To first understand the concepts displayed in this chapter, it is important to understand the overall goal and aim of these procedures. We are trying to use sample data to infer understandings about larger populations. We are going to be trying to infer conclusions about
In order to do this, we are going to have to understand the process of statistical inference.
The whole notion of statistical inference is based on the concept of hypothesis testing. The basic concept is that we are seeking to construct an exploratory hypothesis about a given populaton parameter (the mean, proportion, or variance). This could be the hypothesis that the population mean is less than a given value, that the proportion of defects would be exactly equal to a given value, or the popoulation variance be greater than a given value.
In order to move forward into the actual verification process of our hypotheses, we are going to need to use the sample data to formulate a test statistic. This test statistic will be used to make an informed decision about the hypothesis proposed. This test statistic will be generated purely from the data set and will be completely unique to the sample data. This value, then, will have to be compared to an objective value that will be generated depending on the size of the sample and the level of significance desired. This region of values is known as the rejection region. If the test statistic falls within the rejection region, there is evidence to reject the presumption of the null hypothesis. If the test statistic does not fall within the rejection region, then there is not sufficient evidence to reject the null hypothesis at the specified level of siginifance, given the specified sample size.
If all goes well, this process should lead us to correct conclusions most of the time. In fact, our confidence in the inferences proposed by the analysis of our data, will be solely dependent on the level of confidence we choose to work with. Commonly, a level of significance such as 95% is chosen. This means that 95% of the time, this process of statistical inference will yield correct conclusions depending on the analysis of the data. Other times, though, there are errors that are unavoidable, due to random chance, and particularly due to flaws in the system. Our systems are designed nearly perfectly, but yield the possibilities of two types of errors
The Type I Error occurs when we are convinced that there is sufficient evidence to reject the proposed hypothesis. This would occur if the test statistic that was generated from the sample data fell into the rejection region that was generated from the specified level of signifance and the sample size. We want to believe that when such a thing occurs, that it was indeed because the sample data provided sufficient evidence to reject the proposed claim. A Type I Error is when the rejected hypothesis was actually true. In fact, there was not sufficient evidence in the sample data to conclude that the proposed hypothesis should be rejected. Due to random chance, the test statistic was a little too large or small, and incorrectly fell into the rejection region.
The Type II Error occurs when we are convinced that there is not sufficient evidence to reject the proposed hypothesis. This would occur if the test statistic that was generated from the sample data did not fall into the rejection region that was generated from the specified level of significance and the sample size. We want to believe that when such a thing occurs, that it was indeed because the sample data did not provide sufficient evidence to reject the proposed claim. A Type II Error is when the accepted hypothesis was actually false. In fact, there was sufficient evidence in the sample data to conclude that the proposed hypothesis should be rejected. Due to random chance, the test statistic did not fall into the rejection region, and was incorrectly accepted.
The probability of rejecting true hypothesis is known as ALPHA.
The probability of accepting a false hypothesis is known as BETA
Alpha is an important number, as it allows us to conclude that our inferences should be correct a given % of the time. There is a ALPHA % chance that the hypothesis would be rejected if true.
Beta is an important number. BETA is the % of the time that a false hypothesis would be accepted. 1- BETA is known as the power of the test. It is thought of as power, since the number (1 - BETA) represents the probability of accepting only true hypotheses. The power of the test would be said, for example, to represent that 95% of the time only true hypotheses are accepted.
Therefore, there are two very important values in hypothesis testing
Since our test statistic is generated specifcally from our data set, it will yield a unique description of our data set. In fact, this test statistic will teach us some unique information that we otherwise could not infer only from a rejection region. For this reason, it is common to refer to a value known as the 'p-value', which is obtained from the test statistic. Once the test statistic is obtained, the probability of getting at least the observed value is known as the p-value. If the p-value is lower than the desired level of significance, there is sufficient evidence to reject the hypothesis proposed. This is because the data is speaking to us and telling us that there is a lower than ALPHA% chance of getting these specific sample data values at random. Therefore, we cannot conclude with 1-ALPHA% significance that the hypothesis is true. The data is telling us to reject it, in favor of another conclusion. We can easily compare the p-value to any given level of significance. If the p-value is higher than the level of significance, this is the data speaking to us and telling us that there is actually a higher than ALPHA% chance of getting these specific sample data values at random. Therefore we cannot conclude with 1-ALPHA% significance that the hypothesis is false. The data is telling us to accept the hypothesis, for lack of evidence towards its rejection.
When trying to infer about the population mean from the sample data, we must use a sample point estimator known as the sample mean. XBAR, the sample mean, will be used as a point estimator for the population mean. We can use the XBAR to also construct a confidence interval (depending on our desired level of confidence) to predict within what range the true population mean should fall. When the variance of the population is known, the Z-Test statistic can be generated and compared to a generated rejection region [alpha,n] to verify our proposed claims about the population mean. We can also take the generated test statistic to find out the p-value and easily conclude whether or not we should accept or reject the null hypothesis.
If the variance of the population is unknown, the point of estimator of the population mean can be generated using XBAR and a t-statistic. This is likewise so for the construction of the confidence interval of the population mean when the variance of the actual populaton is unknown.
When trying to infer about the populaton proportion from the sample data, we must use a sample point estimator known as the sample proportion. PHAT, the sample proportion, will be used as a point estimator for the population proportion. We can use the PHAT to also construct a confidence interval (depending on our desired level of confidence) to predict within what range the true population proportion should fall. The Z-test statistic can be generated and compared to a generated rejection region to verify our proposed claims about the population proportion. We can also take the generated test statistic to find out the p-value and easily conclude whether or not we should accept or reject the null hypothesis.
When trying to infer about the populaton variance from the sample data, we must use a sample point estimator known as the sample variance. S-SQUARED, the sample variance, will be used as a point estimator for the population variance. We can use the S-SQUARED value to construct a confidence interval (depending on our desired level of confidence) to predict within what range the true population variance should fall. The CHI-SQUARED-test statistic can be generated and compared to a generated rejection region to verify our proposed claims about the population variance. We can also take the generated test statistic to find out the p-value and easily conclude whether or not we should accept or reject the null hypothesis.
** It is important to note that there is an assumption of normality for the statistical test about the population variance.
The same procedures would follow for inferences about
#1 can be done using the Z-TEST (normality assumption and/or large sample required)
#2 can be done using the T-TEST (normality assumed)
#3 can be done using the T- (normality assumed)
#4 can be done using the Z-TEST
#5 can be done using the F-TEST
We can also use the procedures of statistical inference to conclude what the level of signicance for a given test should be.
Likewise, we can use the procedures of statistical inference to conclude what the power of a given test should be.
A final, important procedure that must be considered when trying to infer about the differences between more than two population means is the ANALYSIS OF VARIANCE.
In the ANOVA, we test the hypothesis that the population means are all equal. We use the data set to infer whether or not the variation within the treatments or between the treatments yields sufficient information to help us reject or accept the specified hypothesis as the given level of significance. The ratio of the mean square of the treatments to the mean square error is generated as an F-STATISTIC. This F-STATISTIC is compared to a rejection region that is unique to the specified level of significance and the degrees of freedom.