Independent Samples Ttest

• With previous tests, we were interested in comparing a single sample with a population • With most research, you do not have knowledge about the population -- you don’t know the population mean and standard deviation

INDEPENDENT SAMPLES T-TEST: • Hypothesis testing procedure that uses separate samples for each treatment condition (between subjects design) • Use this test when the population mean and standard deviation are unknown, and 2 separate groups are being compared Example: Do males and females differ in terms of their exam scores? • Take a sample of males and a separate sample of females and apply the hypothesis testing steps to determine if there is a significant difference in scores between the groups

1

Formula:

t=

(x1 − x2 )− (µ1 − µ 2 )

s x1 − x2

• We are interested in a difference between 2 populations (females, µ1, and males, µ2) and we use 2 samples (females, x1, and males, x2) to estimate this difference

ESTIMATED STANDARD ERROR OF THE DIFFERENCE: • Gives us the total amount of error involved in using 2 sample means to estimate 2 population means. It tells us the average distance between the sample difference (x1-x2) and the population difference (µ1-µ2) • As we’ve done previously, we have to estimate the standard error using the sample standard deviation or variance and, since there are 2 samples, we must average the two sample variances.

2

POOLED VARIANCE: The average of the two sample variances, allowing the larger sample to weighted more heavily Formulae:

s

2 pooled

(df1 ) s 21 + (df 2 ) s 2 2 = df1 + df 2

SS1 + SS 2 OR s 2 pooled = df1 + df 2

df1=df for 1st sample; n1-1 df2=df for 2nd sample; n2-1

Estimated Standard Error of the Difference

s x1 − x2 =

sx1 −x 2

s2 pooled n1

+

s2 pooled n2

book formula

SS1 + SS2 1 1 = + n1 + n 2 − 2 n1 n 2

€

Degrees of freedom (df) for the Independent t statistic is n1 + n2 - 2 or df1+df2

3...

• With previous tests, we were interested in comparing a single sample with a population • With most research, you do not have knowledge about the population -- you don’t know the population mean and standard deviation

INDEPENDENT SAMPLES T-TEST: • Hypothesis testing procedure that uses separate samples for each treatment condition (between subjects design) • Use this test when the population mean and standard deviation are unknown, and 2 separate groups are being compared Example: Do males and females differ in terms of their exam scores? • Take a sample of males and a separate sample of females and apply the hypothesis testing steps to determine if there is a significant difference in scores between the groups

1

Formula:

t=

(x1 − x2 )− (µ1 − µ 2 )

s x1 − x2

• We are interested in a difference between 2 populations (females, µ1, and males, µ2) and we use 2 samples (females, x1, and males, x2) to estimate this difference

ESTIMATED STANDARD ERROR OF THE DIFFERENCE: • Gives us the total amount of error involved in using 2 sample means to estimate 2 population means. It tells us the average distance between the sample difference (x1-x2) and the population difference (µ1-µ2) • As we’ve done previously, we have to estimate the standard error using the sample standard deviation or variance and, since there are 2 samples, we must average the two sample variances.

2

POOLED VARIANCE: The average of the two sample variances, allowing the larger sample to weighted more heavily Formulae:

s

2 pooled

(df1 ) s 21 + (df 2 ) s 2 2 = df1 + df 2

SS1 + SS 2 OR s 2 pooled = df1 + df 2

df1=df for 1st sample; n1-1 df2=df for 2nd sample; n2-1

Estimated Standard Error of the Difference

s x1 − x2 =

sx1 −x 2

s2 pooled n1

+

s2 pooled n2

book formula

SS1 + SS2 1 1 = + n1 + n 2 − 2 n1 n 2

€

Degrees of freedom (df) for the Independent t statistic is n1 + n2 - 2 or df1+df2

3...