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Statistical Tests for Significance

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Define each of the following and provide a specific example for each type of statistical test or concept.

- The null and alternative hypotheses
- Type I and Type II errors
- The proper use of statistical software
- Cross-tabulation
- Chi-Square
- Single sample t-test
- Independent t-test
- One-Way ANOVA

Show your work (either your hand calculations or your statistical program output). You can either scan your work and submit it as a low-resolution graphic, type your answers directly into the document, or cut and paste your work into a Word file.

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• The null and alternative hypothesis
The null hypothesis reflects that there will be no observed effect for our experiment. The alternative hypothesis reflects that there will be an observed effect for our experiment. For example, a theory proposes that the yield of a certain chemical reaction is normally distributed, X ~ N( , 16). Past experience indicates that = 10 if a certain mineral is not present, and = 11 if the mineral is present. Our experiment would be to take a random sample of size n. On the basis of that sample, we would try to decide which case is true. That is, we wish to test the "null hypothesis" H0: = 0 = 10 against the "alternative hypothesis" Ha: = 1 = 11.
In our example above, is a sufficient statistic for , so we may conveniently express the critical region directly in terms of the univariate variable , and we will refer to as the test statistic. Because 1 > 0, a natural form for critical region in this problem is to let C = {(x1, ..., xn)| c}, for some appropriate constant c. The critical region for a test of hypothesis is the subset of the sample space that corresponds to rejecting the null hypothesis. We will reject H0 if c, and we will not reject H0 if <c, There are two possible errors we may make under this procedure. We might reject H0 when H0 is true, or we might fail to reject H0 when H0 is false. These errors are referred to as:
1. Type I error: Reject a true H0
2. Type II error: Fail to reject a false H0
We hope to choose a test statistic and a critical region so that we would have a small probability of making these two errors. We will adopt the following notations for these error probabilities:
1. P[Type I error] =
2. P[Type II error] =
For a simple H0 the significance level is also the size of the test. The standard approach is to specify or select some acceptance level of error such as = 0.05 or = 0.01 for the significance level of the test, and then to determine a critical region that will achieve this . Among all critical regions of size we would select the one that has the smallest . In our example above, if n = 25, then = 0.05 gives c = 0 + z1- / = 10 + 1.645(4)/5 = 11.316.
Thus, a size 0.05 test of H0 : = 10 against the alternative Ha : = 11 is to reject H0 if the observed value 11.316. Note that this critical region provides a size 0.05 test for any alternative value = 1, but the fact that 1 > 0 means that we will get a smaller Type II error by taking the critical region as the right-hand tail of the distribution of rather than as some other region of size 0.05. For an alternative 1 < 0 the left-hand tail would be preferable. Thus, the alternative affects our choice for the location of the critical region, but it is otherwise determined under H0 for specified .
The probability of Type II error for the critical region C is
= P[ < 11.316| = 1 = 11]
= P[ < | = 11]
=P[Z < 0.395] = 0.654

At this point, there is no theoretical reason for choosing a critical region of the form C over ...

Solution Summary

This solution explains each of the following statistical tests and concepts, including a specific example for each one: the null and alternative hypotheses, Type I and Type II errors, the proper use of statistical software, cross-tabulation, chi-square, single sample t-test, Independent t-test, and one-way ANOVA.

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