Question

The manager of a fleet of automobiles is testing two brands of radial tires. She assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data are shown here (in kilometers). Find the lower bound of a 999 Cl on the difference in mean life Car Brand 1 Brand 2 Car Brand 1 Brand 2 1 36,925 34 318 45.300 42.280 3 36.240 35.500 4 32.100 31.950 3 6 7 8 37,210 48.360 38.200 33.500 38,015 47.800 37.810 33.215 Use two decimal digits

Answer 1

Solution:

Given

Cars	Brand 1 (x)	(X - $\bar{x}$)²	Brand 2 (y)	(Y - $\bar{y}$ )²
1	36925	2416081.64	34318	10843849
2	45300	46520925.39	42280	21799561
3	36240	5014800.39	35500	4456321
4	32100	40696425.39	31950	32046921
5	37210	1611312.89	38015	163216
6	48360	97626750.39	47800	103815721
7	38200	78050.39	37810	39601
8	33500	24794175.39	33215	193249816

Let, $\mu_{1}$ be the mean of the Brand 1 and $\mu_{2}$ be the mean of the Brand 2. And $\sigma_{1}$ and $\sigma_{2}$ be the standard deviation of Brand 1 and Brand 2 respectively.

Hypothesis test:

Null hypothesis H_{​​​​​​0} : $\mu_{1}= \mu_{2}$

Alternative hypothesis H_{​​​​​1} : $\mu_{1}\neq \mu_{2}$

Test statistic:

  $Z= \frac{\mu_{1} - \mu_{2}}{\sqrt{\sigma_{1}^{2}/n1 + \sigma_{2}^{2}/n2}}$

  $\sum X_{i}$ = 307835

$\bar{x}=\sum X_{i}/n$

= 307835/8

= 38479.375

  $\sum Y_{i}$ = 300888

  $\bar{y}=\sum Y_{i}/n$

= 300888/8

= 37611

  $\sigma_{1}^{2}= \sum (X_{i} - \bar{x})^{2} /n$

= 218758521.9/8

= 27344815.23

  $\sigma_{2}^{2}= \sum (Y_{i} - \bar{y})^{2} /n$

= 192490006/8

= 24061250.75

S = $\sqrt{\sigma_{1}^{2}/n1 +\sigma_{2}^{2}/n2}$

= 2534.9079

Z = $\frac{\bar{x} - \bar{y}}{S}$

=  (38479.375-37611)/ 2534.9079

= 868.375/2534.9079

  Z = 0.3426.

Under, $\alpha$ = 0.05, Z= 0.3426 < Z(observed) = 1.645

  We failed to reject the Null hypothesis.

As there are less difference in mean of two different brands.

We can choose anyone of the brand. As Brand 1 mean > Brand 2 mean we recommend Brand 1.

  As HOMEWORKLIB RULES guidelines, i have answered only one question.

So, please give positive response.

Thank you.