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Understanding Hypothesis Testing: t-Test and U-Test for Population Mean Evaluation

This guide explores hypothesis testing for population means using both t-tests and U-tests. We examine an example involving hemoglobin levels in 280 healthy male adults to determine if the population mean is 140.0 g/L. We define the null and alternative hypotheses and discuss the logic of testing—the significance of p-values—and provide examples of t-test applications with sample data. This comprehensive overview aims to clarify the hypothesis testing process and the significance level, helping researchers make informed decisions based on statistical analysis.

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Understanding Hypothesis Testing: t-Test and U-Test for Population Mean Evaluation

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Presentation Transcript

  1. 9.4 t test and u test

  2. 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?

  3. 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!

  4. 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)

  5. 1. Comparing to a given population mean 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 . CaCo3

  6. 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 .

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

  8. When holds, • Based on the current sample:

  9. (3) Determine P value, and make decision

  10. 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 .

  11. 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.

  12. α=0.05 • υ=8-1=7 • t > t0.05,7=2.365, P < 0.05, is rejected at significance level α=0.05.

  13. 3. Comparison between Two Sample Means

  14. 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.

  15. α=0.05 • The pooled estimation of sample variance is

  16. υ=n1+n2-2=12+7-2=17

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

  18. 4. Attention for Hypothesis Test 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.

  19. 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

  20. 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.

  21. Thank You

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