LIR 493: Quantitative Methods Professor Wallace Hendricks

SPSS EXAMPLES:

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null hypothesis (H₀): A maintained hypothesis that is held to be true until sufficient evidence to the contrary is obtained.

ALTERNATIVE HYPOTHESIS (H_a): A hypothesis against which the null hypothesis is tested and which will be held to be true if the null is held false.

SIMPLE HYPOTHESIS: A hypothesis that specifies a single value for a population parameter of interest.

COMPOSITE HYPOTHESIS: A hypothesis that specifies a range of values for a population parameter .

ONE-SIDED ALTERNATIVE: An alternative hypothesis involving all possible values of a population parameter on either one side or the other of (that is, either greater than or less than) the value specified by a simple null hypothesis.

TWO-SIDED ALTERNATIVE: An alternative hypothesis involving all possible values of a population parameter other than the value specified by a simple null hypothesis .

HYPOTHESIS TEST DECISIONS: A decision rule is formulated, leading the investigator to either accept (fail to reject) or reject the null hypothesis on the basis of sample evidence.

type I error : The rejection of a true null hypothesis (also called error).

type II error : The acceptance (failure to reject) of a false null hypothesis (also called error).

significance level : The probability of rejecting a null hypothesis this is TRUE. (This probability is often expressed as a percentage, so a test of significance level is referred to as a 100%-level test.

power : The probability of rejecting a null hypothesis that is false (1-).

P-Value - The significance level.

Definitions:

type I error - Rejecting a true null hypothesis

type II error - Failing to reject a false null hypothesis

Interpretation:

type I error - The researcher concludes that the treatment does have an effect when, in fact, there is no treatment effect.

type II error - The researcher concludes that there is no evidence for a treatment effect when, in fact, the treatment does have an effect.

How does it happen?

type I error - By chance, the sample consists of individuals with extreme scores. As a result, the sample looks different than we might have anticipated under the null hypothesis . The treatment had no impact, but the individuals were different on average than other individuals in the population .

type II error - There are several explanations. Perhaps the treatment effect is too small to be picked up by this test (e.g. the test isn't power ful enough). There may be errors in the measurement of the outcome, etc.

Consequences

type I error - The researcher will claim an effect in a published report. This has serious consequences. Many other researchers will spend resources trying to replicate the result. Others will spend time trying to explain the result. Companies may spend money on a worthless project.

type II error - The researcher might decide that the treatment has an effect, but that it is too small to find with this particular experiment. The experiment may be redone with better instruments or larger sample sizes. Alternatively, the researcher may give up on the idea and fail to pursue a worthy area of inquiry.

Intuition Behind Type I vs Type II Errors (MS 311F)

(Percentage error in Parenthesis)		ACTUAL	SITUATION
		No Effect, H0 True	Effect Exists, H0 False
Researcher's	Reject Ho	type I error ()	Decision Correct (1-)
Decision	Accept Ho	Decision Correct (1-)	type II error ()

Examples:

(1) Single sample test on the mean

Number of Valid Observations (Listwise) = 32.00

	Variable	Mean	Std Dev	Minimum	Maximum	N	Label
	EDU	11.66	4.11	4.000	18.000	32	YEARS OF EDUCATION

is at least graduation from H.S.

.lt. graduation from H.S.

Do we reject or fail to reject the null?

What is the critical value of the t-statistic?

(2) Differences of Means Tests

(a) Do married people have less education than single people?

T-TEST /GROUPS mar (1,3) /VARIEABLES edu,hwage.

independent samples of MAR MARITAL STATUS

Group 1: MAR EQ 1 Group 2: MAR EQ 3

t-test for: EDU YEARS OF EDUCATION

		Number of Cases	Mean	Standard Deviation	Standard Error
Married	Group 1	851	12.3149	3.068	.105
Single	Group 2	249	13.1165	2.716	.172

Pooled Variance Estimate

Separate Variance Estimate

F Value	2-Tail Prob	t Value	Degrees of Freedom	2-Tail Prob	t Value	Degrees of Freedom	2-Tail Prob
1.28	.021	-3.72	1098	.000	-3.97	449.47	.000

The pooled variance estimate assumes that the population variances for the two groups are equal. Therefore, the estimate averages the two samplestandard deviations to estimate the (common) population variance.

The separate variance estimate is rarely used by researchers. It does not assume that the population variances are equal. It seems a little strange to test for equality of means when the variances are different.

(b) Do married people make higher wages than single people?

independent samples of MAR MARITAL STATUS

Group 1: MAR EQ 1 Group 2: MAR EQ 3

t-test for: HWAGE HOURLY WAGE

	Number		mean	Standard	Standard
	of Cases			Deviation	Error
	Group 1		854	6.6242	5.160		.177
	Group 2		249	4.6763	2.985		.189
Pooled Variance Estimate				Separate Variance Estimate

F Value	2-Tail Problem	t Value	Degrees of Freedom	2-Tail Prob	t Value	Degrees of Freedom	2-Tail Prob
2.99	.000	5.68	1101	.000	7.53	711.39	.000

(3) More SPSS examples

T-TEST /GROUPS BIG10 (0,1) /VARIABLES SAT GRADRATE BASKETCH TUITIONI.

t-test for: SAT

		Number of Cases	Mean	Standard Deviation	Standard Error
	Group 1	276	960.0449	164.459	10.065
	Group 2	10	1072.5000	101.641	32.142

Pooled Variance Estimate

Separate Variance Estimate

F Value	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.
2.62	.114	-2.14	275	.033	-3.34	10.85	.007

t-test for: GRADUATE

		Number of Cases	Mean	Standard Deviation	Standard Error
	Group 1	251	51.8088	19.538	1.233
	Group 2	10	59.3000	15.621	4.940

Pooled Variance Estimate

Separate Variance Estimate

F Value	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.
1.56	.474	-1.20	259	.233	-1.47	10.16	.171

t-test for: BASKETCH Years in NCAA BB

		Number of Cases	Mean	Standard Deviation	Standard Error
	Group 1	269	1.9777	2.430	.148
	Group 2	10	5.1000	3.381	1.069

Pooled Variance Estimate

Separate Variance Estimate

F Value	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.
1.94	.094	-3.93	277	.000	-2.89	9.35	.017

t-test for: TUITIONI Instate Tuition

		Number of Cases	Mean	Standard Deviation	Standard Error
	Group 1	265	4514.6491	4154.729	255.223
	Group 2	10	3468.5000	3382.776	1069.728

Pooled Variance Estimate

Separate Variance Estimate

F Value	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.	t Value	Degrees of Freedom	2-Tail Prob.
1.51	.516	.79	273	.433	.95	10.05	.364