Practical Statistics Mean Comparisons

There are six statistics that will answer 90% of all questions! Descriptive Chi-square Z-tests Comparison of Means Correlation Regression

interval and ratio scales t-test and ANOVA are for the means of interval and ratio scales They are very common statistics….

T-test Why is it called a t-test?

William S. Gosset 1876-1937 Published under the name: Student

t-test come in three types: A sample mean against a hypothesis.

t-test come in three types: A sample mean against a hypothesis. Two sample means compared to each other.

t-test come in three types: A sample mean against a hypothesis. Two sample means compared to each other. Two means within the same sample.

t-test The standard error for means is:

Each t value comes with a certain degree t-test Hence for one mean compared to a hypothesis: Each t value comes with a certain degree of freedom df = n - 1

IQ has a mean of 100 and a standard deviation of t-test IQ has a mean of 100 and a standard deviation of 15. Suppose a group of immigrants came into Iowa. A sample of 400 of these immigrants found an average IQ of 98. Does this group have an IQ below the population average?

The test statistic looks like this: t-test The test statistic looks like this: There are n – 1 = 399 degrees of freedom. The results are printed out by a computer or looked up on a t-test table.

The critical value for 399 degrees of freedom is about 1.97.

Of course, we could look this up on the internet…. http://www.danielsoper.com/statcalc/calculator.aspx?id=8 For the IQ test: t(399) = 2.67, p = 0.00395

t-test Since the test was “one-tailed,” the critical value of t would be 1.65. Therefore, t(399) = 2.67 would indicate that the immigrants IQ is below normal.

t-test come in three types: A sample mean against a hypothesis. Two sample means compared to each other. Two means within the same sample.

t-test The standard error of the difference between two means looks like this:

t-test Therefore the test statistic would look like this: With degrees of freedom = n(1) + n(2) - 2

t-test Usually this is simplified by looking at the difference between two samples; so that:

Where:

Suppose that a new product was test marketed in the United States and in Japan. The company hypothesizes that customers in both countries would consume the product at the same rate. A sample of 500 in the U.S. used an average of 200 kilograms a year (SD = 20), while a sample of 400 in Japan used an average of 180 kilograms a year (SD = 25). Test the hypothesize…..

The test would start be computing: = 500 (a SD = 22.36)

The results would be written as: (t(898) = 0.89, ns), and the conclusion is that there is no difference in the consumption rate between the U.S. and Japanese customers.

But this is wrong! Can you see why? It is caused by a common mistake of confusing the sampling distribution with a the sample distribution. 25

The results are written as: (t(898) = 13.33, p < .0001), and the conclusion is that there is a large difference in the consumption rate between the U.S. and Japanese customers. 26

t-test come in three types: A sample mean against a hypothesis. Two sample means compared to each other. Two means within the same sample.

t-test come in three types: 3. Two means within the same sample. This t-test is used with correlated samples and/or when the same person or object is measured twice in the same sample.

Student T1 T2 d Tom 89 90 1 Jan 88 91 3 Jason 87 86 -1 Halley 90 90 0 Bill 75 79 4 The measurement of interest is d.

That is… the average difference between test 1 and test 2 is zero. H0 : Average of d = 0 That is… the average difference between test 1 and test 2 is zero.

t-test The sampling error for this t-test is: Were d = score(2) – score(1)

t-test The t-test is: The degrees of freedom = n - 1

Examples can be found at these sites: http://en.wikipedia.org/wiki/T-test http://blog.minitab.com/blog/adventures-in-statistics/understanding-t-tests:-1-sample,-2-sample,-and-paired-t-tests

Suppose there are more than two groups that need to be compared. The t-test cannot be utilized for two reason. The number of pairs becomes large. The probability of t is no longer accurate.

Analysis of Variance (ANOVA) Hence a new statistic is needed: The F-test Or Analysis of Variance (ANOVA) R.A. Fisher 1880-1962

Compares the means of two or more groups The F-test Compares the means of two or more groups by comparing the variance between groups with the variance that exists within groups. According to the Central Limit Theorem there is a relationship between the variance of a statistic and the variance of the population. If that relationship is violated, it is likely that the statistics did not come from the same population as the other statistics.

https://en.wikipedia.org/wiki/Analysis_of_variance https://www.youtube.com/watch?v=0Vj2V2qRU10

F is the ratio of variance:

The F-test http://www.statsoft.com/textbook/distribution-tables/

The F-test The probability distribution is dependent upon the degrees of freedom between and the degrees of freedom within.

The F-test Typical output looks like this:

In SPSS ANOVA looks like this: