You compare two sample variances using an F-test and get F = 0.15 with (6, 10) degrees of freedom. To find the lower-tail critical value using a standard F-table that gives only upper-tail values, you should:
AUse the F-table with F(6, 10) directly — standard tables cover both tails
BCompute 1/F ≈ 6.67 and look up the upper-tail critical value for F(10, 6)
CDouble the upper-tail p-value, since the F-distribution is symmetric
DSquare the F-statistic and use a chi-square table with 16 degrees of freedom
The F-distribution is NOT symmetric. Standard tables give only upper-tail critical values. The reciprocal relationship F_{α, k₁, k₂} = 1/F_{1−α, k₂, k₁} handles lower-tail values: invert F and swap the degrees of freedom, then look up the upper tail of that new F-distribution. Here 1/0.15 ≈ 6.67, looked up in F(10, 6). Option C is wrong because the F-distribution is right-skewed and asymmetric. Option D conflates F with chi-square.
Question 2 Multiple Choice
In a one-way ANOVA, an F-statistic much larger than 1 most likely indicates:
AThe within-group variance substantially exceeds the between-group variance
BThe between-group variance substantially exceeds the within-group variance
CThe sample sizes across groups are unequal
DThe assumption of equal population variances has been violated
The F-statistic in ANOVA is the ratio of between-group variance (MSB) to within-group variance (MSW). When all group means are equal, both estimate the same underlying variance and their ratio should be near 1. When group means truly differ, MSB inflates — it now captures genuine mean differences plus sampling noise — while MSW remains stable. A large F signals real differences between group means, pushing the statistic into the right tail of the F-distribution.
Question 3 True / False
The F-distribution always takes positive values and is right-skewed, especially when the degrees of freedom are small.
TTrue
FFalse
Answer: True
The F-distribution is defined as [χ²(k₁)/k₁] / [χ²(k₂)/k₂]. Both chi-square variables are sums of squared normals, so they are non-negative — making their ratio non-negative (positive almost surely). Chi-square distributions are right-skewed, especially at small degrees of freedom, which the F inherits. As both degrees of freedom grow large, the distribution concentrates near 1 and the skew decreases, but it remains non-symmetric.
Question 4 True / False
The shape of the F-distribution is determined by a single degrees-of-freedom parameter, just like the t-distribution.
TTrue
FFalse
Answer: False
Unlike the t-distribution (one df parameter), the F-distribution has two separate parameters: numerator degrees of freedom k₁ and denominator degrees of freedom k₂. Because F is a ratio of two chi-square variables each divided by their own df, the shape depends on both. This is why F-tables are two-dimensional. The t-distribution is a special case (t(k)² = F(1, k)), but the general F is fundamentally biparametric.
Question 5 Short Answer
Explain why the F-statistic in ANOVA should be close to 1 when all population group means are equal, and why it tends to exceed 1 when means differ.
Think about your answer, then reveal below.
Model answer: When all group means are equal, both the between-group estimate (MSB) and the within-group estimate (MSW) are estimating the same population variance σ². Their ratio F = MSB/MSW should therefore be near 1. When group means truly differ, MSB inflates because it captures both sampling variability and the actual spread of group means, while MSW continues to estimate only within-group variability. The numerator grows relative to the denominator, pushing F above 1 and into the right tail.
This is why F-tests for ANOVA are one-tailed (right-tail only): we reject when F is too large, because large F means between-group variation is too great to plausibly arise from groups with the same mean. An F near 1 is consistent with the null hypothesis; an F much greater than 1 is evidence against it.