Statistical hypothesis testing
Suggest and explain possible limitations of performing appraise tests to determine whether there is a difference between population average completion times of laps in a race - Statistical hypothesis testing introduction. 2. Instead of performing a large number of paired t-tests to compare the five lap times in a mm race, it is desirable to be able to compare all the five lap times simultaneously in one analysis to determine whether there are significant differences in the lap times. The t-statistic for a paired t-test has the fool lowing structure: If there are more than two sample means (more than two laps) then it is no anger possible to compute a sample mean difference.
Instead, we can look at the variance of the sample means (how spread out the sample means are) to determine whether there are significant differences using a new measure, the F-statistic: A simple random sample of 20 athletes running mm was taken and the excel file below contains details on their completion times for each of the five mm laps. A. Match the independent (cause) variable and the dependent (effect) variable below: Variable Nature Variable Definition Independent Lap completion time Dependent Number of the lap within the mm race .
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The variance of all the laps times in the dataset can be computed using the formula: Where degrees of freedom is given by: using these formulae, determine the variance of the set of all lap times in the dataset [Hint: You may wish to make use of the excel function SUMS(… )]. The sum squares ( computed in part (b) can be split up into the following parts to account for different sources of variation in the lap times: Where: Notation Meaning Variability due to different lap number (1, 2, 3, 4 or 5) Variability within times for the same lap Variability due to different athlete
Variability due to chance Degrees of freedom can be similarly split up: Number of athletes Number of laps in the race. C. Use the above to complete the following table: Source of variation Sum Squares Degrees of freedom () Mean Square Total 738. 16 99 7. 456 Between treatments 58. 68 14. 67 Within treatments (Total) 679. 47 95 7. 152 Within treatments (Between Subjects) 591 . 23 31. 117 Within treatments (Error) 88. 24 76 1. 161 d. The F-statistic is given by: Determine the F-statistic for the data. 12. 64 e.
We can conduct a hypothesis test using the F-statistic with: HO: Population average times for each of the five laps are the same HI: The population average time is different for at least one of the laps. Suppose that the null hypothesis, HO, is true. I. Explain whether you would expect the sample mean times for each lap to be similar or different from each other. Ii. Hence explain whether you would expect the F-statistic to be larger or smaller. F. Fifth null hypothesis is true, the F-statistic will follow an F-distribution with degrees Of freedom on the numerator and degrees of freedom on the denominator.
This distribution is illustrated below: I. Using your answer to part e(ii), explain whether the critical region is Region 1 or Region 2. Ii. Use the following table to determine the critical value of the F-statistic at the 5% significance level. Iii. Making reference to the F-statistic that you computed in part (d) and the critical region you have Identified, explain whether the null hypothesis should be rejected.