Chapter 9: Testing Hypotheses Overview Research and null hypotheses One and two-tailed tests Errors Testing the difference between two means t tests

You already know how to deal with two nominal variables Overview You already know how to deal with two nominal variables Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers the distribution of one variable across the categories of another variable Considers the difference between the mean of one group on a variable with another group Considers how a change in a variable affects a discrete outcome Considers the degree to which a change in one variable results in a change in another

You already know how to deal with two nominal variables Overview You already know how to deal with two nominal variables Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Lambda Dependent Variable Interval Nominal TODAY! Testing the differences between groups Considers the difference between the mean of one group on a variable with another group Considers the degree to which a change in one variable results in a change in another

You already know how to deal with two nominal variables Overview You already know how to deal with two nominal variables Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Lambda Dependent Variable Interval Nominal TODAY! Testing the differences between groups Considers the degree to which a change in one variable results in a change in another Confidence Intervals t-test

Example Draw a random sample of 100 Africa American from GSS 1998. Calculate the mean earnings--$24,100 Based on census information, the mean earnings for Americans is $28,985. Is the observed gap ($28,985 - $24,100) large enough to convince us that the sample we drew is not representative of the population?

Example The average earnings of the Africa Americans are indeed lower than the national average The average earnings of the Africa Americans are about the same as the national average, and this sample happens to show a particularly low mean.

General Examples Is one group scoring significantly higher on average than another group? Is a group statistically different from another on a particular dimension? Is Group A’s mean higher than Group B’s?

Specific Examples Do people living in rural communities live longer than those in urban or suburban areas? Do students from private high schools perform better in college than those from public high schools? Is the average number of years with an employer lower or higher for large firms (over 100 employees) compared to those with fewer than 100 employees?

Testing Hypotheses Statistical hypothesis testing – A procedure that allows us to evaluate hypotheses about population parameters based on sample statistics. Research hypothesis (H1) – A statement reflecting the substantive hypothesis. It is always expressed in terms of population parameters, but its specific form varies from test to test. Null hypothesis (H0) – A statement of “no difference,” which contradicts the research hypothesis and is always expressed in terms of population parameters.

Research and Null Hypotheses One Tail — specifies the hypothesized direction Research Hypothesis: H1: 2 1, or 2 1 > 0 Null Hypothesis: H0: 2 1, or 2 1 = 0 Two Tail — direction is not specified (more common) H1: 2 = 1, or 2 1 = 0

One-Tailed Tests One-tailed hypothesis test – A hypothesis test in which the alternative is stated in such a way that the probability of making a Type I error is entirely in one tail of a sampling distribution. Right-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the right tail of the sampling distribution. Left-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the left tail of the sampling distribution.

Two-Tailed Tests Two-tailed hypothesis test – A hypothesis test in which the region of rejection falls equally within both tails of the sampling distribution.

Probability Values Z statistic (obtained) – The test statistic computed by converting a sample statistic (such as the mean) to a Z score. The formula for obtaining Z varies from test to test. P value – The probability associated with the obtained value of Z.

Probability Values

Probability Values Alpha ( ) – The level of probability at which the null hypothesis is rejected. It is customary to set alpha at the .05, .01, or .001 level.

Five Steps to Hypothesis Testing Making assumptions (2) Stating the research and null hypotheses and selecting alpha (3) Selecting the sampling distribution and specifying the test statistic (4) Computing the test statistic (5) Making a decision and interpreting the results

Type I and Type II Errors Type I error (false rejection error)the probability (equal to ) associated with rejecting a true null hypothesis. Type II error (false acceptance error)the probability associated with failing to reject a false null hypothesis. Based on sample results, the decision made is to… reject H0 do not reject H0 In the true Type I correct population error () decision H0 is ... false correct Type II error decision

One-Sample z Test When we know population parameters μ and σ, how likely we could draw a random sample whose mean (y bar) differs from μ? Null Hypothesis Population mean μy equals to population mean μ.

One-Sample z Test Test statistic

One-Sample z Test Compare z we calculate to the critical value Make a decision

Example how likely we could draw a random sample from a population whose mean is differ from μ? id GPA 7 3.6 1 3.2 4 3.4 5 3.5 6 3 3.3

Example Is the observed gap ($28,985 - $24,100) large enough to convince us that the sample we drew is not representative of the population?

Five-step Testing Hypothesis-1 Making Assumptions: A random sample is selected. Because N>50, the assumption of normal population is not required. The level of measurement of the dependent variable is interval-ratio.

Five-step Testing Hypothesis-2 Stating the Research and the Null Hypotheses The research hypothesis is The null hypothesis is

Five-step Testing Hypothesis-3 Selecting the Sampling distribution and Specify the Test Statistic We use the z distribution and the z statistic to test the null hypothesis

Five-step Testing Hypothesis-4 Computing the z Test Statistic

Five-step Testing Hypothesis-5 Making a Decision and Interpreting the Results Our obtained |z| statistic of 2.09 is greater than 1.96 or probability of obtaining a z statistic of 2.09 is less than .05. This P value is below .05 alpha level. The probability of obtaining the difference of $4885 ($28,985 - $24,100) between the income of African Americans and the national average for all, if the null hypothesis were true, is extremely low.

Five-step Testing Hypothesis-5 We have sufficient evidence to reject the null hypothesis and conclude that the average earnings of African American are significantly different from the average earnings of all. The difference is significant at the .05 level.

t Test t statistic (obtained) – The test statistic computed to test the null hypothesis about a population mean when the population standard deviation is unknown and is estimated using the sample standard deviation. t distribution – A family of curves, each determined by its degrees of freedom (df). It is used when the population standard deviation is unknown and the standard error is estimated from the sample standard deviation. Degrees of freedom (df) – The number of scores that are free to vary in calculating a statistic.

One-Sample t Test t test A test of significance similar to the z test but used when the population’s standard deviation is unknown.

t distribution

t distribution table

Example how likely we could draw a random sample whose mean (Y bar) differs from μ? id GPA 7 3.6 1 3.2 4 3.4 5 3.5 6 3 3.3

The Earnings of White Women We drew a sample of white females (N=371) from GSS 2002. The mean earnings is $28,889 with a standard deviation 21,071. In 2002, the national average earnings for all women is $24,146.

Five-step Testing Hypothesis-1 Making Assumptions: A random sample is selected. The sample size is large. The level of measurement of the dependent variable is interval-ratio.

Five-step Testing Hypothesis-2 Stating the Research and the Null Hypotheses The research hypothesis is The null hypothesis is

Five-step Testing Hypothesis-3 Selecting the Sampling distribution and Specify the Test Statistic We use the t distribution and the t statistic to test the null hypothesis

Five-step Testing Hypothesis-4 Computing the Test Statistic Firstly, calculate the degree of freedom associated with test

Five-step Testing Hypothesis-5 Making a Decision and Interpreting the Results Our obtained t statistic of 4.33 is greater than 1.980 or probability of obtaining a t statistic of 4.33 is less than .05. This P value is below .05 alpha level. The probability of obtaining the difference of $4743 ($28889-$24146) between the income of white women and the national average for all women, if the null hypothesis were true, is extremely low.

Five-step Testing Hypothesis-5 We have sufficient evidence to reject the null hypothesis and conclude that the average earnings of white women are significantly different from the average earnings of all women. The difference is significant at the .05 level.

Exercise Can you do a one-tail test see if the mean earnings of white women is significantly higher than the average for all women?

Two-Sample t Tests The t-test assesses whether the means of two populations statistically differ from each other. The 2 independent sample t-test is used when testing 2 independent groups..

t-test for difference between two means Is the value of 2 1 significantly different from 0? This test gives you the answer: If the t value is greater than 1.96, the difference between the means is significantly different from zero at an alpha of .05 (or a 95% confidence level). The difference between the two means  the estimated standard error of the difference The critical value of t will be higher than 1.96 if the total N is less than 122. See Appendix C for exact critical values when N < 122.

Test Statistic Equal population variance assumed

Test Statistic Unequal population variance assumed

t-test and Confidence Intervals The t-test is essentially creating a confidence interval around the difference score. Rearranging the above formula, we can calculate the confidence interval around the difference between two means: If this confidence interval overlaps with zero, then we cannot be certain that there is a difference between the means for the two samples.

Why a t score and not a Z score? Use of the Z distribution has assumes the population standard error of the difference is known. In practice, we have to estimate it and so we use a t score. When N gets larger than 50, the t distribution converges with a Z distribution so the results would be identical regardless of whether you used a t or Z. In most sociological studies, you will not need to worry about the distinction between Z and t.

What can we conclude about the difference in wages? t-Test Example 1 Mean pay according to gender: N Mean Pay S.D. Women 46 $10.29 .8766 Men 54 $10.06 .9051 Equal population variances assumed What can we conclude about the difference in wages?

What can we conclude about the difference in wages? t-Test Example 2 Mean pay according to gender: N Mean Pay S.D. Women 57 $9.68 1.055 Men 51 $10.32 .9461 Equal population variances assumed What can we conclude about the difference in wages?

In-Class Exercise Using these GSS income data, calculate a t-test statistic to determine if the difference between the two group means is statistically significant. Unequal population variances assumed

Steps Making assumptions (2) Stating the research and null hypotheses and selecting alpha (3) Selecting the sampling distribution and specifying the test statistic (4) Computing the test statistic (5) Making a decision and interpreting the results

Example Suppose we have obtained # of years of education from one random sample of 38 police officers from City A and # of years of education from a second random sample of 30 police officers from City B. The average years of education for the sample from City A is 15 with a standard deviation of 2. The average years of education for the sample from City B is 14 with a standard deviation of 2.5. Is there a statistically significant difference between the education levels of police officers in City A and City B?

1.Making Assumptions Two random samples are selected. The sample sizes are large. Because N>50, the assumption of normal population is not required. The level of measurement of the dependent variable is interval-ratio. Population variances are assumed to be equal.

2.State Hypotheses H0: There is no statistically significant difference between the mean education level of police officers working in City A and the mean education level of police officers working in City B.

2.State Hypotheses For a 2-tailed hypothesis test H1: There is a statistically significant difference between the mean education level of police officers working in City A and the mean education level of police officers working in City B.

2.State Hypotheses For a 1-tailed hypothesis test H1: The mean education level of police officers working in City A is significantly greater than the mean education level of police officers working in City B.

2. Set the Rejection Criteria Determine the degrees of freedom df = (n1+n2)-2 df = 38+30-2=66 Determine level of confidence -- alpha (1 or 2-tailed test) Use the t-distribution table to determine the critical value If using 2-tailed test Alpha.05, tcv= 1.997 If using 1-tailed test Alpha.05, tcv= 1.668

3. Specifying the test statistic Because the population variances are unknown, t-distribution should be used. t-statistic.

4. Compute Test Statistic

4. Compare the t-cal with t-cri

5. Make a decision If using 2-tailed test the test statistic 1.835 does not meet or exceed the critical value of 1.997 for a 2-tailed test. There is no statistically significant difference between the mean years of education for police officers in City A and mean years of education for police officers in City B.

If using 1-tailed test the test statistic 1.835 does exceed the critical value of 1.668 for a 1-tailed test. Police officers in City A have significantly more years of education than police officers in City B.

Another Example http://www.gallup.com/poll/111703/Final-Presidential-Estimate-Obama-55-McCain-44.aspx

Test for two sample proportions

Interpreting a t test