Independent samples difference for the means

Case I (null hypothesis)
No significant difference in the sample means. Both samples have means that could have come from the same population. Put another way, the potential population means implied by each sample could be the same.

Population
population mean difference is zero
µx1 = µx2
µx2 − µx1 = 0

Sample1
sample size n1
sample mean x1
sample sx1

Sample2
sample size n2
sample mean x2
sample sx2

When we say the sample means are "equal" we mean "statistically equal," not mathematically equal. We mean that there is no statistically significant difference between the two sample means. Statistically speaking we say that the two samples could come from the same population.

Independent samples difference for the means

Case II (alternate hypothesis)
Significant difference in the sample means. The samples have means that did not come from the same population. Put another way, the potential population means implied by each sample is different.

Populationx1
population mean µx1

Sample1
sample size n1
sample mean x1
sample sx1

Populationx2
population mean µx2

Sample2
sample size n2
sample mean x2
sample sx2

In case II the difference in the sample means is too large for that difference to likely be zero. Statistically speaking we say that the two samples come from different populations.

Paired difference for the means of two samples

Case I (null hypothesis)
No difference between before and after means

Population
population mean difference is zero
µbefore = µafter
µafter − µbefore = 0

Samplebefore
sample size nbefore
sample mean xbefore
sample sxbefore

Sampleafter
sample size nafter
sample mean xafter
sample sxafter

When we say the sample mean before is "equal" we mean "statistically equal," not mathematically equal. We mean that there is no statistically significant difference between the before and after at some level of confidence. Statistically speaking we say that the two samples could come from the same population.

Paired difference for the means of two samples

Case II (alternate hypothesis)
Significant difference between before and after means

Populationbefore
population mean µbefore

Samplebefore
sample size nbefore
sample mean xbefore
sample sxbefore

Populationafter
population mean µafter

Sampleafter
sample size nafter
sample mean xafter
sample sxafter

In case II the difference in the sample means is too large for that difference to likely be zero. Statistically speaking we say that the two samples come from different populations.

Sampling distribution of the mean

Case I
the sample comes
from the population

Population
population mean µ

Sample1
sample size n1
sample mean x1
sample stdev sx1

Case II
the sample does not come from the population

Population
population mean µ

Sample1
sample size n1
sample mean x1
sample stdev sx1

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