Sample can test the hypothetical mean of normally distributed population {t test} {one-sample t test}. Hypothesize that sample and population means are equal. Set significance level to 5%. Sample size less one gives independent-value number {degrees of freedom, t test}. Calculate distribution of same-size-sample means with same degrees of freedom. Result is similar to normal distribution, except distribution includes degrees of freedom {t value} {t distribution}: t = (x - u)/e, where x is sample mean, u is hypothetical population mean, and e is sample-mean standard error. If calculated t value is less than actual t value for significance level and degrees of freedom, do not reject hypothesis.
two samples
Testing two independent samples from population can show if samples are from same population. Hypothesize that first and second sample means are equal. Set significance level to 5%. Degrees of freedom involve both sample sizes: (N1 - 1) + (N2 - 1) = N1 + N2 - 2. Calculate t value: t = (x1 - x2)/e, where x is sample mean. e is standard error of difference, which equals ( ( (v1 * (N1 - 1) + v2 * (N2 - 1)) / (N1 + N2 - 2) )^0.5) * ((1 / N1 + 1 / N2)^0.5), where v is sample variance and N is sample degrees of freedom. If t value is less than t-distribution value with same degrees of freedom at significance level, do not reject hypothesis.
paired samples
Testing two paired samples, or matched pair samples, can show if they are from same population. Hypothesize that first and second sample means are equal. Set significance level to 5%. Degrees of freedom are sample size minus one. Calculate t value: t = sum from i = 1 to i = N of (n1 - n2)/e, where N is sample size and n is sample value. e is standard error of difference, which equals (N * (sum from i = 1 to i = N of (n1 - n2)^2) - (sum from i = 1 to i = N of (n1 - n2)^2) ) / (N - 1)^0.5. If t value is less than t-distribution value with same degrees of freedom at significance level, do not reject hypothesis.
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Date Modified: 2022.0224