Practical Statistics Mean Comparisons

There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison of Means 5. Correlation 6. Regression

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

William S. Gosset Published under the name: Student

t-test come in three types: 1. A sample mean against a hypothesis.

t-test come in three types: 1. A sample mean against a hypothesis. 2.Two sample means compared to each other.

t-test come in three types: 1. A sample mean against a hypothesis. 2.Two sample means compared to each other. 3.Two means within the same sample.

t-test The standard error for means is:

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

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

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…. For the IQ test: t(399) = 2.67, p =

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

t-test come in three types: 1. A sample mean against a hypothesis. 2.Two sample means compared to each other. 3.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

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.

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.

t-test come in three types: 1. A sample mean against a hypothesis. 2.Two sample means compared to each other. 3.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.

StudentT1T2d Tom89901 Jan88913 Jason Halley90900 Bill75794 The measurement of interest is d.

H 0 : 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:

Suppose there are more than two groups that need to be compared. The t-test cannot be utilized for two reason. 1.The number of pairs becomes large.

Suppose there are more than two groups that need to be compared. The t-test cannot be utilized for two reason. 1.The number of pairs becomes large. 2.The probability of t is no longer accurate.

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

The F-test Compares the means of two or more groups by comparing the variance between groups with the variance that exists within groups.

F is the ratio of variance:

The F-test

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:

Service Encounter The average age of Iowans over 18 is approximately 47. Is the sample a cross-section of this population by age? A sample mean against a hypothesis.

Service Encounter Is the measure of personality different between men and women? Two sample means compared to each other.

Service Encounter Is the measure of personality different between men and women? Two sample means compared to each other.

Service Encounter Is the measure of personality different between men and women?

Service Encounter Do respondents like themselves better than the service provider? Two means within the same sample.

Service Encounter Do respondents like themselves better than the service provider? Two means within the same sample.

Service Encounter Is the measure of personality different between shopping times?

Service Encounter Is personality difference by perception of service encounter? More than two sample means compared to each other.

Service Encounter Is personality difference by perception of service encounter? More than two sample means compared to each other.