The following data was reported on total Fe for four types of iron formation (1 = carbonate,2 = silicate,3 = magnetite,4 = hematite).

Group 1:20.4,28.1,27.8,27.0,28.1,25.2,25.3,27.1,20.5,31.1

Group 2:26.2,24.0,26.2,20.2,23.9,34.0,17.1,26.8,23.7,25.2

Group 3:29.4,34.0,27.5,29.4,28.1,26.2,29.9,29.5,30.0,36.0

Group 4:37.2,44.2,34.1,30.3,31.6,33.1,34.1,32.9,36.3,25.8

To carry out an analysis of varianceFtest at significance level 0.01, state the appropriate hypotheses.

H₀:x₁=x₂=x₃=x₄

H_a: allx_i's are unequal

H₀:₁=₂=₃=₄

H_a: all_i's are unequal

H₀:x₁=x₂=x₃=x₄

H_a: at least twox_i's are unequal

H₀:₁=₂=₃=₄

H_a: at least two_i's are unequal

Use software to make side by side boxplots of the four groups. Which of the following statements is true?

There is at least one mild outlier, proceed with caution.

None of the groups has any outliers.

There are several outliers, conditions are not met.

All histograms must look approximately normal to proceed with the ANOVA, but sample sizes are too small to make definitive statements about the normality of the groups. In addition, all groups must be random samples from populations that are much larger than the samples (which is a reasonable assumption). The final condition for an ANOVA is equal variation. Use software to calculate the standard deviation for each group; is this condition met?

Yes, the largest standard deviation is less than twice the smallest.

No, the largest standard deviation is less than twice the smallest.

No, the largest standard deviation is more than twice the smallest.

Yes, the largest standard deviation is more than twice the smallest.

The results of an F test are summarized in an ANOVA table.

SourcedfSum of

SquaresMean

SquaresfTreatments3520.87173.6211.08Error36564.3315.68Total391085.20

Give the test statistic. (Round your answer to two decimal places.)

f=

What is theP-value for the test?

State the conclusion in the problem context.

RejectH₀. There is insufficient evidence to conclude that the total Fe differs for the four formations.

Fail to rejectH₀. There is insufficient evidence to conclude that the total Fe differs for the four formations.

RejectH₀. There is sufficient evidence to conclude that the total Fe differs for at least two of the four formations.

Fail to rejectH₀. There is sufficient evidence to conclude that the total Fe differs for at least two of the four formations.

Which type of error is possible with this test conclusion? Explain the error in the problem context.

Type II error; we conclude that total Fe does not differ between groups, but in reality it does.

Type I error; we conclude that total Fe does not differ between groups, but in reality it does.

Type II error; we conclude that total Fe differs between groups, but in reality it doesn't.

Type I error; we conclude that total Fe differs between groups, but in reality it doesn't.

What next steps should a researcher take if their work depends on the outcome of this test?

Use a different statistical procedure (like bootstrapping); conditions were not met for the ANOVA, so we cannot trust the conclusion.

Remove the outliers and redo the ANOVA with the smaller sample sizes.

Collect larger samples so that outliers are no longer a concern when doing the ANOVA.