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Understanding Single-Sample z and t Tests: Testing the Significance of a Single Mean

This seminar chapter explores the significance testing of a single mean using z and t-tests. It focuses on Case I research, highlighting the calculations of z and t statistics, standard deviations, and sampling errors. Key concepts covered include acceptance and rejection regions, critical values, and Type I and Type II errors. An illustrative example involving lunch consumption times sets the stage for hypothesis testing. The findings emphasize the importance of statistical analysis in understanding how individual cases compare to population means.

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Understanding Single-Sample z and t Tests: Testing the Significance of a Single Mean

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  1. Statistic Seminar,Chapter.7TESTING THE SIGNIFICANCE OF A SINGLE MEAN-The Single-Sample z and t tests- Jun.13 2008 Yoshioka

  2. Chapter 7 • In this chapter, we deal with case I research. (cf. pp.126) Case I Case II

  3. z score = the mean of the sample = the hypothesized mean = the standard deviation of the population

  4. z score (reminder) e.g. For alpha levels .05 68% Normal distribution 95% z -3.0 -2.0 -1.0 0 +1.0 +2.0 +3.0

  5. z statistics = the mean of the sample = the hypothesized mean = the standard error of the mean

  6. The sampling distribution Due to sampling error?? Hypothesized μ True μ 100 104 105 Sample

  7. The acceptance and rejection regions e.g. For alpha levels .05 (Critical region) 95% 2.5% 2.5% z -1.96 0 +1.96 Reject H0 Accept H0 Reject H0

  8. t statistics (Formula 7.2 pp.154) The mean of the sample The hypothesized mean The sample standard deviation Sample size

  9. Type I and Type II Errors True state of affairs H0 is true H0 is false Type II Error Correct Retain H0 Your Decision Type I Error Correct Reject H0

  10. Degree of Freedom • e.g. You can 4 numbers whatever you like. note the mean oh them should be 10. a, b, c, 10-(a+b+c)

  11. Standard Deviation Sample SD Population SD

  12. Example • Each Okadaken member takes 20 minutes to eat lunch on average. According 2weeks of investigation, Ikeuchi takes 22 minutes to eat lunch on average. Standard deviation of him was 2 minutes. • Compared to other members, is Ikeuchi a slow eater or not? note: this is an imaginary story.

  13. Solution • Identify the null and alternative hypotheses H0 : μ＝ 20 H1 : μ≠ 20 • Set alpha as .05

  14. Solution • Compute tobt using formula

  15. Solution • Find the critical value for and • The critical value is 2.262. • Compare the tobt of 3.16 to the critical value of 2.262. tobt is larger than the critical value, so H0 is rejected. • So Ikeuchi is ….

  16. Thank you for your attention!

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