12.3 Facts About the F Distribution

Step 1 — Identify the properties that hold for every F curve: The section lists five facts; the ones that hold regardless of the degrees of freedom are the shape and range facts.

Step 2 — State the three: 1. The curve is not symmetrical — it is skewed to the right. 2. The F statistic is always greater than or equal to zero. 3. There is a different curve for each pair of degrees of freedom (this is what makes the dfs essential to identifying the distribution).

Answer: The three universal properties are: (1) the F curve is skewed to the right (not symmetrical), (2) \(F \geq 0\) always, and (3) there is a separate F curve for each set of degrees of freedom.

Step 1 — Explain why a different curve exists for each df pair: The F statistic is a ratio of two mean squares, and each mean square depends on its own degrees of freedom. Changing either df changes how much variability that mean square estimates, so the whole shape of the ratio's distribution changes.

Step 2 — Name the two identifying numbers: A specific F curve is identified by the ordered pair \((df_{num}, df_{denom})\): the numerator degrees of freedom (groups − 1) and the denominator degrees of freedom (total observations − groups). Order matters — \(F_{3,20}\) and \(F_{20,3}\) are different curves.

Answer: Each pair of dfs changes the ratio of two mean squares' distribution, so every \((df_{num}, df_{denom})\) pair has its own curve; the two numbers are the numerator df and denominator df, written as subscripts in order.

Step 1 — Recall fact 4 from the section: As both the numerator and denominator degrees of freedom get larger, the sharp right skew of the F curve flattens out.

Step 2 — Name the limiting distribution: With both dfs large, the curve becomes tall and humped, centered near 1, approximating the normal distribution. This is visible in Figure 12.3.1 with \(F_{100,100}\).

Answer: As both degrees of freedom grow very large, the F curve comes to resemble the normal distribution.

Step 1 — Look at how F is constructed: The F statistic is

where each mean square is a sum of squares divided by its degrees of freedom.

Step 2 — Check whether either part can be negative: Sums of squares are built from squared deviations, so every term is nonnegative, and dividing by positive degrees of freedom keeps them nonnegative. A ratio of two nonnegative quantities (with nonzero denominator) cannot be negative.

Answer: No. An F statistic can never be negative, because it is a ratio of two mean squares, and mean squares come from squared deviations, which are always ≥ 0. A negative F indicates an arithmetic error.

Step 1 — Compute the numerator df: The numerator degrees of freedom equal the number of groups minus one:

Step 2 — Compute the denominator df: The denominator degrees of freedom equal the total number of observations minus the number of groups:

Step 3 — Name the distribution: The test uses an F distribution with those two dfs.

Answer: \(df(num) = 5\), \(df(denom) = 24\); the test uses the \(F_{5,24}\) distribution.

Step 1 — Compare α to the p-value: Here \(\alpha = 0.05\) and p-value = 0.008. Since

we reject \(H_0\).

Step 2 — Write the conclusion in plain language: Rejecting \(H_0\) means the data give sufficient evidence that at least one group mean differs from the others. In context: at the 5% significance level, there is strong evidence that the means of the groups being compared are not all equal — the differences among the group means are unlikely to be due to chance alone.

Answer: Decision: reject \(H_0\). Conclusion: At the 5% significance level, there is sufficient evidence that at least one of the group means differs from the rest.

Step 1 — Compute \(MS_{between}\): A mean square is its sum of squares divided by its own degrees of freedom:

Step 2 — Compute \(MS_{within}\):

Step 3 — Compute the F statistic: Divide the between mean square by the within mean square:

Answer: \(MS_{between} = 150\), \(MS_{within} \approx 33.33\), and \(F \approx 4.50\).

Step 1 — Interpret what a small F means: The F statistic is the ratio

Here \(F = 0.87 < 1\), which forces the numerator to be smaller than the denominator:

Step 2 — Translate into variation language: Since the between-group mean square is smaller than the within-group mean square, the variation among the four teaching-method group means is small relative to the natural variation of students within each method. That is exactly why the p-value is large (0.46): the observed differences among group means look like ordinary chance variation.

Answer: Because \(F = \frac{MS_{between}}{MS_{within}} = 0.87 < 1\), we must have \(MS_{between} < MS_{within}\); the between-group variation is smaller than the within-group variation, so no new calculations are needed.

Step 1 — Verify the numerator df: With \(k = 4\) groups,

Step 2 — Verify the denominator df: With 4 groups of 8 observations each, the total is \(n = 32\):

(Equivalently, total df = 31, minus 3 for the factor, leaves 28.)

Step 3 — Explain "balanced": A design is balanced when every group contains the same number of observations. Here each of the 4 groups has exactly 8 observations, so the design is balanced — this makes the arithmetic tidier because each group contributes equally to the pooled within-group estimate.

Answer: Verified: \(df(num) = 3\) and \(df(denom) = 28\). "Balanced" means all 4 groups have the same sample size (8 each).

Step 1 — Enter the data: Press STAT, then 1: EDIT, and type the three data sets into lists \(L_1\), \(L_2\), and \(L_3\).

Step 2 — Run ANOVA: Press STAT, arrow over to TESTS, arrow down to ANOVA (on some models F:ANOVA), press ENTER, then enter (L1, L2, L3) and press ENTER.

Step 3 — Read the decision-making outputs: The screen displays the F statistic and the p-value, along with the full ANOVA table (SS, df, MS for Factor and Error). The two outputs used for the decision are:

• the p-value: compare it to \(\alpha\) — if p-value ≤ \(\alpha\), reject \(H_0\);
• the F statistic: the test statistic corresponding to that p-value on the appropriate F curve.

Answer: Keystrokes: STAT → 1: EDIT (enter data in \(L_1, L_2, L_3\)) → STAT → TESTS → ANOVA → (L1, L2, L3) → ENTER. The two decision outputs are the F statistic and the p-value.

Step 1 — Write the entry: To find the area to the right of 2.15 under the \(F_{5,20}\) curve, use

Fcdf(2.15, E99, 5, 20)

(accessed via 2nd DISTR, arrow down to Fcdf().

Step 2 — Explain each argument: 1. 2.15 — the lower bound of the interval (the observed F value). 2. E99 (\(10^{99}\)) — the upper bound, standing in for infinity since the F distribution extends indefinitely to the right. 3. 5 — the numerator degrees of freedom. 4. 20 — the denominator degrees of freedom.

Answer: Enter Fcdf(2.15, E99, 5, 20); the arguments are the lower bound (2.15), the upper bound (E99 ≈ ∞), the numerator df (5), and the denominator df (20).

Step 1 — Recall fact 5 from the section: Besides testing equality of several means (one-way ANOVA), the F distribution has other uses listed in the text.

Step 2 — Name the two other uses: (1) comparing two variances, and (2) two-way Analysis of Variance (two-way ANOVA).

Step 3 — State which this chapter covers: Two-Way Analysis is explicitly noted as beyond the scope of this chapter, so the use covered here (besides one-way ANOVA itself) is comparing two variances.

Answer: The two other uses are comparing two variances and two-way Analysis of Variance; two-way ANOVA is beyond the scope of this chapter, so comparing two variances is the one covered.