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Week 7 Homework Numbers 9,10 and 12
This document provides step-by-step instructions for completing homework problems related to hypothesis testing using z-tests. It includes instructions for finding critical values, performing left-tailed, right-tailed, and two-tailed z-tests using Minitab software. Examples are provided for problems testing claims about population means, finding test statistics, determining p-values, and interpreting results to either reject or fail to reject the null hypothesis. Guidance is given to carefully consider the wording of claims and hypotheses and set up tests accordingly.
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Similar to Week 7 Homework Numbers 9,10 and 12
Week 7 Homework Numbers 9,10 and 12
- 1. WEEK 7 HOMEWORKHELP NUMBERS 9,10 & 12 B Heard Not to be copied, posted, shared without my permission. Students can download one copy for their personal use.
- 2. WEEK 7 HOMEWORKHELP (MORE) Find the critical value(s) for a left tailed z test with α = 0.11. Include a graph… Easy In Minitab Graph >> Probability Distribution Plot >> Click View Probability, Click OK On Menu, Make sure Distribution is set to Normal, Enter 0 as the Mean and 1 as the Standard Deviation
- 3. WEEK 7 HOMEWORKHELP (MORE) Continued On Menu, Make sure Distribution is set to Normal, Enter 0 as the Mean and 1 as the Standard Deviation
- 4. WEEK 7 HOMEWORKHELP (MORE) Continued Click Shaded Area Tab, Click Probability Radial Button, Choose Left Tail, Enter 0.11 as the Probability
- 5. WEEK 7 HOMEWORKHELP (MORE) Continued Click OK button(s), See the Graph Answer is -1.227, rounded to two decimals, answer is -1.23 0.4 0.3 0.2 0.1 0.0 X Density -1.227 0.11 0 Distribution Plot Normal, Mean=0, StDev=1
- 6. WEEK 7 HOMEWORKHELP (MORE) Test the claim about the population mean µ, at the given level of significance using the given sample statistics. Claim: µ = 60, α = 0.05. Sample Statistics: x-bar = 59.1, s = 4.11, n =90 Easy The claim would be H0 for this one (because it contains equality). H0: µ = 60 Ha: µ ≠ 60 .
- 7. WEEK 7 HOMEWORKHELP (MORE) Continued Since Ha: µ ≠ 60 (or contains the not equal to sign), this is a two- tailed test. You can find the critical values in Minitab. Remember the that α = 0.05. In Minitab, Graph >> Probability Distribution Plot >> Click View Probability Normal Distribution, Mean = 0, Standard Deviation = 1 Click Shaded Area Tab .
- 8. WEEK 7 HOMEWORKHELP (MORE) Continued Click Probability button, Click Both Tails button, Enter 0.05 as your probability .
- 9. WEEK 7 HOMEWORKHELP (MORE) Continued You can see your Critical values of +/- 1.96 and the rejection region in red. . 0.4 0.3 0.2 0.1 0.0 X Density -1.960 0.025 1.960 0.025 0 Distribution Plot Normal, Mean=0, StDev=1
- 10. WEEK 7 HOMEWORKHELP (MORE) Continued Find your test statistic using Minitab and the specifics of the problem. Claim: µ = 60, α = 0.05. Sample Statistics: x-bar = 59.1, s = 4.11, n =90 Stat >> Basic Statistics >> 1 Sample Z .
- 11. WEEK 7 HOMEWORKHELP (MORE) Continued Click Summarized data button, the specifics including checking Hypothesized Mean box and entering it… Claim: µ = 60, α = 0.05. Sample Statistics: x-bar = 59.1, s = 4.11, n =90 .
- 12. WEEK 7 HOMEWORKHELP (MORE) Continued Click Options button and enter confidence. Since α = 0.05, the confidence is 1-0.05 or 0.95 (95%). .
- 13. WEEK 7 HOMEWORKHELP (MORE) Continued Print out in Session Window shows test statistic, which is in the rejection region, so REJECT H0: µ = 60 and Accept Ha: µ ≠ 60… Remember you never “Accept the null, you either reject it or “do not reject it.” Also remember the claim can be either… . 0.4 0.3 0.2 0.1 0.0 X Density -1.960 0.025 1.960 0.025 0 Distribution Plot Normal, Mean=0, StDev=1 -2.08 is in the rejection region
- 14. WEEK 7 HOMEWORKHELP (MORE) H0: µ ≤ 290 Ha: µ > 290 (claim) The claim in this one is actually since it was “more than 290.” Understand that if they would have said it is “at least 290,” it would have been the null (H0: µ ≥ 290) Always watch the wording
- 15. WEEK 7 HOMEWORKHELP (MORE) Continued Same approach as in number 10 but this one is right tailed… Why? Always look at the Ha Ha: µ > 290 (Pretend it is an arrow, > is “pointing right) The only time you have two tails is when Ha contains an ≠ sign. Ha has a >, it’s right tailed. Ha has a <, it’s left tailed. .
- 16. WEEK 7 HOMEWORKHELP (MORE) In Minitab, they ask us FIRST to find the standardized test statistic z and the corresponding area… Next Chart…
- 17. WEEK 7 HOMEWORKHELP (MORE) Stat >> Basic Statistics >> 1 Sample Z
- 18. WEEK 7 HOMEWORKHELP (MORE) Continued Results The standardized test statistic z is 0.93
- 19. WEEK 7 HOMEWORKHELP (MORE) Continued To find the Area, you can use tables or Minitab In Minitab Graph >> Probability Distribution Plot > Click View Probability Normal, Mean of 0, Standard Deviation of 1 Click Shaded Area Tab CLICK X VALUE RADIAL BUTTON, LEFT TAIL, ENTER VALUE FOR STANDARDIZED TEST STATISTIC
- 20. WEEK 7 HOMEWORKHELP (MORE) Continued 0.4 0.3 0.2 0.1 0.0 XDensity 0.93 0.8238 0 Distribution Plot Normal, Mean=0, StDev=1 They are asking for this area rounded to three decimal places. So the answer would be 0.824
- 21. WEEK 7 HOMEWORKHELP (MORE) Continued To find the p value, just look at the session window again This is the p value
- 22. WEEK 7 HOMEWORKHELP (MORE) Continued This can get confusing. But Minitab sorts it out… If you chose the correct values for the hypothesis when using Minitab, it gives you the correct p value. H0: µ ≤ 290 Ha: µ > 290 The example shows you subtracting the area from 1 or 1 – 0.824 = 0.176 (this is the same p value in your session window). .
- 23. WEEK 7 HOMEWORKHELP (MORE) Continued Now compare your p value to your α. p = 0.176 > α = 0.15 So “p is not less than alpha” - If p is not less than alpha, we do NOT reject. If “p is less than alpha” – If p is less than alpha, pLess or Please Reject. That’s the way I remember it… .
- 24. WEEK 7 HOMEWORKHELP (MORE) Continued Our final conclusion is that we Fail to Reject and say there is not enough evidence to accept our claim. Remember the claim for this one was the alternative because of the wording. H0: µ ≤ 290 Ha: µ > 290 (claim) .