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29 Δεκ 2023 · In summary, the key differences between the two mean formulas are µ vs. x̅ (mu vs. x bar symbols) and N vs. n. In each case, the former relates to the population, while the latter is for the sample mean formula. Summing up values and dividing by the number of items is consistent in both formulas.
One-sample t-tests: Compare the sample mean with a known value, when the variance of the population is unknown; Two-sample t-tests: Compare the means of two groups under the assumption that both samples are random, independent, and normally distributed with unknown but equal variances
You would create very messy notation using the $\mu$-Notations, whereas $$E\left(X+\frac{1}{2}Y\right)=E(X)+\frac{1}{2}E(Y)$$ is as clear as it can get. Note that the expected value above is potentially unknown, or at least not explicitly given.
In fact, the t-value that the t-test relies on is a ratio between the signal (difference between mean (\(\bar{x}\)) and threshold (\(\mu_{0}\))) and noise (variability, standard error of the mean (\(s/ \sqrt{n}\))): \[t = \frac{\bar{x}-\mu_{0}} {s/ \sqrt{n}}\]
One-sample: Compares a sample mean to a reference value. Two-sample: Compares two sample means. Paired: Compares the means of matched pairs, such as before and after scores. In this post, you’ll learn about the different types of t tests, when you should use each one, and their assumptions.
5 ημέρες πριν · Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other. In particular, you can use this test to check whether the two groups are different from one another. Examples:
The good news is that by using a special, adjusted degrees of freedom value, the distribution of $$t = \frac{(\overline{x}_1 - \overline{x}_2) - 0}{\displaystyle{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}}$$ is so close to a $t$-distribution that nobody will be able to tell the difference.
