Exam 3 Review: Hypothesis Testing Concepts

Exam 3 Review

Chapters 9-11 Overview

• Focus on Hypothesis Testing.

Hypothesis

• Hypothesis refers to assertions about PARAMETERS, not statistics.

  • Ho (Null Hypothesis): Represents the starting assumption that nothing has changed (i.e., it has stayed the same) or that there is no relationship (independence).

  • Always involves the equality symbol "= "

  • The value is an assumption derived from past information, not from the sample.

  • Ha (Alternative Hypothesis): A statement indicating that something has changed.

  • Always involves a change symbol (> , < , ≠).

  • The value is the same as in the null hypothesis.

Hypotheses for Two Sample Tests

• Consider the sign of the difference before choosing the symbol for the alternative hypothesis:

  • If ext{µ1} - ext{µ2} > 0 (positive), then ext{µ1} is larger than ext{µ2}.

  • If ext{µ1} - ext{µ2} < 0 (negative), then ext{µ2} is larger than ext{µ1}.

• Avoid solely relying on the wording of the problem (e.g., “increase” or “decrease”) to select the symbol for the inequality.

• Instead, consider what sign the difference would have if the desired change occurred:

  • Example: If expecting ext{µ2} to be higher than ext{µ1}, then the difference ext{µ1} - ext{µ2} would be negative.

  • If the difference is ext{µ2} - ext{µ1}, expectation of a positive difference.

Test Statistics

• A test statistic transforms the observed data into a standardized value that can be compared to a probability distribution (Normal (Z), T, or Chi-square).

• Formulas for test statistics include:

  • z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}

  • t = \frac{\bar{x} - \mu}{\sqrt{\frac{s^2}{n}}}

  • \chi^2 = \sum \frac{(O - E)^2}{E}

  • Where:

  • O = Observed frequency

  • E = Expected frequency

Finding t-Critical Values

• Critical values (z or t) can be found in the t-distribution table and require three pieces of information:

  • Ha (Alternative Hypothesis)

  • α (Alpha level)

  • Sample size (n)

• Compute Degrees of Freedom (df) as df = n - 1 to find the correct row.

• If Ha is two-tailed (≠), then use \frac{α}{2}; otherwise, use α to find the appropriate column.

• Additional Notes:

  • Include a negative sign if the test is left-tailed.

  • A two-tailed test will always have both a positive and negative value (same absolute value, different signs).

Finding p-values (based on Z)

• p-values correspond to the area in the tail beyond the test statistic (Z) in the Normal (Z) probability table.

• Steps to find p-values:

  1. Negate the given Z-value (test statistic) if it is not already negative and find its corresponding area in the normal table.

  2. Check if Ha is two-tailed (≠):

  • If it is, double the area calculated from the Normal table for your Z-value.

Making Decisions

• Two methods for decision-making in hypothesis testing:

  1. Compare the test statistic to the critical value:

  • The critical value defines the rejection region in the appropriate tail based on the alternative hypothesis (Ha).

  • Reject the null hypothesis (H0) when the test statistic ("the runner") reaches the rejection region in the tail by surpassing the critical value boundary.

  1. Compare the p-value to alpha (α):

  • The p-value is the probability of observing the data (test statistic) or something more extreme if the null hypothesis is true.

  • Alpha represents the significance level and the risk of making a Type I error.

  • Reject the null hypothesis when the p-value is less than alpha.

Interpreting Decisions

• Conclusions in hypothesis testing focus on the Alternative hypothesis (indicating that something has changed):

  • If the null hypothesis is rejected, state: "There is enough evidence that something has changed."

  • If the null hypothesis is not rejected, state: "There is not enough evidence that something has changed."

• Important Note:

  • We NEVER ACCEPT THE NULL or proclaim evidence of its truth.

Types of Errors

• Type I Error: Occurs only when we reject H0.

  • This means that we “find evidence” or “conclude” that something has changed, implying H0 was rejected.

• Type II Error: Occurs only when we fail to reject H0.

  • This means we “do not find evidence” or “are unable to conclude” that something has changed, implying H0 was not rejected.

• Considerations:

  • Before declaring an error, ensure that the decision conflicts with reality, if known.

Confidence Intervals for Differences

• When estimating a difference in means (or proportions) with a confidence interval, there are three possible outcomes:

  1. Bounds of the form (-, +) indicate no significant difference, as zero is a potential value within this interval.

  2. Bounds of the form (-, -) indicate a significant difference, showing that the second population has a larger mean (or proportion).

  3. Bounds of the form (+, +) indicate a significant difference, showing that the first population has a larger mean (or proportion).

Chi-Square Tests

• Goodness of Fit Test:

  • Ha: The distribution differs from the claimed distribution.

  • Expected counts: Calculated as n * \pi if specific percentages are provided; otherwise, divide n by the number of categories (k).

  • Degrees of Freedom (DF): Calculated as Number of categories - 1.

• Independence Test:

  • Ha: The variables are dependent.

  • Expected counts: Computed as (\text{row}) * (\text{col}) / n.

  • Degrees of Freedom (DF): Calculated as (\text{# rows - 1}) * (\text{# columns - 1}).

Chi-Square Tests Interpretations

• Reject the null when the chi-square statistic exceeds the critical value, which is determined from the table based on the degrees of freedom and alpha levels.

• Interpretations are as follows:

  • Reject H0: There is enough evidence that:

  • The distribution differs from what was claimed.

  • The variables are dependent.

  • Fail to Reject H0: There is not enough evidence that:

  • The distribution differs from what was claimed.

  • The variables are dependent.