sampling errors).
• The formulas will give you the OBSERVED VALUE of t as well as the _DEGREES of FREEDOM (*df* )_, used to find the CRITICAL VALUE of *t*
  
  • You know that if a value of z as extreme as 1.96 is obtained, the result is declared unlikely to occur by chance because the odds are less than .05 that this is a chance deviation in a normal distribution. However, this is a t-test, which is based on the fact that the underlying distributions are NOT normal in shape when the sample size is small. Thus, instead of using constants such as 1.96 to evaluate the value of t, we use the appropriate _CRITICAL VALUE of t_ found in the t-table in Table 4 near the end of this book.
    
    • NOTE: If you have a Degree of Freedom that is NOT in the table, use the next LOWER Degree of Freedom on the table.
  • Once You’ve calculated the OBSERVED VALUE of t and then found the CRITICAL VALUE of t using the calculated DEGREES of FREEDOM df and the desired ALPHA LEVEL (.05, .01, .001), then you compare the two.
    
    • If the OBSERVED VALUE of t is GREATER THAN the CRITICAL VALUE of t, then we can REJECT THE NULL HYPOTHESIS and declare the results to be STATISTICALLY SIGNIFICANT at that ALPHA LEVEL.
    • If the OBSERVED VALUE is NOT GREATER, then you can NOT REJECT THE NULL HYPOTHESIS nor declare the results STATISTICALLY SIGNIFICANT.
• As with many formulae, this one looks daunting, but again, breaking things down into their components makes them much easier to handle.
  
  • In order to do this, please refer to the “t-Test Independent” tab in the “Quant Psych Tool” Spreadsheet.
    
    • There is also a full example in there to compare to the results you find in the spreadsheet calculations.

Question 3

Reporting the Results of t-Tests

Answer 3

Reporting the results of the _*t*-test_ to test the difference between two sample means for significance.

• The values of the means, standard deviations, and the number of cases in each group should be reported in a table (see Table 1. Below)
• Alternatively, we might include a bar graph (see Figure 1 below) to display the mean difference visually. While a bar graph may not communicate the standard deviations and sample sizes as clearly as a table does, Figure 1 (below) includes 95% confidence intervals.
  
  • Since we are working with the difference between MEANS, we use the STANDARD ERROR of the MEANS to calculate the Confidence intervals (See “Stnd Err” tab in Spreadsheet).
  • Because the two confidence intervals do not overlap, we might conclude that each sample mean is drawn from a different population. Upon examination, the figure should agree with the results of the statistical test.
• The samples that formed Groups A and B (Table 1 and figure 1 below) were drawn at random. The NULL HYPOTHESIS states that the 3.50-point difference between the means of 2.50 and 6.00 is the result ONLY of sampling errors and that the true difference in the population is zero.
  
  • Because the sampling error for each sample mean is represented by the 95% confidence intervals in the figure, and they do NOT overlap, we know that the null hypothesis is unlikely to be true.

_There are THREE ways to REPORT the FINDINGS of a *t*-TEST_:

1. The difference between the means is statistically significant (t = 3.22, df = 10, p < .01, two-tailed test).
2. The difference between the means is significant at the .01 level (t = 3.22, df = 10, two-tailed test).
3. The null hypothesis was rejected at the .01 level, t (10) = 3.22, p < .001, two-tailed test.

• If the findings were NOT statistically significant, then you only need insert the word “NOT” in the appropriate locations.
• # 3 is the format consistent with the guidelines of the American Psychological Association (APA).
• Journals always refer to “Statistical Significance” and rarely mention the NULL HYPOTHESIS.
• Always precede “Significance” with “Statistical”