Question

The z-tests

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The z-test 10.1 Assume that a treatment does have an effect and that the treatment effect is being evaluated with a z hypothesis test. If all factors are held constant, how is the outcome of the hypothesis test influenced by sample size? To answer this question, do the following two tests and compare the results. For both tests, a sample is selected from a normal population distribution with a mean of u 30 and a standard deviation of σ 8, After the treatment is administered to the individuals in the sample, the sample mean if found to be M- 33. In each case, use a two-tailed test with a 05 a. For the first test, assume that the sample consists of n = 16 individuals b. For the second test, assume that the sample consists of n 64 individuals c. Explain in your own words how the outcome of the hypothesis test is influenced by the sample size. Note: Be sure and show a fiull diagram of the research design. Also show all steps and calculations you made for each test following the process outlined in the z-test formula sheet handout. What statistical decision do you make in each case? Finally report your results professionally in APA format (see last step of the formula sheet for this module) Researchers at a National Weather Center in the northeastern United States recorded the number of 90 degree days each year since records first started in 1 numbers form a normal shaped distribution with a mean of u -9.6 and a standa devia also computed the mean number of 90 degree days for the most obtained M-11.85 days. Do the data indicate that the past four years have had ignificantly more 90 degree days than would be expected for a random sample from this populaton? Use a one-tailed test with alpha 05 10.2 875 19. To see if the data showed any evidence of global warming, they 4 years and tion ofơ recent n Note: Be sure and show a diagram of the research design. Also show all steps and calculations you made for each test following the process outlined in the s-test sheet handout What statistical decision do you make in each case? Finally report your results professionally in AP.A format (see last step of the formula sheet for this module) 1. Assumptions for z-test A one sample, randomly selected B. know population mean and population standard deviation ahead of time C. standard deviation is unchanged by treatment or experiment D. sample means are normally distributed; take all the possible sample means that could happen by chance without treatment (usually normally distributed for behavioral sciences if sample is greater than or equal to 30) Diagramming your research (show the whole logic and process of hypothesis testing) a. ll. Draw a picture of your research design (see diagramming your research handout). b. There are always two explanations (i.e. hypotheses) of your research results, the wording of which depends on whether the research question is directional (one- tailed) or non-directional (two-tailed). State them as logical opposites For statistical testing, ignore the alternative hypothesis and focus on the null hypothesis, since the null hypothesis claims that the research results happened by chance through sampling error Assuming that the null is true (i.e. that the research results occurred by chance through sampling error) allows one to do a probability calculation (i.e. all statistical tests are nothing more than calculating the probability of getting your research results by chance through sampling error). Observe that there are two outcomes which may occur from the results of the probability calculation (high or low probability of getting your research results by chance, depending on the alpha (a) level). Each outcome will lead to a decision about the null hypothesis, whether the null is probably true (i.e. we then accept the null to be true) or probably not true (i.e we then reject the null as false) c. I Hypotheses (i.e. the two explanations of your research results) A. Two-tailed (non-directional research question) 1. Alternative hypothesis (H,): The independent variable (i.e. the treatment) 2 Null hypothesis (Ho): The independent variable (i.e. the treatment) does One-tailed (directional research question) does make a difference in performance not make a difference in performance B. 1. Alternative hypothesis (Hi): The treatment has an increased (right tail) or a decreased (left tail) effect on performance Null hypothesis (Ho): The treatment has an opposite effect than expected or no change in performance 2. non-tx Step 3: Using Part Ill of formula sheet, determine the critical value(s) Single Sample z test Diagram Sample Population σ ? Step 1: Diagram your study noting the data you are given) Two tailed test both tails One tailed test left or right tail 2 explanations of your data lusing directional or non-directional language depending on the research question) tx H: IV had an effect and half α or fulla n ? M-? s ? 2 outcomes 2 decisions single sample High Prob z test Ho: Ⅳ had no effect (of results are (easy)Accept Ho (due due to chonce primarily througcalcul to chance) sompling error) Low Prob.Reject H,& (hard) Accept H, (due to real effect) Step 2: Create a table of calculations for each of the terms which you will need to solve the formulas (not needed in this problem) Step 4: Calculate the z statistic population ybefore tx sample after Step 5: Compare critical z to the calculated z & make a decision about the Ho hypothesis tx Vn The independent variable is being manipulated with two treatments (hint: with an abstract problem like this, sometimes it helps to put in a real treatment, such as IV Pain Relievers with tx Ibuprofen and tx non-treatment) (doto for sample) 2, z= Step 6: Report results professionally (seeloast step of formula sheet) (x)%

Answer 1

10.1)

a)

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ha: 30 This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used (2) Rejection Region Based on the information provided, the significance level is 0.05, and the critical value for a two- tailed test isc1.96. The rejection region for this two-tailed test is R-z >1.96 (3) Test Statistics The z-statistic is computed as follows: X-o 33 -30 1.5 (4) Decision about the null hypothesis Since it is observed tha1.51.96, it is then concluded that the null hypothesis is not rejected Using the P-value approach: The p-value is p 0.1336, and since p 0.13362 0.05, it is concluded that the null hypothesis is not rejected. (5) Conclusion tis concluded that the nulil ypothesis Ho is not rejected Theretore, there is not enough evidence to claim that the population mean μ is different than 30, at the 0.05 significance level. Confidence Interval The 95% confidence interval is 29.08 < < 36.92.

Graphically Z-Test Results: t-stats 1.5, p-value -0.1336 = 1.5, p-va 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 4.0-3.5 -3.0 -2.5 -2.0 -1.5-1.0 -0.5 0.00.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

b)

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: 30 This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. Based on the information provided, the significance level is α-0.05, and the critical value for a two- tailed test is ze1.96 The rejection region for this two-tailed test is R-z: >1.96 The z-statistic is computed as follows: X-o 33 -30 2 (4) Decision about the null hypothesis Since it is observed thatz3ze1.96, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p 0.0027, and since p- 0.00270.05, it is concluded that the null hypothesis is rejected. (») <rciessicrt It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean u is different than 30, at the 0.05 significance level. er The 95% confidence interval is 31.04 < μ < 34.96

Graphically Z-Test Results: t-stats 3, p-value-0.0027 0.40 0.35- 0.30 0.25- 0.20 0.15 0.10 0.05 0040 35 30 35 30 3s 10 a5 0 as 10 15 20 35 30 55 4.0 -3.5 -3.0 2.5 -2.0 -1.5 1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

c)

as sample size increase, |TS| increases , p-value decrease, it becomes easier to reject the null hypothesis

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Answer 2

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