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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...