Z-Test with Examples
- 1. Z-TEST BY GROUP 04 B.Sc. (Hons.) Agriculture 2nd Semester Assignment presented as the partial fulfillment of the requirement of Course STAT-102 College of Agriculture BZU, Bahadur Sub-Campus Layyah
- 2. DEFINATION Z test is a statistical procedure used to test an alternative hypothesis against a null hypothesis. Z-test is any statistical hypothesis used to determine whether two samples’ means are different when variances are known and sample is large (n ≥ 30). It is Comparison of the means of two independent groups of samples, taken from one populations with known variance.
- 3. The null hypothesis (H0) is a hypothesis which the researcher tries to disprove, reject or nullify. The 'null' often refers to the common view of something, while the alternative hypothesis is what the researcher really thinks is the cause of a phenomenon.
- 4. Z-TEST Formula to find the value of Z (z-test) Is: x̄ = mean of sample μ0 = mean of population σ = standard deviation of population n = no. of observations
- 5. WHY WE USE LARGE SAMPLE…? When we perform a statistical test we are trying to judge the validity of the null hypothesis. We are doing so with an incomplete view of the population. Our sample is our window into the population. The larger the sample size the bigger our window. However without a full view of the population there is always the chance that our sample will lead us to the wrong conclusion.
- 6. When do we use Z-Test When samples are drawn at random. When the samples are taken from population are independent. When standard deviation is known. When no. of observation is large (n ≥ 30)
- 7. EXAMPL E
- 8. EXAMPLE A principal at a school claims that the students in his school are above average intelligence. A random sample of thirty students’ IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.•
- 9. EXAMPLE • Step 1: State the Null hypothesis. The accepted fact is that the population mean is 100, so: H0: μ=100. • Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H1: μ > 100.
- 10. EXAMPLE • Step 3: Draw a picture to help you visualize the problem.
- 11. EXAMPLE • Step 4: State the alpha level. If you aren’t given an alpha level, use 0.05, An alpha level of 0.05 is equal to a z-score of 1.645. To calculate z-score use TI-83 calculator.
- 12. EXAMPLE • Step 5: Find the Z using this formula: For this set of data: Z= (112.5-100) / (15/√30) = 4.56 • Step 6: If Step 5 (4.56) is greater than Step 4 (1.645), reject the null hypothesis. If it’s less than Step 4, you cannot reject the null hypothesis. In this case, it is greater, so you can reject the null and principal’s claim is right.
- 13. Z-TEST FOR TWO SAMPLES • Requirements: Two normally distributed but independent populations, σ is known • Formula: • where x1 and x2 and are the means of the two samples, μ 1 – μ 2 is the hypothesized difference between the population means , σ 1 and σ 2 are the standard deviations of the two populations, and n 1and n 2are the sizes of the two samples.
- 14. EXAMPLE • The amount of a certain trace element in blood is known to vary with a standard deviation of 14.1 ppm (parts per million) for male blood donors and 9.5 ppm for female donors. • Random samples of 75 male and 50 female donors yield concentration means of 28 and 33 ppm, respectively. What is the likelihood that the population means of concentrations of the element are the same for men and women?
- 15. EXAMPLE • Null hypothesis: H 0: μ 1 = μ 2 or H 0: μ 1 – μ 2= 0 • alternative hypothesis: H a : μ 1 ≠ μ 2 or: H a : μ 1 – μ 2≠ 0
- 16. EXAMPLE • The computed z‐value is negative because the (larger) mean for females was subtracted from the (smaller) mean for males. • But the order of the samples in this computation is arbitrary— it could be in opposite order, in which case z would be 2.37 instead of –2.37. An extreme z‐score from alpha level (which is 5% = 1.654 (plus or minus) will lead to rejection of the null hypothesis.
- 17. NO RELEGION IS HIGHER THAN HUMANITY…! (ABDUL SATTAR EDHI)
Editor's Notes
- #12: Probability of rejecting a null hypothesis; R. A. Fisher. Z test is a statistical procedure used to test an alternative hypothesis against a null one.