Stat 1000: Tips for Assignment 5
Published: Wed, 11/23/11
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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 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:
First, be sure
to note whether a question gives you σ, the population standard
deviation, or s, the sample standard deviation. That dictates whether
you will use z or t when making your confidence interval or testing your hypothesis.
Question 1
is using the formula I introduce in question 6 of "The Confidence Intervals for the Mean" lesson. Make sure
you look through my questions 6, 7 and 8 before attempting this
question.
Question 2 is a challenging algebra question. Study my question 13 in Lesson 9 for some understanding of what you might do here. You will have to solve for z* algebraically. Once you know z* you can establish the level of confidence using Table A. Make sure you look at my question 10 in Lesson 8 to understand how z* relates to the level of confidence.
Question 3 requires you go to http://bcs.whfreeman.com/ips6e/
and click the "Statistical Applets" button and then
click "Confidence Interval". To graph your results, use JMP. Open a "New Data
Table" and name your first column something like "Confidence
Intervals". In that column you will type 10, 20, 30, ... 200.
Double-click at the top to the right of your first column to create a
second column and name it "Percent Hit" and record the percent hits for
your intervals as given by the applet. You are expecting the Percent
Hit to eventually be 95% by the end. Graph your data as a time series.
Here's how:
To make a Time Series: Select Analyze in the toolbar, then select Modeling in the drop-down list and finally select time series. Select your time variable ("Confidence Intervals") and click "X, Time ID" and select the other variable you are tracking ("Percent Hit") and click "Y, Time Series". Click OK. Just ignore that other pop-up menu asking about time lags or autocorrelations or whatever, click OK and move on. None of that has anything to do with the time series.
Personally, I think they should have had you click
the "Sample 50" button instead and made you do that several times (say
up to a total of 1000 or 2000 confidence intervals). Try that
yourself. Notice, in the long run, you reach a 95 percent hit rate.
That is because, we know a 95% confidence interval for the mean μ will
catch μ 95% of the time. Which is to say, if we make 2000 confidence
intervals for μ, then we would expect about 95% of them to actually
contain μ.
Question 4 is good practise at testing a hypothesis like I teach in Lesson 9.
Question 5 is a runthrough of hypothesis testing and confidence intervals. Use Stat mode on your calculator to get the sample mean. Be sure to look at question 10 in my Lesson 8 as an example of using an unusual level of confidence and my question 6 in Lesson 9 as examples of how to interpret a P-value and question 1 in Lesson 8 as an example of how to interpret a confidence interval.
To use JMP, click "New Data Table", then enter the data into Column 1. Double-click Column 1 and name it something appropriate, like "Fat Content". Now select "Analyze",
"Distribution" and highlight "Fat Content" 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 "Fat Content" then select "Test Mean" from the
drop-down list. Enter in
the
mean from your null hypothesis and enter in the given standard
deviation. Click
"OK" and JMP gives you the hypothesis test at the bottom of the
printout. Note that:
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
To make the confidence interval. Click the red
triangle next to "Fat Content" then select "Confidence Interval" from the
drop-down list. Select "Other" to get a pop-up menu. It probably
already has 0.95 typed in (for 95% confidence interval), but, if not, be
sure to type in 0.95. 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. I believe σ = 2.5 in your case. Click "OK" and JMP gives you the
Confidence Interval at the bottom of the printout. You can now select, copy and print your output to a file ready for upload as usual.
Again, take note of my question 13 in Lesson 9 when deciding if you can use the confidence interval to test the hypothesis.
Question 6 is another runthrough of hypothesis testing. 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 (b).
To use JMP, click "New Data Table", then enter the data into
Column 1. Double-click Column 1 and name it something appropriate, like
"Wait Time". Now select "Analyze",
"Distribution" and highlight "Wait 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 "Wait Time" then select "Test Mean" from the
drop-down list. Enter in
the
mean from your null hypothesis and enter in the given standard
deviation. 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. JMP gives you the P-values, as I discussed in question 5 above, and you make your decision from that.
Study Lesson 3: Designing Samples and Experiments in the current edition of my book, if you have it, to prepare for this assignment. Note that, in older editions of my study book, this was Lesson 4.
You will need Table B to help answer some of the questions on this assignment. That is in the textbook or can be downloaded from the resources section of Stats Portal. Here is a link where you can download the table immediately, if you prefer:
Thankfully, there is no JMP needed for this stuff.
Question 1. Be sure to read my section teaching the different types of experimental design (after question 6 in Lesson 3 of my book) for some ideas on how to answer this question.
Question 4 can be frustrating because you don't know whether you are to describe what is definitely being done in your example, or what you also assume should be done. I recommend you first select the options you are certain have been done, and, if they mark you wrong, add other options that you think might have been done, and cross your fingers. Remember the three principles of experimental design while you answer this question.
Question 5
They want two diagrams in this problem. A table that shows how you are determining the treatments (like my question 7(b) in Lesson 3) and then an arrow chart of the experiment. To draw one of those arrow charts and upload it, there are many
ways to do it and it depends on how fancy you want to get and what
program you are using. A pretty simple way might be to insert a table
into your document (In Word, select Insert, and choose Table) and give
yourself lots of rows and columns to work with (I'd say at least 4
columns and 5 rows; you can always add extra rows if you need them by
just pressing the Tab button when your curser is in the last cell of the
last row). You could also use a spread sheet program like Excel.
Type in the various words you want in the approximate
positions throughout the table. To connect words together with arrows
insert arrows (in Word or Excel, select Insert, Shape, and select the Arrow).
You then use your mouse to click where you want the arrow to start and
where you want it to stop. Get rid of the lines on your table by
right-clicking your mouse while pointing anywhere in the table and
selecting Table Properties. Then, select Borders and Shading and select
None. Don't get rid of the lines until you have made your whole chart
since the lines will help you place your words throughout.
By the way, I show you how to make a table in
question 7 to identify the treatments needed for a two-factor
experiment. When you have a three-factor experiment, make a table for
just two of the factors first, then, give all of those treatments to the
first level of the third factor. You then give those exact same
treatments to the second level of the third factor, and so, until you
have used all the levels of the third factor. Make sure you number the treatments in each cell of your table, and if you have to make two or more table, use different treatment numbers for each cell.