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9.4 t test and u test

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Hypothesis testing for population mean Example : Hemoglobin of 280 healthy male adults in a region: Question: Whether the population mean in this region is 140.0 g/L?

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Two possibilities: (1) The population is 140, the sample mean =136.0 is due to the sampling error (Null hypothesis) (2) The population is not 140 at all so that the sample mean =136.0 (Alternative hypothesis) Question: Which is the truth? -- problem of hypothesis test!

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Basic logic: Basic logic: Under the null hypothesis How possible to occur the current situation and even more unfavorable situation to ? -- Calculate a probability ( -value) If it is less possible to occur the current situation and even more unfavorable situation to, then reject ; otherwise, not reject. -- Given a small, compare and ( is called the level of the test)

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Example 9-15: The content (mg/L) of within a material was independently measured 15 times, resulting in: 20.99, 20.41, 20.62, 20.75, 20.10, 20.00, 20.80, 20.91, 22.60, 22.30, 20.99, 20.41, 20.50, 23.00, 22.60. Please check whether the true value was 20.7mg/L. 1. Comparing to a given population mean CaCo 3

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(1)Set hypotheses and the level of test To make decision: reject or accept ? If reject, the probability of miss reject should not be greater than.

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(2) Select an appropriate test and calculate the test statistics If X follows a normal distribution Then

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When holds, Based on the current sample:

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(3) Determine P value, and make decision

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When holds, the probability of the current situation (sample mean=21.13) and even more unfavorable situation (sample mean>21.13) to is greater than 0.05. The probability of the current situation and even more unfavorable situation to is called P value. Now P >, no reason to reject.

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2. Comparison for Paired Data Example 9-13 (A paired design) 8 patients with hypertension were treated with a medicine and the DBP was measured before and after the treatment. Data list in the table 9-10.

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α=0.05 υ =8-1=7 t > t 0.05,7 =2.365, P < 0.05, is rejected at significance level α=0.05.

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3. Comparison between Two Sample Means Two Sample Means

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Example9-18: Two group of rats were fed by different food. One contains high protein, another contains low protein. Comparing the effects of different food on increasing weight.

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α =0.05 The pooled estimation of sample variance is

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υ=n 1 +n 2 -2=12+7-2=17

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P>0.05, the null hypothesis is not rejected at the significance level α=0.05.

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a. What does P-value mean? P-value is the area of the tail(s) in the distribution of the test statistic beyond the value(s) of the test statistic calculated based on the sample. If the null hypothesis is rejected, the probability of mistake = P -- A smaller P-value implies the better quality of your rejection. If the null hypothesis is not rejected, the bigger P-value implies the better quality of your acceptation. 4. Attention for Hypothesis Test

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b. What does the significance level α mean? αshows the quality of the inference. If you reject the null hypothesis, the probability of making mistake is limited by α

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c. What is the situation that t-test could be applied? The variable follows a normal distribution; Sample size is small; The variances are equal.

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Thank You

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter.

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