To understand this chapter you have to understand the processes of Chapter 10.
The t-distribution is a lot like the normal (z) distribution. It is much more forgiving (look for the references in the book to robustness) than the normal and we use it mostly when we have only a sample to work from--no population standard deviation.
The formulas involving t start out a lot like the z formulas.
t-statistic = (x-bar - mean)/(sample std dev/sqrt n)
and t-interval boundaries are x-bar +/- t* (sample std dev/sqrt n)
We use n-1 degrees of freedom because we "lost " one when we used x-bar to create the estimator s.
The sample std dev / sqrt n is called the standard error of the mean.
The value we use for t*, in fact the line of the table we use when considering probabilities, is based on the number of degrees of freedom (df). You can't use a line with a df = some number if you don't have at least that number of degrees of freedom. It's kind of like buying stuff. If you don't have the money, you can't buy the product. Do you realize what this means??? If you have 990 degrees of freedom and the closest choices in the text are 100 and 1000, you are supposed to select the conservative number, the one you can afford, 100 df. Now, if you can get a closer number from your calculator, use it.
How can you get the value from your calculator? (1) Use the Inv T program or function. Ti-84s with system 2.41 have it. If you have an '84, upgrade your system. If you have something else, get the program.
(2) Use the trick we demonstrated in class: Use T-INT with x-bar = 0, sx = sqrt of n, and n = n. The upper bound of the interval you generate is the estimate for t*.
Paired t-test
This is a routine t-test that is done on matched-pairs data. When you can load the first data set into L1 and the second into L2 and the following two conditions hold, you are looking at a matched-pairs design. (1) Each row of the data has to be naturally linked, as in data coming from the same person--and a different person from the rest of the rows. The two lists are DEFINITELY NOT independent of each other. (2) The variable of interest is the difference between the two values, like L1 - L2. The null hypothesis is usually mu(of the differences) = 0.
To perform the test, just do the regular t-procedures on the column of differences. DF still equals n-1.
If the two sets of data are two independent samples, that's something different. . ..
Two-sample tests
Note: The t-statistic for the difference betwen two means IS NOT t-distributed, but it is pretty close under most conditions.
We use two-sample procedures when we are looking at two separate, independent samples and trying to make an inference about the difference between the two population means.
While most of the procedure is intuitive, the standard error and the number of degrees of freedom require a little explanation.
Std Error of the difference of the means:
Do you remenber how we can't add std deviations? And how the variance of the difference of two variables is the sum of the variances? Put it together for this problem.
Find each sample variance--(s/sqrt(n))^2. Add the two sample variances together. Take the square root. In these formulas, s1 is the sample std dev for the first sample, n1 is the size of the firs sample, etc.
Then the std error of the difference = sqrt( (s1^2/n1) + (s2^2/n2) ).
Degrees of freedom:
For the number of degrees of freedom, either use the number that the calculator or the computer calculates for you or use the more conservative minimum of n1-1 or n2-1.
Hypothesis:
Ho: mu1 = mu 2 which is equivalent to Ho: mu1 - mu2 = 0
Other than these little changes, the procedures are similar to those you've already practiced.
Pooled vs unpooled
This refers to the situations when you believe that the variances of the two populations should really be equal. Using a concept similar to our Law of Large Numbers, combining the standard deviations from the samples in a clever way creates an even stronger estimate for the ONE estimated standard deviation. This is pooling of variances.
Just because the means are the same we cannot assume that the variances are equal also.
We almost never pool variances of X-bar. You can generally leave your calculator set on UNPOOLED and forget about memorizing the formula. You can only pool variances if you are really sure that the variances are equal.
Wednesday, February 21, 2007
Tuesday, February 06, 2007
Chapter 10 Beginning of Inference
This chapter introducecs important methods under the highly unrealistic conditions where we know the population standard deviation but not the population mean.
Point estimates for the average value of X found through samples are generally good estimates, but they are wrong. You can generate a better estimate by creating a confidence interval.
The confidence interval =
x-bar +/- Z* times sigma of x / (Sqrt n).
We get Z* from the t-table for a specific confidence level, for instance when we want a 95% we use 1.96.
In creating a complete solution we first write down all of the given information. Define your variable. Then we determine whether the central limit theorem has "kicked in" or if the underlying data were already normally distributed. Be sure to address whether the data were from a SRS. Graph them if you have them to make sure there are no gaps or outliers. Is the sample size less than 1/10 of the population size???
Identify what you are trying to produce-- a 95% Z interval for mu and give the formula. Show how the numbers are plugged in and calculate the interval.
Write the interpretation of your interval.
We are 95% confident that the true population mean value of [insert the contextual information here] falls between [lower bound] and [upper bound].
Refer to your notes for all of the baaaaaaaaaaaaad interpretations of a confidence interval and NEVER use them. :)
If a value of mu had been proposed before we collected our sample, we could see if the value falls within our interval. If the proposed value does fall in the interval, then it is a reasonable value, although not necessarily correct. If it does not fall in the interval, then it is not a reasonable value according to our sample values.
Hypothesis tests
For hypothesis tests, you develop a null and alternate hypothesis BEFORE you collect data. Both hypotheses use the parameter (NEVER THE STATISTIC) and they are considered logical opposites. The null hypotheses ALWAYS has an "equals" aspect to it: the alternate hypothesis is always <, >, or not equal to.
For instance: H0: mu = 15
Ha: mu > 15.
Although these are not actually opposites, finding evidence that mu is less than 15 provides no support of the alternative hypothesis. You can think of the null hypothesis in this case as mu<=15, which still has an "equals" in it. This is the way I learned hypothesis-writing back in the day and it is still acceptable, but not as common.
Alpha, Beta, Type I error, Type II error, and Power
Alpha is the likelihood of a Type I error--accidentally rejecting the null hypothesis when it was actually correct. (Like convicting the wrong guy.)
Beta is the likelihood of a Type II error--failing to reject the null when it was wrong. (Kind of an error of omission, or not enough evidence to convict.)
Power is the likelihood that the test would have been sensitive enough to pick up the difference between the hypothesized mu and the actual mu (given some other new value for mu). This is the complement of Beta. Yes, 1 - Beta = Power. 1 - power = beta. Power + beta = 1.
Notice that alpha and beta are NEVER added together. They don't live under the same conditions--one assumes that the null was true and the other that the null was false. DO not fall into the trap of EVER adding alpha and beta together (unless you are TOLD to do it and then only if they offer you a lot of money or a passing grade on a test).
Calculating beta is easier than people think on the calculator.
(1) Figure out what the critical values are for rejection of Ho in terms of x-bar.
(2) Find the area under the curve centered at the NEW mu that falls between these critical values. You can use normalcdf(left_critical_value, right_critical_value, new mean, standard dev or error of x-bar).
Point estimates for the average value of X found through samples are generally good estimates, but they are wrong. You can generate a better estimate by creating a confidence interval.
The confidence interval =
x-bar +/- Z* times sigma of x / (Sqrt n).
We get Z* from the t-table for a specific confidence level, for instance when we want a 95% we use 1.96.
In creating a complete solution we first write down all of the given information. Define your variable. Then we determine whether the central limit theorem has "kicked in" or if the underlying data were already normally distributed. Be sure to address whether the data were from a SRS. Graph them if you have them to make sure there are no gaps or outliers. Is the sample size less than 1/10 of the population size???
Identify what you are trying to produce-- a 95% Z interval for mu and give the formula. Show how the numbers are plugged in and calculate the interval.
Write the interpretation of your interval.
We are 95% confident that the true population mean value of [insert the contextual information here] falls between [lower bound] and [upper bound].
Refer to your notes for all of the baaaaaaaaaaaaad interpretations of a confidence interval and NEVER use them. :)
If a value of mu had been proposed before we collected our sample, we could see if the value falls within our interval. If the proposed value does fall in the interval, then it is a reasonable value, although not necessarily correct. If it does not fall in the interval, then it is not a reasonable value according to our sample values.
Hypothesis tests
For hypothesis tests, you develop a null and alternate hypothesis BEFORE you collect data. Both hypotheses use the parameter (NEVER THE STATISTIC) and they are considered logical opposites. The null hypotheses ALWAYS has an "equals" aspect to it: the alternate hypothesis is always <, >, or not equal to.
For instance: H0: mu = 15
Ha: mu > 15.
Although these are not actually opposites, finding evidence that mu is less than 15 provides no support of the alternative hypothesis. You can think of the null hypothesis in this case as mu<=15, which still has an "equals" in it. This is the way I learned hypothesis-writing back in the day and it is still acceptable, but not as common.
Alpha, Beta, Type I error, Type II error, and Power
Alpha is the likelihood of a Type I error--accidentally rejecting the null hypothesis when it was actually correct. (Like convicting the wrong guy.)
Beta is the likelihood of a Type II error--failing to reject the null when it was wrong. (Kind of an error of omission, or not enough evidence to convict.)
Power is the likelihood that the test would have been sensitive enough to pick up the difference between the hypothesized mu and the actual mu (given some other new value for mu). This is the complement of Beta. Yes, 1 - Beta = Power. 1 - power = beta. Power + beta = 1.
Notice that alpha and beta are NEVER added together. They don't live under the same conditions--one assumes that the null was true and the other that the null was false. DO not fall into the trap of EVER adding alpha and beta together (unless you are TOLD to do it and then only if they offer you a lot of money or a passing grade on a test).
Calculating beta is easier than people think on the calculator.
(1) Figure out what the critical values are for rejection of Ho in terms of x-bar.
(2) Find the area under the curve centered at the NEW mu that falls between these critical values. You can use normalcdf(left_critical_value, right_critical_value, new mean, standard dev or error of x-bar).
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