Introducing Statistics: Are My Results Unexpected - AP Statistics

1. What is a two-tailed test?

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

2. What is the purpose of a paired t-test?

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

3. Find the sample mean of the dataset: $5, 10, 15$.

   $\bar{x} = 10$. $(5+10+15)/3 = 30/3 = 10$.

4. What is the formula for the confidence interval for a population mean?

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

5. What is the null hypothesis in an ANOVA test?

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

6. Find the critical z-value for a 95% confidence level.

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

7. What does it mean if a p-value is less than $\alpha$?

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

8. Identify the standard normal distribution.

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

9. What is the formula for variance?

   $\sigma^2$ or $s^2$. Variance equals standard deviation squared.

10. What type of variable is gender?

    Categorical variable. Qualities or labels that classify observations.

11. What is the purpose of a hypothesis test?

    To determine if there is enough evidence to reject the null hypothesis. Provides systematic framework for decision-making with data.

12. What does it mean if a result is statistically significant?

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

13. Identify the symbol for the sample standard deviation.

    $s$. Lowercase s measures sample spread.

14. What is an outlier in a dataset?

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

15. State the formula for calculating a z-score.

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

16. What is the significance level commonly denoted as?

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

17. What is a one-tailed test?

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

18. Identify the symbol for the population mean.

    $\mu$. Greek letter mu represents the true population average.

19. Which statistical test is used for comparing means of more than two groups?

    ANOVA. Analysis of variance extends t-tests to multiple groups.

20. Identify the symbol for the sample mean.

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

21. Find the z-score for $x = 10$, $\mu = 8$, $\sigma = 2$.

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

22. What is the alternative hypothesis in hypothesis testing?

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

23. Which error occurs if the null hypothesis is incorrectly accepted?

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

24. Identify the degrees of freedom in a t-test.

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

25. What does a 95% confidence interval mean?

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

26. Which error occurs if the null hypothesis is incorrectly rejected?

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

27. What type of variable is height?

    Quantitative variable. Numerical measurements that can be meaningfully ordered.

28. What does a p-value represent in hypothesis testing?

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

29. Identify the symbol for standard deviation.

    $\sigma$. Greek letter sigma measures population spread.

30. What is the chi-square test used for?

    Testing independence or goodness of fit. Compares observed and expected frequencies in categories.

31. What does a z-score indicate?

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

32. What does a high F-statistic in ANOVA indicate?

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

33. What is the assumption for a t-test?

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

34. What is the null hypothesis in hypothesis testing?

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

35. What is the formula for calculating a sample size for a given margin of error?

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

36. What is the typical significance level used in hypothesis testing?

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

37. What does a p-value of 0.03 indicate at $\alpha = 0.05$?

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

38. What is the formula for a test statistic in a t-test?

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

39. What is the power of a test?

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

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