One-Way Analysis of Variance · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Many means, one test

Mathematics · Data & Statistics · ages 21-22
Name ______________________   Date ____________

  1. Between group sum of squares is sixty and within group is forty. What is the total sum of squares?

    Answer: ______________

  2. The total sum of squares splits into which two parts?

    • Between group plus within group
    • Mean plus variance
    • First df plus second df
    • F statistic plus p value

  3. Between group sum of squares is 60 and within group is 40. What is the total sum of squares?

    Answer: ______________

  4. Four groups are compared. How many pairwise t tests would that take?

    • 4
    • 3
    • 6

  5. With three groups, the between groups df is groups minus one. What is it?

    Answer: ______________

  6. Why is comparing four groups with six t tests worse than one analysis of variance?

    • It uses less arithmetic
    • The chance of at least one false rejection grows fast
    • It needs larger samples

  7. Mean square between is nine and mean square within is three, so F is three. What does a large F mean?

    • All group means are exactly equal
    • The null hypothesis is certainly true
    • Group differences exceed what noise explains
    • The data cannot be normal

  8. With five groups there are ten pairwise tests, and at five percent each the chance of at least one false rejection far exceeds five percent. True or false?

    Circle one:   True   False

  9. After analysis of variance rejects, what comes next?

    • Rerun the same six t tests at full level
    • Stop, nothing more is allowed
    • Careful follow ups with protection for the extra looks

  10. F on (2, 12) df is 8.1 against a critical value of 3.9. What is the decision?

    • Accept that all means are equal
    • The test assumptions make it invalid
    • Collect fewer data and retry
    • Reject equal means since F exceeds the cutoff
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Many means, one test W1-mt_WYGK3Ktnqz-s1

  1. 100 · Sixty plus forty is one hundred.
  2. Between group plus within group · Variation between means plus variation inside groups rebuilds the total.
  3. 100 · The two piles add back to the total: 60 plus 40.
  4. 6 · Four groups make 6 pairs, and each test would add its own error risk.
  5. 2 · Three minus one is two.
  6. The chance of at least one false rejection grows fast · Each test risks a false rejection, and with many pairs the family rate balloons far past five percent.
  7. Group differences exceed what noise explains · F compares signal to noise. Large values mean the means spread beyond luck.
  8. True · True. Ten unprotected looks inflate the family error rate toward forty percent.
  9. Careful follow ups with protection for the extra looks · Rejection opens the door to asking which pairs differ, but the extra looks need protection.
  10. Reject equal means since F exceeds the cutoff · 8.1 lands deep in the right tail, so equal means are rejected.
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