What makes it challenging? It is a LOT of interpretation. Most people are comfortable with all the calculations. We test you on your understanding of the concepts and definitions.
Basically, this means know your definitions. Know them in and out. Know how to interpret them and recognize them.
Main Topics:
1. Tests of Significance
2. Confidence Interval Estimations
a. For both 1&2, need to know t and z tests, four step process
3. ANOVA
Side Topics
1. Type I/Type II errors
2. What type of procedure is this?
3. Two-sided confidence intervals
4. Sample size
5. Symbols
1. Tests of Significance
Definitions to KNOW:
- Test of significance: An outcome that is unlikely to happen if a claim is true is good evidence that the claim is not true. (This is the theory of a test of significance. Remember the coin example?)
- p-value: The probability of getting an x-bar as extreme or more extreme if the null hypothesis is true. (KNOW THIS. You will need to be able to INTERPRET this as well. Meaning if I give you an actual situation, you could put numbers into the right locations. You can see previous posts for a more in depth explanation of this).
- Parameter: The mean of what you are finding out about the population. Okay, so this isn't really a definition, but be comfortable writing parameters. (Remember we need the MEAN and the POPULATION).
- Null/Alt Hypothesis: Null Hypothesis: Statement of no change. Alternative: What we want to prove.
Obviously you need to be comfortable with every part of the four step process for a test of significance (for both t and z tests).
Write the parameter, null and alternative hypothesis and state the level of significance. I've talked about these previously, but make sure you can do them.
Conditions
For a Z-test the conditions are (and are met by):
1. Randomization: Met through SRS OR RAT.
2. Normality: Met through CLT OR graph displaying approximately normality.
3. Sigma is known: Yes or no. They give it to you or they don't.
For a t-test conditions are (and are met by):
1. Randomization: Met through SRS OR RAT.
2. Normality: Met through CLT OR graph displaying no extreme skewness or outliers.
For a z-test, we use the equation z=x-bar - mu/ (sigma/ sqrt(n)). This is called the test statistic. We then go to the z-table and get a value for the p-value.
Things to remember about how to find z-test p-values:
- One-sided test with Ha: Mu<#: Read p-value directly off table.
- One-sided test with Ha: Mu>#: 1-table value = P-value.
- Two-sided test with x-bar < null hypothesis mu: 2*(table value)= p-value.
- Two-sided test with x-bar > null hypothesis mu: 2*(1-table value)=p-value.
For a t-test, we use the equation t=x-bar-mu/( s/sqrt(n). This is called the t test statistic. We then go to the t-table and get a value for p-value.
We like the t table because it already accounts for if it's a one-sided or two sided or if it is greater than or less than. The basic process to find the p-value is as follows:
- Take your t test-statistic
- Find your degrees of freedom (df) (n-1)
- Enter the table on your df row.
- Find the two values that sandwich your t test stat.
- Follow those two columns down to the bottom.
- Decide if you have a one-sided or two-sided test
- Read the two p-value values off
- Say "P-value = Number on right < P-value
Conclude
- Compare p-value with alpha
- Reject/Fail to Reject Null (p-value
- Conclude in context.
2. Confidence Interval Estimation
Definitions to KNOW:
- What is a confidence interval: It is used to estimate the mean. Gives reasonable values for the mean, etc.
- Confidence Level: If the procedure were repeated many times, confidence level is the amount of INTERVALS we would expect to contain the true mean. (This is an important one. Realize what confidence level is NOT: It is NOT how often our interval will contain mu, or x-bar or the percentage of time we are right).
- Margin of Error: the amount we expect our mean (mu) to differ from our sample mean (x-bar).
Procedure
Write the parameter, choose confidence level. Conditions are the same as for test of hypothesis.
Z confidence interval
Use equation x-bar +/- z* (sigma/sqrt(n)).
Finding z*
- Go to the t-table
- Find the row with your confidence level in it (top of chart)
- Follow it down to the third row from the bottom labeled "z*".
t confidence interval
Use equation x-bar +/- t* (s/sqrt(n))
Finding t*
- Find degrees of freedom
- Go to where your df and your confidence level intersect
- That is your t*
Conclude
Use the cookie-cutter answer to conclude for confidence intervals.
"We are _______% confident that the true mean _____________ lies between (_____,_____)"
a. Difference between Z and t tests and four step process
We basically stepped through the four step process for both confidence intervals and tests of significance above. Wouldn't hurt to go over the different sections, though.
Remember, we use a z-test is Sigma (population standard deviation) is KNOW, we use a t-test if sigma is UNKNOWN.
For multiple choice, it's helpful to really remember if you are using a t and z test. Remember, a z-test will give you a one-number p-value. A t-test will give you a range of values. Keep this in mind for your answers.
It may help you to remember the differences between the distributions (we talked about this briefly in class, check your notes). For example, the t-distribution has more areas in the tail (less precise). As n increases, the t-distribution becomes more like the z-distribution in shape.
3. ANOVA (Analysis of Variance)
Anova is all about reading the output. Be sure you know how to:
- Write Hypothesis
- Find the p-value
- check conditions
- Conclude IN CONTEXT based on the confidence intervals
If you want to go through an example, you can see my post about Assignment 24.
SIDE TOPICS
1. Type I/ Type II Errors
You should know the definitions of these/ what they are. I can't really give you the graph on the blog, but check over your notes and be sure to understand/know where everything is on the graph.
Be able to know the connection between alpha and beta, and when we might make alpha big or small (relatively).
Type I Error: Rejecting a true null hypothesis
Type II Error: Failing to Reject a false null hypothesis
Power: Rejecting a false null hypothesis or accepting a true alternative hypothesis
Also be able to do something like this:
Ho: Hillary is not addicted to "drawsomething"
Ha: Hillary is addicted to "drawsomething"
If she is not addicted, she gets to keep the app. If she is addicted, the app gets deleted. Be able to write the errors and power for something like that.
2. What type of procedure is this?
Be able to recognize procedures (aka is this a "one sample t-test" or a "two sample z-test" etc).
Think of the chart we drew in class. You can choose one thing from each column.
One Sample | z | test (of significance)
Two Sample | t | confidence interval estimation
Matched Pairs | |
3. Two-sided confidence interval estimation
I caution to even write about this. This was a question on the homework we didn't have time to go over in class. This is when you can use a confidence interval to prove/disprove hypothesis.
BE AWARE: This can ONLY be used in a TWO-SIDED TEST. And it should ONLY be used when specifically asked for. You should never, ever conclude a confidence interval this way. Concluding confidence intervals means using the "cookie-cutter" answer. Only if they ask you to do this should you, regardless if it is a two-sided test or not.
Okay. Read that warning a few times. Make sure you understand.
Anyways, Let's say we have the following hypothesis:
Ho: Mu=15
Ha: Mu does not equal 15.
Since we have a two-sided test, we could use a confidence interval to decide whether to reject or fail to reject the null hypothesis.
For example, let's say we got the interval (20, 28). Because the purpose of a confidence interval is to estimate the true mean, what could we say in this case? Clearly, 15 is not in the interval. So we could reject the null in this case.
Let's say we got an interval of (13,20)? Well, because fifteen is IN the interval, there is not sufficient evidence that mu is NOT 15. Thus, we fail to reject the null.
4. Sample Size Estimation
This is super easy. It's just an equation. You did it once on the homework. The equation you use is on the second row, far right side, of your equation sheet.
The only thing you need to know is that you always round UP. No matter what. The reason is that this is solving for a specific number of people you need to sample in order to meet specifications given to you. So if, for example, your equation gave you:
n= 24.3 people.
If you use 24 people, you don't meet the requirements. So you must round up to 25.
5. Symbols
So symbols are on every exam. But a few of you brought them up in class, and since I like you guys, I'm going to do a section on them. Here's symbols you should know:
µ (possible answers: mean of the sampling dist. of x-bar, population mean)
σ (population standard deviation)
x-bar (sample mean)
σ/√n (standard deviation of the sampling distribution of x-bar)
s/√n (standard error of the sampling distribution of x-bar, or just standard error)
s (standard deviation of a sample)
This is not necessarily a comprehensive list, but it should help.
Finally, don't forget the written portion. Be sure you:
- Go over assignment 21. It was there for a reason.
- Know how to do the four step for both two-sample and matched pairs. (this includes things like, how does the parameter change? When do you do each one? What do you graph? What equations do you use? We went over all this in class).
Remember, this is not necessarily a comprehensive list, just things I think will help.
GOOD LUCK!!!
-Hillary
You're the best TA ever!!!
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