Introducing Statistics: Are My Results Unexpected - AP Statistics

Type I error. False positive: rejecting a true null hypothesis.

Type II error. False negative: failing to reject a false null hypothesis.

The probability of correctly rejecting a false null hypothesis. Higher power means better ability to detect real effects.

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$. Uses critical value and standard error of the mean.

95% of intervals will contain the true population parameter. It's about the method's long-run capture rate.

$n = \left(\frac{z^* \sigma}{E}\right)^2$. Balances desired precision with critical value and variability.

Mean $0$, standard deviation $1$. The reference distribution for z-scores.

$0.05$. This is the standard cutoff for statistical significance.

The result is unlikely due to chance, given $\alpha$. The observed difference is unlikely to be due to chance.

The null hypothesis states there is no effect or difference. It's the baseline assumption we test against.

The alternative hypothesis states there is an effect or difference. It's what we're trying to prove or find evidence for.

$\bar{x}$. X-bar represents the calculated average from sample data.

The probability of observing data at least as extreme assuming the null hypothesis is true. It measures how unusual our data would be under $H_0$.

$\alpha$. Greek letter alpha sets our threshold for significance.

Reject the null hypothesis. Evidence is strong enough to conclude $H_a$ is true.

$z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by SD.

A test where effects in only one direction are considered. Alternative hypothesis specifies direction of the effect.

$z = 1$. $(10-8)/2 = 1$ standard deviation above mean.

A data point that differs significantly from other observations. Unusual values that don't fit the general pattern.

A test where effects in both directions are considered. Alternative hypothesis allows for effects in either direction.

Reject the null hypothesis. P-value is less than significance level.

At least one group mean is different. Large F suggests between-group variation exceeds within-group.

All group means are equal. ANOVA tests if all population means are identical.

To compare means from the same group at different times. Controls for individual differences by using matched pairs.

$1.96$. Corresponds to 2.5% in each tail of standard normal.

Data are normally distributed. Required for t-distribution to be appropriate.

$n - 1$. Reflects loss of information from estimating population SD.

How many standard deviations an element is from the mean. It standardizes position relative to the distribution.

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$. Uses sample SD when population SD is unknown.