Lesson #20b
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Lesson #20b. ~ Small Sample Test for a Mean ~. Assumptions Use of t-distribution as reference distribution Relationship to confidence interval. Main Ideas. 1. 1. Assumptions A random sample is taken from a population with mean m and standard deviation s .

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Lesson #20b

~ Small Sample Test for a Mean ~

  • Assumptions

  • Use of t-distribution as reference distribution

  • Relationship to confidence interval

Main Ideas

1


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1. Assumptions

  • A random sample is taken from a population with mean m and standard deviation s.

  • The population has a normal distribution.

  • The sample size may be small or large, but the technique was developed especially for small sample sizes.

  • We are interested in testing the same types of hypotheses about m we did in the large sample case.

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2. Test Statistic for a Mean

  • The test statistic has the form

  • This behaves as a t random variable with n - 1 degrees of freedom if H0 is true.

  • Values of t “too large” or “two small” indicate that the alternative hypothesis is true.

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3. Critical region for two-sided test, level of significance .05, sample size n = 10, d.f. = 9

If t falls here, reject H0

If t falls here, reject H0

- 2.262

2.262

If t falls here, accept H0

H0: m = m0, Ha: m not equal m0

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Example #1. The average length of a certain jawbone measurement of a moose is 11.5 cm. Jawbone measurements of 10 samples from an unknown species are examined to see if they are consistent with that of the moose. The data are:

14.0, 15.1, 13.6, 12.9, 11.0, 12.5, 14.9, 9.6, 15.0, 14.2

mean = 13.28, s.d. = 1.82, n = 10, m0 = 11.5

test statistic: t = (13.28 - 11.5)/[1.82/sqrt(10)]

= 3.09

Since 3.09 > 2.262, we conclude that the mean length of the jawbones of the unknown species are not consistent with that of the moose.

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4. Relationship to Confidence Interval measurement of a moose is 11.5 cm. Jawbone measurements of 10 samples from an unknown species are examined to see if they are consistent with that of the moose. The data are:

  • It is possible to do a two-sided test of hypothesis using the confidence interval approach.

  • Make a confidence interval for the population mean. If the interval “captures” the hypothesized mean m0, then accept H0. If not, reject H0.

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Example #2. Refer to the moose example. The sample mean is 13.28 and the sample standard deviation is 1.82. The 95% confidence interval for the mean is

13.28 2.262(1.82)/sqrt(10) or

11.98 to 14.58

Since the interval does not capture the hypothesized mean of 11.5, we conclude that the jawbones came from a population whose mean is different from that of the moose.

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Is it better to do a hypothesis test or make a confidence interval?

Most statisticians would agree that the confidence interval is more informative. However, research literature relies heavily on tests of hypotheses to summarize findings. Know both.

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