Question

A researcher wants to test if there is an association between the type of cardboard box (four different types) and its compression strength. Six boxes of each type were tested to see how much compression was needed in order to crush the box.

(a) What will be the distribution of the statistic needed to get the p-value for a test of this type?

• F_6,24
• F_3,23
• F_3,20
• ?²₁₅
• ?²₂₄

_?

(b) The researcher collects the data and obtains the side-by-side boxplots given above. Based on this plot, which of the assumptions would be most likely to have been violated?

• One of the samples was not a SRS.
• One of the samples does not appear to be normally distributed.
• One of the samples does not have the same standard deviation as the others do.
• Two of the above assumptions appear to have been violated.
• None of the assumptions appear to have been violated.

Answer 1

(a)

The numerator and denominator degrees of freedom are,

$$ \begin{aligned} d f_{1} &=k-1 \\ &=4-1 \\ &=3 \\ d f_{2} &=n-k \\ &=4(6)-4 \\ &=24-4 \\ &=20 \end{aligned} $$

The distribution of the statistic needed to get the \(p\) -value for a test of this type is,

$$ F_{d_{1}, d f_{2}}=F_{3,20} $$

(b)

Based on the side by side boxplots, the assumption of the equal standard deviation is not satisfied. The standard deviations are varying for each brand. So, the correct option is, "One of the samples does not have the same standard deviation as the others do."