Question

A tire company hopes to adopt a new tread pattern for its tires if the pattern allows a car traveling at 60 miles per hour to come to a complete stop within an average of 125 feet after the brakes are applied. Let denote the relevant population mean distance. The distances (in feet) for 5 stops on a test track are: 129; 128; 123; 128; 135: Should the company adopt the new tread pattern? Using a confidence interval as evidence, what would you recommend to the company on whether the tire performs as they hope? Explain how you dealt with the outlier and why you made the recommendation you did.

Answer 1

If the confidence interval contains the value of 125 yes the company should adopt the new read pattern

The 95% confidence interval would be given by (123.279;133.921)

For both cases with and without the outlier the confidence interval contains the value of 125, but we notice that the values obtained are a little above the specification of 125, so the company should check the system in order to be sure that works properly.

Explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the mean and the sample deviation we can use the following formulas:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

`s=\sqrt{(\sum_(i=1)^n (x_i-\bar X))/(n-1)}` (3)

The mean calculated for this case is
`\bar X=128.6`

The sample deviation calculated
`s=4.28`

Should the company adopt the new tread pattern?

First we cacluclate the margin of error given by:

`Me=t_(\alpha/2)(s)/(√(n))`

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=5-1=4`

We can use a 95% confidence interval. Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,4)".And we see that
`t_(\alpha/2)=2.78`

`Me=2.78 (4.28)/(√(5))=5.32`

Should the company adopt the new tread pattern?

If the confidence interval contains the value of 125 yes the company should adopt the new read pattern

Using a confidence interval as evidence, what would you recommend to the company on whether the tire performs as they hope?

Now we have everything in order to replace into formula (1):

`128.6-2.78(4.28)/(√(5))=123.279`

`128.6+2.78(4.28)/(√(5))=133.921`

So on this case the 95% confidence interval would be given by (123.279;133.921)

Explain how you dealt with the outlier and why you made the recommendation you did.

Since we can consider as an outlier the value of 135, because it's the highest value, we can discard this value and calculate a new interval with 4 observations.

`df=n-1=4-1=3`

We can use a 95% confidence interval. Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,3)".And we see that
`t_(\alpha/2)=3.18`

The average and the sample deviation changes since we have just 4 values : 129; 128; 123; 128;

`127-3.18(2.71)/(√(4))=122.694`

`127+3.18(2.71)/(√(4))=131.306`

For both cases with and without the outlier the confidence interval contains the value of 125, but we notice that the values obtained are a little above the specification of 125, the company should check the system in order to be sure that works properly.