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Q: A tire manufacturer wishes to compare the tread wear of tires made of a new material with that of conventional material. 10 Cars are driven 40,000 miles as the sample set. the following data is obtained:

(To avoid some confusion and messy tables I've done some of these computations myself)

$\mu_{conventional}$ =4.11
$\mu_{new}$ = 4.814
$s$ = .6699 (To clarify this is the sample standard deviation of the "new" material dataset)

The question then is to test at $\alpha$=0.05 that the true mean of the new material exceeds that of the old material.

This is a one-sided t-test. (Right?)

So I've set it up the following way:

$H_0 : \mu = 4.11$
$H_1 : \mu$ > 4.814

$ t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}} = \frac{4.814-4.11}{.6699/\sqrt{10}}$ = 3.3233

Now to look for a .05 confidence using the t-test I obtained the value 1.833 from the t-distribution table. Since the value from our test statistic is 3.3233 > 1.833, I'd reject the Null hypothesis. Can anyone check my work to verify this?

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    @fra$n$kli$n$, the reason I asked, is that if you are in a low level college course, your solution may be fine; but if you are in a High School AP course, they are far more careful with the criteria (they have more time!) and there you must indicate in your answer how you verified the normality conditions (e.g., a normal-quantile plot, or...). You should also, in AP, address if the sample is representative...2012-05-08

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As per @Ronald, my working above seems to be correct.

I believe @ChrisK.Caldwell is referring to the criteria under which a t-test can be a reliable instrument for hypothesis testing. Check this out here for some necessary assumptions for t-tests

In this case, a sample s.d. is used rather than the population deviation $\sigma$, also the sample size is small $n$ < 30. In this case a t-distribution would seem like a sound substitute for a standard normal curve.

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    Indeed I was, as that page states: The assumptions underlying a t-test are that Z [$T=Z/s$] follows a standard normal distribution under the null hypothesis... However, it is common for text books to ignore this in the exercises (even after warning about it in the text of the section). So your answer may be exactly what is required.2012-05-08