Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other a different brand. The brand of shoe they wear in which race is determined at random. All runners are timed and are a
Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other a different brand. The brand of shoe they wear in which race is determined at random. All runners are timed and are a
Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other a different brand. The brand of shoe they wear in which race is determined at random. All runners are timed and are a
Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other a different brand. The brand of shoe they wear in which race is determined at random. All runners are timed and are asked to run their best in each race. The results (in minutes) are given below.
Runner
Brand 1
Brand 2
1
31.23
32.02
2
29.33
28.98
3
30.50
30.63
4
32.20
32.67
5
33.08
32.95
6
31.52
31.53
7
30.68
30.83
8
31.05
31.10
9
33.00
33.12
10
29.67
29.50
11
30.55
30.57
12
32.12
32.20
Note that in this case the data are paired (the same subject is used to generate a pair of observations.) The data are analyzed by focusing on the sample of paired differences (paired diff. = obs. before ‘minus’ obs. after). This essentially reduces the problem to a one-sample t-procedure under the right assumptions.
Make a scatterplot of the running times of brand 1 vs. brand 2. Was it a good idea to pair data? Explain.
Use a statistical test to test the hypothesis that that athletes tend to run faster in a 10-kilometer race using brand 1 than using brand 2.
Construct a 95% confidence interval for the difference between the mean time to run a 10-kilometer race using brand 1 and the mean time to run a 10-kilometer race using brand 2.
Assess the validity of the t procedure you used to analyze the data.
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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