Question

Durability of shock absorbers. A manufacturer of automobile shock absorbers was interested in comparing the durability of its shocks with that of the shocks produced by HULL SHOCK, its biggest competitor. To make the comparison, one of the manufacturer’s and one of the competitor’s shocks were randomly selected and installed on the rear wheels of each six cars. After the cars had been driven 20,000 miles, the strength of each test shock was measured, coded, and recorded. Results of the examination are shown in the table.

Card number	Manufacture’s Shock	Competitor’s Shock
1	8.8	8.4
2	10.5	10.1
3	12.5	12.0
4	9.7	9.3
5	9.6	9.0
6	13.2	13.0

a. Explain why the data are collected as matched pairs.

Short answer

a. The Manufacturer’s shock and the Competitor’s shocks are dependent samples. So, these two samples are collected as matched pairs.

Step-by-step solution

Given information

The number of cars is 6.

The dataset for the Manufacture’s and Consumer’s shock is given as follows:

Card number	Manufacture’s Shock	Competitor’s Shock
1	8.8	8.4
2	10.5	10.1
3	12.5	12.0
4	9.7	9.3
5	9.6	9.0
6	13.2	13.0

State about small sample paired difference.

The Student’s t-test statistic for the small sample paired difference is,

\(t = \frac{{\bar d - {D_0}}}{{{s_d}/\sqrt {{n_d}} }}\)

Where \(\bar d\) is the sample mean difference, \({s_d}\) is the sample standard deviation difference, and \({n_d}\) is the number of pairs.

The confidence interval for the small sample paired difference for \({\mu _d}\)is:

\(C.I = \left( {\bar d \pm {t_{\frac{\alpha }{2}}}\frac{{{s_d}}}{{\sqrt {{n_d}} }}} \right)\)

Where \({t_{\frac{\alpha }{2}}}\)is based on \(\left( {{n_d} - 1} \right)\) degrees of freedom. \(\)

Explain why the data are collected as matched pairs.

a.

The given two samples are:

• Manufacturer’s shock
• Competitor’s shock.

Here, it is the study of the durability of its shocks.

The Manufacturer’s shock and the Competitor’s shocks are dependent samples. So, these two samples are collected as matched pairs.