Question

A treadmill manufacturer has developed a new machine with softer tread and better fans than its current model. The manufacturer believes these new features will enable runners to run for longer times than they can on its current machines. To determine whether the desired result is achieved, the manufacturer randomly sampled 10 runners. Each runner was measured for one week on the current treadmill and for one week on the new treadmill. The weekly total number of minutes for each runner on the two types of treadmills was collected, and is provided in the table below.

Runner	New treadmill (minutes run)	Current treadmill (minutes run)
1	269	270
2	280	268
3	260	254
4	271	256
5	273	258
6	264	264
7	263	262
8	260	251
9	263	264
10	274	258

Construct a 98% confidence interval estimate of the mean difference in running time (in minutes) on the new and current treadmills.

ANSWER:  (Click to select)-±+  ≤ (Click to select)(μ1 - μ2)(n1 - n2)(σ1 - σ2)μd(p1 - p2)(s1 - s2)(π1 - π2)(x-bar1 - x-bar2) ≤ (Click to select)±+-  (report your answers to 2 decimal places, using conventional rounding rules.)

Answer 1

Since we are doing repeated measure on same subject, this is a paired sample study design.

The 98% confidence interval for the the mean of difference in time between new and current treadmill is obtained using the formula,

Where,

From the data values,

Runner	New treadmill (minutes run)	Current treadmill (minutes run)	Difference, D
1	269	270	-1	-8.2	67.24
2	280	268	12	4.8	23.04
3	260	254	6	-1.2	1.44
4	271	256	15	7.8	60.84
5	273	258	15	7.8	60.84
6	264	264	0	-7.2	51.84
7	263	262	1	-6.2	38.44
8	260	251	9	1.8	3.24
9	263	264	-1	-8.2	67.24
10	274	258	16	8.8	77.44
Sum					451.6
Average			7.2

The t-critical value is obtained from t distribution table for significance level = 0.02 and degree of freedom = n - 1 = 9 (In excel use function =T.INV.2T(0.02,9))

Now,