Math 137

Hypothesis Testing:The t Test

Fresno StateDr. Burger

• In all of the previous examples, we assumed that we knew the population standard deviation

• In practice, this is an extremely rare situation!

• Instead, the sample standard deviation S is used in lieu of

• This approximation is called the bootstrap estimate.

Example. The manufacturer of a new fiberglass tire claims that its average life will be at least 40,000 miles. To verify this claim, a sample of 12 tires is tested, and the lifetimes were found to be

36,100 40,200 33,800 38,50042,000 35,800 37,000 41,00036,800 37,200 33,000 36,000

Test the manufacture’s claim using = 0.05.

36100 Mean 37283.3340200 SD 2731.9133800385004200035800370004100036800372003300036000

Solution.

H0: The average life is at least 40,000 miles.

Ha: The average life is less than 40,000 miles.

We choose = 0.05.

Notice that we have to compute the sample SD this time; it’s not given.

36100 Mean 37283.3340200 SD 2731.9133800385004200035800370004100036800372003300036000

We now can compute the test statistic

But there’s a catch: since we used S instead of , the test statistic t does NOT follow the normal curve.

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nS

Xt

Since we used S instead of , the test statistic t does NOT follow the normal curve.

For the current problem, instead of using the normal curve to compute the observed significance level, we will use the Student t distribution with 11 degrees of freedom.

For the sake of completeness, here’s the pdf of the Student’s t-distribution with r degrees of freedom:

If you find this intimidating, don’t worry: we will never use it.

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Student T Distribution with 1 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 2 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 3 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 4 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 5 degrees of freedom

Red: t distribution Blue: standard normal curve

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Student T Distribution with 10 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 20 degrees of freedom

Red: t distribution Blue: standard normal curve

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Student T Distribution with 30 degrees of freedom

Red: t distribution Blue: standard normal curve

-4 -2 2 4

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Student T Distribution with 40 degrees of freedom

Red: t distribution Blue: standard normal curve

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Student T Distribution with 50 degrees of freedom

Red: t distribution Blue: standard normal curve

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Student T Distribution with 100 degrees of freedom

Red: t distribution Blue: standard normal curve

For the t distribution with 11 df, we can compute the critical value for = 0.05 using Table 4 (p. 510).

In Excel, be careful: the command is TINV(0.1,11).(The default is for two tails, not one tail.)

-3.4448 -1.79588

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In Excel, the command is TDIST(3.448,11,1).(The third entry specifies the number of tails.)

0.002724

Observed Significance Level

Conclusion: We reject the null hypothesis. There is good reason to believe that the average lifespan of thetires is less than 40,000 miles.

Note: It is possible to compute power with theStudent’s t-distribution, but the computations aremuch, much more complicated than the normal case(Larsen & Marx, 3rd ed., p. 447). Many statistical software packages are able to compute power for the t-test automatically.

Remember: If the sample is small (n < 30) and thepopulation variance is unknown, then we use thet-test and not the z-test.

On the other hand, if either is known or the sampleis sufficiently large (n > 30), then we may safely usethe z-test instead.

Also, we must be careful about stating the null and alternative hypotheses so that we correctly choose whether to use a left-tail, a right-tail, or both tails.

Example. Before a substance can be deemed safe for landfilling, its chemical properties must be assessed.In a sample of six replicates of sludge from a NewHampshire wastewater treatment plant, the mean pHwas 6.68 with a standard deviation of 0.20. Can weconclude than the mean pH is less than 7.0?

J. Benoit, T. Eighmy and B. Crannell, Journal of Geotechnical andGeoenvironmental Engineering 1999, pp. 877--888.

Example. Certain rectangles appear more pleasing to the eye than others. The ancient Greeks called a rectangle with

the golden rectangle, and this ratio was called the golden ratio. The golden ratio has been claimed to be a deliberate design of various artand architecture.

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The data below shows the width-to-length ratios of beaded rectangles used by the Shoshone Native Americans to decorate their leather goods. Does it appear that the golden rectangle is also an aesthetic standard for the Shoshones?

0.693 0.662 0.690 0.606 0.5700.749 0.672 0.628 0.609 0.8440.654 0.615 0.668 0.601 0.5760.670 0.606 0.611 0.553 0.933

C. Dubois, ed., Lowie’s Selected Papers in Anthropology(UC Press, Berkeley, 1960), pp. 137--142