B1. [Total for Question B1 = 15 marks]
A school tests the effectiveness of a new type of athletics training
by comparing the average times of 10 students running the 100 meters. Shown below is the 100m time in seconds before and after training for each athlete.
The summary statistics are as follows:
(a) Perform a paired t-test to determine if the new training methods resulted in a significant improvement in the running times.
(i) State the null and alternative hypothesis in words and/or symbols.
Standard deviation of differences
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(ii) Calculate the test statistic. Show working.
(iii) Report the degrees of freedom for the test and the critical value for a significance level of
(iv) Is there evidence to reject the null hypothesis? Explain.
Degrees of freedom: Critical value:
(v) State your conclusion in context.
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(b) After an investigation, it was discovered that instead of recording two times (before and after training) for each of the ten students, the two sets of times were recorded on two different groups of ten students. Given this new information, a two-sample t-test was performed. The results are shown below.
t-Test: Two-Sample Assuming Equal Variances
Mean 13.17 13.68
Variance 3.27 2.91
Observations 10.00 10.00
Pooled Variance 3.09
Hypothesized Mean Difference 0.00
t Stat -0.65
P(T<=t) one-tail 0.26
t Critical one-tail 1.73
P(T<=t) two-tail 0.52
t Critical two-tail 2.10
(i) Interpret this output and state your conclusions (with reasons). Compare your answer to your answer from part (a). If you have a different conclusion, explain why this occurs.
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