Question

List the three different cases that we studied for comparison of means, and write the equations used in each case.

Short answer

Case 1: Comparing a Measured Result with an Accepted Result that is known

$t_{calc}=\left(\overline{x}-\mu \right)\frac{\sqrt{n}}{s}$

Case 2: Comparing Two Samples/Methods Means is$s_{pooled}=\sqrt{\frac{s_1^2\left(n_1-1\right)+s_2^2\left(n_2-1\right)}{n_1+n_2-n_t}}$

Case 3: Individual differences are compared using a paired t test; standard deviation of differences is a necessary computation$s_d=\sqrt{\frac{\sum _{}{\left(d_i-\overline{d}\right)}^2}{n-2}}$

Step-by-step solution

Definition of mean and standard deviation of difference

• Average value derived by summing all values and dividing by the total number of values (science: statistics).
• The standard deviation (SD) is a measure of the variability, or dispersion, between individual data values and the mean.
• Whereas the standard error of the mean (SEM) is a measure of how far the sample mean (average) of the data is expected to differ from the genuine population mean. Always, the SEM is smaller than the SD.

Determine a measured result with an accepted result and compare two samples means

Case 1: Comparing a Measured Result with an Accepted Result that is known

$t_{calc}=\left(\overline{x}-\mu \right)\frac{\sqrt{n}}{s}$

The confidence interval (CI) formula is used to derive the expression above.

Case 2: Comparing Two Samples/Methods Means

$t_{calc}=\frac{\left|{\overline{x}}_1-{\overline{x}}_2\right|}{s_{pooled}}\sqrt{\frac{n_1{n}_2}{n_1+n_2}}$

Calculate the prerequisites:

$s_{pooled}=\sqrt{\frac{s_1^2\left(n_1-1\right)+s_2^2\left(n_2-1\right)}{n_1+n_2-n_t}}$

Determine the individual difference and comparing by using the paired t- test

Case 3: Individual differences are compared using a paired t test.

$t_{calc}=\frac{\left|d\right|}{8_d}\sqrt{n}$

Standard deviation of differences is a necessary computation

$s_d=\sqrt{\frac{\sum _{}{\left(d_i-\overline{d}\right)}^2}{n-2}}$

Where: $d_3=$difference of each sample and$\overline{d}=$ mean of differences