When analyzing the variance from a completely randomized experiment, if the null hypothesis from the F test is true

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To analyze this question, begin by restating what is being compared in a completely randomized experiment using an F test. The core idea under the null hypothesis is that all treatment means are equal, so the variability attributed to treatments should be just random sampling variability around a common mean, not a real treatment effect.

Option 1: 'the mean square due to treatments and the mean square due to error are estimating the same thing.' Under H0, both mean squares estimate the common underlying population variance (the error variance) because there is no systematic treatment effect. Therefore, their expected values are equal, which makes this statement true in the null context.

Option 2: 'the mean square due to treatments will be less than the mean square due to error.' This is not generally guaranteed by the null hypothesis. If there is no true effect, the two mean squares should be similar in expectation, not systematically smaller for treatments. Observed values can vary due to random sampling, but there isn’t a theoretical expectation that MS_treatments < MS_error under H0.

Option 3: 'the test statistic is distributed approximately normal with degrees of freedom equal to the sample size minus the number of treatments.' The F test statistic is the ratio of MS_treatments to MS_error and follows an F distribution with (k−1, N−k) degrees of freedom under H0, where k is the number of treatments and N is the total sample size. It is not a normal distribution, and the df are not simply (N−k) for both components.

Option 4: 'the mean square due to treatments will be greater than the mean square due to error.' While MS_treatments can be greater or smaller by chance, the null hypothesis does not mandate MS_treatments > MS_error. The key under H0 is not a directional relationship but that they estimate the same variance; thus this statement is not universally true and is not the defining property of the null.

In summary, the most accurate reflection of the null hypothesis in a variance analysis via the F test is that the two mean squares are estimating the same underlying variance (the error variance), making Option 1 the correct characterization among the choices. The remaining options describe conditions or distributions that do not align with the null hypothesis framework or with the distributional properties of the F statistic.

When analyzing the variance from a completely randomized experiment, if the null hypothesis from the F test is true

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