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Don't have my book or audio lectures? You can download a sample containing some of these lessons here: Did you read my tips on how to study and learn Stat
2000? 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: Please note that I made major changes to my book in September 2014. If you are using a book older than September 2014, you are missing about 100 pages of new material and an entirely new lesson on Probability.
Study Lessons
1, 2 and 3 in my study book (if you have it) to learn the concepts involved in Assignment 1. Remember my advice in the tips above. Don't start working on the assignment too soon. Study and learn the lesson first, and use the assignment to test your knowledge. 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.
This is a good overview of the concepts I teach in Lesson 1. Especially look at my questions 6, 7 and 8. Use the Inverse-Square Relationship I introduce in question 8 to answer parts (c) and (d).
Parts
(a) and (b)
If you want to show work but don't want the hassle of using the equation editor, it would be enough to just write something like this (in my opinion):
n=(z* sigma/ m)^2 = (whatever numbers you are using)^2 = ...
Note that the ^ symbol is usually above 6 on a keyboard, and is traditionally used to denote "raised to the power of" or
"superscript".
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).
Careful, in part (c), they only want the multiplier. Which is to say, if you want to divide the margin of error by 10, you want to multiply the sample size by 100 (10-squared). Just enter 100 in the box.
In part (d), where they don't give you the multiplier, take the old margin of error given in (a), and divide it by the new margin of error in (d) to get the appropriate factor. Now merely square
that factor. That factor is what they want. (Do not compute the actual sample size,n.) They don't want n, they just want the factor, just like (c) above.
Note that, in part (e), you should just use the sample size formula, just like you did in parts (a) and (b).This is a good run-through of errors and power in hypothesis testing as I teach in Lesson 3. Make sure you round off to four decimal places as required. Don't just trim your answers. For example, if you get 1.23456
and you are rounding to four decimal places, you would round that off to 1.2346. They do not make it clear, so I would assume that once you have computed the critical value for x̅ (what I call x̅*), and you have entered it as your answer to part (a), use this rounded off value for any future calculations.
BE CAREFUL! Make sure you are using the correct value for z*. Is the test
upper-tailed, lower-tailed, or two-tailed? What does that mean about z*? Keep that in mind for all the problems in this assignment.
To show your work, you can use the Equation Editor, or write things like this (in my opinion):
xbar* = z* (sigma/ sqrt n) + mu-naught = ....
z = (xbar - mu)/
(sigma/ sqrt n) = ....
Just like question 2 above.
Yet more of the same. To do part (c) more easily, read my part of Lesson 3 talking about "The Relationship between the Alternative Mean and the Power of a Test." This is just a matter of typing your explanations into the box provided. This is like what I discuss at the start of Lesson 3 where the Type I error is "saying someone does not have cancer when they actually do." Note that they also want you to describe the consequences of
each error. In my example, the consequences of a Type I error would be someone with cancer has gone undiagnosed, which may seriously affect their health, or perhaps even lead to a premature death.
This is a standard hypothesis test question where they have you do all five steps eventually, as taught in Lesson 2 of my book. I suggest that you always round off answers to no less than four decimal places unless specifically instructed to do
otherwise.
Never forget! If sigma, the population standard deviation is given, then you can use z. If you are not given sigma, you must use t.
Don't forget to show your work.
This should be done both by hand (i.e. just using your calculator) and with JMP.
Part (a) is fundamental. Be sure to read the section in Lesson 1 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 earlier in Lesson 1). Although we prefer a population to be normal, that is not a necessary condition to test hypotheses or make confidence intervals for the mean. All that matters is that we can assume the sample mean is normally distributed (or is at least approximately
normal).
Part (b) is using an unusual level of confidence. See my question 10 in Lesson 1 for an example. Note that they want you to explain how you get z*.
Part (c) is standard. I show you how to interpret a confidence interval in Lesson 1, question 1(b).
Part (d) is just taking you through the 5 steps to test a hypothesis (skipping
the P-value step for now). Note that you already know the z* critical value for part (d). You used it in part (b). By saying "differs from 5.0" in this part, they have made it clear what the hypotheses should be. Note that, among other things, they want the z decision rule or the rejection region in terms of z (like what I do in Lesson 2, question 2).
Parts (e) and (f) are standard P-value
stuff. I show you how to interpret a P-value in Lesson 2, question 6.
Don't forget to answer part (h) in the text box! I discuss this concept in Lesson 2, question 13(d) of my book.
Part (g):
Of course, since you are not actually submitting the JMP output, there really is no need to do this part at all. However, you could
use it to check your answers above.
First, enter the data into JMP manually: Click the "New Data Table" icon on the toolbar at top left in the JMP home screen. You are automatically taken to an empty spreadsheet with one column. Double-click "Column 1" and change its name to pH level, or right-click "Column 1" and select "Column Info" and type in the name
pH level and click OK.
Now just type in the various pH level values in the cells using your "Tab" or down arrow button to move to each proceeding cell. You can also hit "Enter" after each piece of data to enter it and move to the next cell.
Once you have entered all the data down your columns, you are ready to test the hypothesis and make your confidence interval.
In the toolbar at the top, select Analyze then select Distribution. In the "Select Columns" part of the pop-up window, click the column you want to analyze (pH level in this case) to highlight it, and click the Y, Columns button. You should see the pH level appear in the section to the right of the "Y, Columns" button. Click
OK.
It now opens yet another pop-up window called "Distributions" where a histogram should appear. Your histogram appears sideways.
To get JMP to make confidence intervals and test hypotheses for the mean:
To get a confidence interval, click the red triangle next to the variable pH level directly above the histogram to get a drop-down list and select "Confidence
Interval". In the pop-up window that appears, select "Other" (even if the level of confidence you desire is in the list) and type in the level of confidence you want (in decimal form, so, for example, 93% is 0.93). Make sure "Two-sided" is selected. You are given a value for sigma, the population standard deviation, so click the "Use known Sigma" checkbox. Click OK. You will get another pop-up window where you can type in your known value for sigma (0,54 in this
case). A Confidence Intervals table will appear in your output screen at the bottom. Your lower and upper limits are shown under "Lower CI" and "Upper CI" in the Mean row. These answers should agree with your computed answers in part (b) above.
To test a hypothesis, click that same red triangle you used to make a confidence interval and select "Test Mean". Type in the value the null hypothesis believes the mean to be
and type in the known value of sigma in the box that says "Enter the True Standard Deviation to do a z-test ..." Click OK. A "Test Mean = Value" table appears in your output where, among other things, JMP gives you the test statistic and three probability values. Those three probabilities are the P-value for the three possible alternative hypotheses. JMP will use a z statistic since you gave it a sigma value. It is up to you to know what the correct
alternative hypothesis is, and so whether the test is two-tailed, upper-tailed, or lower-tailed.
- Prob > |z| is the two-tailed P-value.
- Prob > z is the upper-tailed P-value.
- Prob < z is the lower-tailed P-value.
JMP's answer for the P-value should agree with what you calculated yourself in part (e). There might be a slight disagreement due to the greater accuracy in JMP,
but your P-value should be at least identical to three decimal places. This is basically a repeat of question 7, but now you are using t. Do not use Stat Mode on your calculator to compute the mean and standard deviation. I assume those answers would be more precise and lead to aswers that are too accurate.
They want you to use the values for the sample mean and standard deviation they provided to answer the questions.
Part (a) is standard confidence interval stuff.
Parts (b) to (g) take you through the five steps to test a hypothesis. Note that they want the t decision rule in part (g) or the rejection region in terms of t (like what I do in Lesson 2,
question 2).
Note: If you ever get a t test statistic that is off the charts on Table D because it is so small that it is off the left edge, or so large that it is off the right edge, do this: - Assume that there is a column labeled 0.50 on the far left of the table, and a column labeled 0 on the far right (since the upper tail areas get steadily smaller as you read from left to right).
- If
you have a test statistic that is off the left edge of the table, then the upper tail must be between 0.50 and 0.25 (although get in the habit of listing the smaller one first, so say it is between 0.25 and 0.50).
- If you have a test statistic that is off the right edge of the table, then the upper tail must be between 0.0005 and 0 (again say it is between 0 and 0.0005).
- Don't forget to double these bounds if you are doing a two-tailed test!
For
part (e), follow exactly the same JMP steps I outlined in question 7 above to do the JMP in this question. Make a New Data Table, naming the column Radiation Level. This time, you do not have a value for sigma, so leave the "Enter the True Standard Deviation ..." box in the Test Mean screen empty.
Again, you are given a choice of three probabilities "Prob > |t|," "Prob > t," and "Prob < t" in the Test Mean
= Value part of the output. That gives you the two-tailed, upper-tailed, and lower-tailed P-value, respectively. You should know which of those is appropriate for the hypothesis test you have been doing. (That exact P-Value better agree with the bounds you established earlier.)
If you want to avoid using JMP, but need to know the exact P-value to answer part (e), 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. Yet another hypothesis test as I teach in Lesson 2. Thankfully, no JMP required. Very similar to the previous question.
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