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Final Exam Prep Seminar April 9
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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lesson 1:
Did you read my tips on how to study and learn Stat 1000? If not, here is a link to those important suggestions: Did you read my Calculator Tips? If not, here is a link to those important suggestions: Study Lesson 8: Confidence Intervals for the Mean and Lesson 9: Hypothesis Testing for the Mean in my book, if you have it, to prepare for this topic.
You also need to study the Distribution of the Sample
Proportion section from Lesson 6 of my book (page 404 to the end) and look at question 10 in that Lesson. Note that it is fine with me if you never look at the section preceding this part (The Distribution of X in a Binomial Distribution). Questions 8 and 9 in Lesson 6 show another way of doing the normal approximation of the binomial distribution using the X bell curve rather than the p^
bell curve. These notes were written back at a time when there was no formula sheet, and, since there are two ways to do the normal approximation, I would show both ways, and leave it up to the students which way they found easier to remember. Now, however, the p^ bell curve formula is given on your formula sheet, so you should always use that approach. You can use that formula to solve questions 8 and 9, too.
Of course, always seek
out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test.
To type in formulas you are using and to show your numbers subbed into the formulas you can click the Equation Editor button in the toolbar that looks like the Sigma Summation symbol (you have to click the "..." other options button to see the
sigma formula input button. Then click the various buttons to make your fractions and enter the symbols. However, the Equation Editor is extremely slow and clunky. Personally, I would never use it. Just type ordinary text explaining what you are doing if you think you should show some work.
Exception: Always do any JMP stuff open-book. Have my tips in front of you, and let me guide you step-by-step through any JMP
stuff. JMP is just "busy" work. The sooner you get it done and can move on to productive things like understanding the concepts and interpreting the JMP outputs, the better off you will be. Then again, since you never have to upload the JMP printouts, perhaps you might not even bother to do the JMP at all. Most questions can be answered by hand even when they told you to use JMP.
Important Note about Hypothesis Testing
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Unlike what I instruct in my book, make sure that you compute a P-value every single time that you perform a hypothesis test. They have decided to not teach about critical values this term, so there is no need to use Table D to get z* or t*, the critical value,
for any hypothesis test. Because you are not using critical values, it therefore becomes essential to compute a P-value.
Your steps to test a hypothesis should always be:
- State the Hypotheses and so establish whether the test is upper-tailed, lower-tailed, or two-tailed.
- State the given level of significance, alpha. Let alpha = 5% if none is
given.
- Compute the test statistic using the correct formula for z or t.
- Compute the P-value by marking the test statistic on a bell curve and shading the appropriate region according to your alternative hypothesis.
- State your conclusion knowing that you always reject Ho if the P-value < alpha.
You will be using Table A and Table D while learning Lesson 8 and 9 and doing this assignment. Here is a link where you can download those tables if you have not done
so already:
This is p^ bell curve stuff similar to what I do in Lesson 6, question 10(c) and (d).
Be careful to note which is the true proportion, p, and which is the sample proportion
p^.
Be careful that you don't lose accuracy by rounding off too much. I suggest you round off to no less than 5 or 6 decimal places while computing things like the standard deviation of p^ to ensure that you get accurate z-scores. Better yet, store exact answers in memory in your calculator.
This is standard sample size stuff, like my questions 6 to 8 in Lesson 8.
Parts (a) and (b)
Just apply the sample size formula.
Part
(c)
Your answers for the previous two parts enable you to answer this one. Note they just want to know the multiplier. So, if you discovered that the sample size doubled from (a) to (b) (it doesn't), you would say that you want the sample size to be 2 times larger.
Here is another way to think about the Inverse-Square Relationship. Essentially, if you want your
margin of error to get smaller, then you want your sample size to get larger by the square of the factor. If you want your margin of error to get larger, then you want your sample size to get smaller by the square of the factor.
- This means, if you want to multiply the margin of error, you divide the sample size.
- If you want to divide
the margin of error, you multiply the sample size.
For example, if I want to divide my margin of error by a factor of 7, then I multiply my sample size by a factor of 49 (7-squared). If I want to multiply my margin of error by a factor of 5, then I divide my sample size by a factor of 25 (5-squared).
Part (d)
Don't use
the sample size formula in part (d)! It will be too good an answer, possibly. They want you to use the inverse-square relationship.
If you are having trouble identifying the multiplier they are using in part (d), here is a trick you can use. Divide the larger margin of error by the smaller margin of error as given in parts (a) and (d). That is the factor you want. Then, as
always, square it to establish the factor you want to determine your new sample size. Do you want the multiply by that value, or divide? Do you want the sample size to get larger or smaller?
Part (e)
This just returns to using the sample size formula that you used in parts (a) and (b).
Part (f)
Just look at your results from (a) and (e)
to answer this question.
This is standard confidence interval for the mean stuff. Also take a look at Lesson 8, question 10 for an example of how to deal with an unusual level of confidence. Be sure to state your answer in the form (Lower Limit, Upper
Limit).
Part (c) Compare your answers in (a) and (b) to see what happens to the length (or width) of a confidence interval (the distance between the Upper and Lower Limits) when you change the level of confidence.
This is an algebra problem. They have given you sigma, the population standard deviation. They have given you n, the sample size. From the given interval, you can figure out what m, the margin of error, must be, since m is half the width. - Find the width
of the interval (upper limit - lower limit) and divide by 2 to determine m, the margin of error.
- We know that m = z* sigma/ square root of n, so figure out z*, algebraically.
Hint: z* = m times square root n divide by sigma.
- Once you know z*, read Table D backwards to see what the confidence level is. So, if you get z*=2.326, for example, then Table D tells us that you must be 98% confident. (That is not the
answer.)
This is good practice at my revised five steps to test a hypothesis as I outlined above in the introduction to these tips. Make sure you have studied Lesson 9 before attempting this question. My question 12 is a
similar example. Make sure that you state your hypotheses correctly.
Part (b)
Take a look at my Lesson 9, question 6 for examples of interpreting a P-value. A good rule of thumb when interpreting a P-value is first and foremost, state that you are assuming the null hypothesis is correct (don't say that, say specifically what you are saying in your null hypothesis). Then,
describe in words the shaded region on your bell curve you used to visualize the P-value.
Approach this just like the previous question.
Part (a)
Be sure to read the section in Lesson 8 about "Inferences for the Mean are robust" that I write in the pages leading up to question 1 to understand what they are getting at in part
(a) (or remind yourself about the Central Limit Theorem in Lesson 7). Although we prefer a population to be normal, that is not a necessary condition to test hypotheses or make confidence intervals for the mean.
Part (b)
Look at my Lesson 8, question 10 for a similar question.
Part (c)
Look at my
Lesson 8, question 1(b) for an example of how to interpret a confidence interval for the mean.
Part (d) is running you through the revised five steps to test hypotheses again as I outlined above (you get to skip the critical value). Note that they tell you what to hypothesize about in this part of the question.
Part
(e)
Look at my Lesson 9, question 6 for some examples of how to interpret a P-value.
Part (f)
See my steps below to do the JMP. But, really, why bother? You already know the answer in this question. If you want to get a slightly more exact P-value, compute the z test statistic in part (d) above to 4 decimal places rather than just the usual 2, then type that value
into this handy P-value calculator:
If you want to avoid using JMP, but need to know the exact P-value to answer part (f), there is this nice easy to use P-value calculator on the web. Note that it gives you a two-tailed P-value, but it should be pretty
obvious what the P-value would be if the test is only one-tailed.
Part (g)
Make sure you look at my Lesson 9, question 13(d) for an example of the concept of using confidence
intervals to test hypotheses.
Confidence Intervals and Hypothesis Tests in JMP
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To use JMP, click "New Data Table", then enter the data into Column 1. Double-click Column 1 and name it Waiting Time. Now select "Analyze", "Distribution" and highlight Waiting Time and click "Y, Columns", then click
OK. You are now looking at a histogram and stuff.
To test the hypothesis:Click the red triangle next to Waiting Time then select "Test Mean" from the drop-down list. Enter in the mean from your null hypothesis (180) and enter in the given standard deviation (78.4). Click "OK" and JMP gives you the hypothesis test at the bottom of the printout. Note that you cannot enter the level
of significance they have given. The level of significance is not relevant to JMP, you will use that yourself to make your decision.
Prob > |z| is the P-value for a two-tailed test. Prob > z is the P-value for an upper-tailed test. Prob < z is the P-value for a lower-tailed test.
Select the appropriate P-value from this list.
To make a confidence
interval: You aren't asked to make a confidence interval in this question, but why not do so to check your answer to part (b)?
Click the red triangle next to Waiting Time then select "Confidence Interval" from the drop-down list. Select "Other" to get a pop-up menu. Type in the level of confidence you desire as a decimal. For example, if you want a 97% confidence interval, type in
0.97.
Make sure you click the box saying "Use known Sigma". Click "OK" and you will then get a pop-up menu to type in the sigma value given at the start of the (78.4, in this case). Click "OK" and JMP gives you the Confidence Interval at the bottom of the printout.
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