- Refresher on independent samples t-test
- Deriving a t-test from the descriptive statistics (mean, standard deviation, sample size)
- Calculating the standard error
- Calculating the t-value
- Calculating the p-value
- Calculating the confidence interval
Outline
Sampling distribution for differences of means
If assumptions of the t-distribution are true
- data come from a normal distribution (normality)
- both standard deviations are identical (equality of variance, homoscedasticity)
- and \(\mu_x\) = \(\mu_y\) or \(\mu_x - \mu_y = 0\)
Then, the difference in the means is distributed according to a t-distribution
\[ t(n_x + n_y -2) \sim \frac{\bar{x} - \bar{y}}{\text{se}} \]
with \(n - 2\) degrees of freedom.
Sampling distribution for differences of means
The denominator of the t-statistic is known as the standard error which is calculated according to
\[ \text{se} = \sqrt{\frac{(n_x - 1) \times s_x^2 + (n_y - 1) \times s_y^2}{n_x + n_y - 2}} \times \sqrt{\frac{1}{n_x} + \frac{1}{n_y}} \]
Breaking down the standard error (left term)
Numerator:
\[ (n_x - 1) \times s_x^2 + (n_y - 1) \times s_y^2 \]
Denominator:
\[ n_x + n_y - 2 \] Fraction:
\[ \text{left term} = \sqrt\frac{\text{numerator}}{\text{denominator}} \]
Breaking down the standard error (right term)
\[ \text{right term} = \sqrt{\frac{1}{n_x} + \frac{1}{n_y}} \]
Putting the standard error together again
Product of the left and the right side of the SE equation.
\[ \text{se} = \text{left term} \times \text{right term} \]
Confidence interval for the null hypothesis
The confidence interval for the difference in means is
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \] where \(\pm\tau\) are the lower and upper values in a t-distribution which contains 95%, 99% etc. of the area under the curve.
Again, the shape of the t-distribution depends on the degrees of freedom, so \(tau\) does too.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
Confidence intervals
Set of all hypothetical values that we do not rule out at the \(\alpha < 0.05\) level of significance is known as the 95% confidence interval.
\[ (\bar{x} - \bar{y}) \pm \tau \times \text{se} \]
\(\pm\tau\) are values in t-distribution that embrace 95% of the area under the curve.
