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# Question 2 [21 marks] The 100 metre Sprint is one of the most watched events in any Olympic Games. Sprinters will often go to great lengths to...

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Question 2 [21 marks] The 100 metre Sprint is one of the most watched events in any Olympic Games. Sprinters
will often go to great lengths to improve their times, looking into factors such as the
equipment they use and the coaches they hire. Listed in the table below are the personal
best times for the 100 metre sprint for 27 sprinters that are in training. The 27 sprinters
were chosen as follows: For each of three shoe brands (the Fast, the Quick and the
Speedy brand), 9 sprinters who used that particular brand were randomly selected. The
sample variances of the times for each shoe brand are also listed in the table. Shoe Brand Times 32 Fast 9.82 10.22 10.15 9.77 10.12 9.86 9.87 10.11 10.18 0.0313111
Quick 9.67 9.95 10.10 10.16 9.98 10.09 10.20 10.68 10.29 0.0753278
Speedy 9.69 9.75 9.49 10.03 9.66 10.24 10.04 9.87 10.01 0.0557528 (a) [6 marks] Test whether the mean time for sprinters using the speedy brand is more
than 0.2 seconds faster than for sprinters using the quick brand. Clearly state your
hypotheses and use a signiﬁcance level of a = 5%. Clearly state any assumptions
you have made (without testing them) when performing this test. A one-way AN OVA was performed on this data, using shoe brand as the factor. The
partially ﬁlled ANOVA table is provided below: (b) [2 marks] Calculate the sum of squares for error for the one-way ANOVA.

Source Sum of squares Degrees of freedom Mean squares F Shoe Brand 0.3059 '? ‘? ?
Error ? ? ?
Total ? ? (c) [3 marks] Test whether there is a difference in the mean times between the three
shoe brands. Clearly state your hypotheses and use a signiﬁcance level of a = 5%. For each shoe brand, suppose the ﬁrst 3 sprinters in the table were trained by coach
Andy, the next 3 sprinters were trained by coach Bobby and the last 3 sprinters were
trained by coach Carl. A two-way ANOVA was performed on the same data, using shoe brand and coach as factors. The partially ﬁlled ANOVA table is displayed below: Source Sum of squares Degrees of freedom Mean squares F Coach ? ? ‘? ?
Shoe Brand ? ? ? ?
Interaction 0.2970 ? ? ? Error 0.6781 '7 7 Total ? ? (d) [3 marks] Test whether there is an interaction between coach and shoe brand.
Clearly state your hypotheses and use a signiﬁcance level of a = 5%. (e) [3 marks] Test whether there is a difference in the mean times between the three
shoe brands. Clearly state your hypothases and use a signiﬁcance level of a = 1%. (f) [4 marks] Test whether there is a difference in the mean times between the three
coaches Clearly state your hypotheses and use a signiﬁcance level of a = 1%.

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