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What is a two-sample t-test?
A test to determine if there is a statistically significant difference between the means of two independent groups.
Define 'degrees of freedom' (df) in a t-test.
The number of independent pieces of information available to estimate a parameter. Approximated as min(n1-1, n2-1) for two-sample t-tests by hand.
What is a p-value?
The probability of observing a sample mean difference as extreme as, or more extreme than, what you got, assuming the null hypothesis is true.
Define 'significance level' (α).
The probability of rejecting the null hypothesis when it is true (Type I error).
What is the null hypothesis (H0)?
A statement of no effect or no difference, which we aim to disprove.
What is the alternative hypothesis (Ha)?
A statement that contradicts the null hypothesis, representing what we are trying to find evidence for.
What is the formula for the two-sample t-test statistic?
$\text{t} = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
How do you calculate degrees of freedom (df) by hand?
df = min(n1 - 1, n2 - 1)
What is the general form for calculating a test statistic?
(Observed Difference) / (Standard Error) which is essentially (sample statistic - null hypothesis value) / (standard error of the statistic)
Explain the concept of statistical significance.
A result is statistically significant if the p-value is less than the significance level (α), indicating strong evidence against the null hypothesis.
Explain the importance of checking assumptions before conducting a two-sample t-test.
Violating assumptions (randomness, independence, normality) can invalidate the results of the test, leading to incorrect conclusions.
Explain how the Central Limit Theorem (CLT) relates to two-sample t-tests.
If sample sizes are large enough (n ≥ 30), the CLT allows us to proceed with the t-test even if the populations are not normally distributed.
Explain the meaning of failing to reject the null hypothesis.
It means we do not have enough evidence to support the alternative hypothesis. It does not prove the null hypothesis is true.
Describe the relationship between the t-score and the p-value.
The t-score is used to calculate the p-value. A more extreme t-score (further from zero) generally results in a smaller p-value.
What does the standard error in the t-test formula represent?
It estimates the variability of the difference between the two sample means.