Question

10.4.1 .WP The manager of a fleet of automobiles is testing two brands of radial tires and 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 (in kilometers) follow. Find a 99% confidence interval on the difference in mean life. Which brand would you prefer based on this calculation? Car Brand 1 Brand 2 1 36,925 34,318 2 45,300 42,280 3 36,240 35,500 4 32,100 31,950 5 37,210 38,015 6 48,360 47,800 7 38,200 37,810 8 33,500 33,215

Answer 1

Answer:

[-0.65;2.54]km

Step-by-step explanation:

Hello!

So, you need to find a 99%CI for the difference in mean life of two brands of radial tires. Since he assigned one tire of each brand at random to the two rear wheels of each car, in other words, every car tested had one rear tire of each brand at the same time, this test can be considered to be of paired samples.

           Brand 1    Brand 2    X₁-X₂

car 1:    36,925 ;   34,318   ; 2.607

car 2:   45,300 ;   42,280  ; 3.020

car 3:   36,240 ;    35,500 ; 0.740

car 4:   32,100  ;    31,950  ; 0.150

car 5:   37,210  ;    38,015  ; -.0805

car 6:   48,360 ;    47,800 ; 1.160

car 7:   38,200 ;    37,810  ; 0.390

car 8:   33,500 ;    33,215  ; 0.285

n= 8

With this in mind, we define the study variable as Xd= X₁-X₂

Where X₁ corresponds to the lifespan, in km, of a tire from Brand 1

and X₂ corresponds to the lifespan, in km, of a tire from Brand 2

so Xd will be the difference between the lifespan of the tires from Brand 1 and Brand 2.

This variable Xd~N(μd;δd²) (p-value for normality test is 0.4640)

To calculate the CI the best statistic is the Student's t with the following formula:

t= (xd[bar] - μd)/(Sd/√n) ~t₍ₙ₋₁₎

sample mean: xd[bar]= 0.94

standard deviation: Sd= 1.29

$t_{8;0.995}$ = 3.355

xd[bar] ± $t_{8;0.995}$*(Sd/√n)

⇒ 0.94 ± 3.355*(1.29/√8)

[-0.65;2.54]km

The Confidence Interval can be compared to a pair of bilateral hypothesis. If we were to determine the following hypothesis

H₀:μd=0

H₁:μd≠0

Using the level of significance of 0.01 (complementary to the confidence level)

As the calculated confidence interval contains zero, we do not reject the null hypothesis, that is, there is no significant difference between the two tire brands.

I hope you have a SUPER day!