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# Statistical Consulting

### Blog

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#### Hypothesis Testing

 Posted on February 28, 2016 at 11:43 PM comments (405)
 The Steps to Performing Hypothesis Testing Write the original claim and identify whether it is the null hypothesis or the alternative hypothesis.Write the null and alternative hypothesis. Use the alternative hypothesis to identify the type of test.Write down all information from the problem.Find the critical value using the tablesCompute the test statisticMake a decision to reject or fail to reject the null hypothesis. A picture showing the critical value and test statistic may be useful.Write the conclusion. Null Hypothesis (H0)      Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not equal, >) then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis. Alternative Hypothesis (Ha or H1)      Statement which is true if the null hypothesis is false. The type of test (left, right, or two-tail) is based on the alternative hypothesis. Left Tailed TestH: parameter < value Notice the inequality points to the left Decision Rule: Reject H if t.s. < c.v. Right Tailed TestH: parameter > value Notice the inequality points to the right Decision Rule: Reject H if t.s. > c.v. Two Tailed TestH: parameter not equal value Another way to write not equal is < or > Notice the inequality points to both sides Decision Rule: Reject H if t.s. < c.v. (left) or t.s. > c.v. (right) Type I error Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error. Type II error Failing to reject the null hypothesis when it is false (saying true when false). Probability of committing a Type I error.alpha Probability of committing a Type II error.beta Test statisticSample statistic used to decide whether to reject or fail to reject the null hypothesis. Critical region Set of all values which would cause us to reject H0Critical value(s) The value(s) which separate the critical region from the non-critical region. The critical values are determined independently of the sample statistics. Significance level ( alpha ) The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation. Decision A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis. Conclusion A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative).

#### Type I and Type II Error

 Posted on February 28, 2016 at 11:30 PM comments (27)
 Type I and II errors     There are two kinds of errors that can be made in significance testing: (1) a true null hypothesis can be incorrectly rejected and (2) a false null hypothesis can fail to be rejected. The former error is called a Type I error and the latter error is called a Type II error. These two types of errors are defined in the table. Statistical Decision True State of the Null Hypothesis H True H False Reject H Type I error Correct Do not Reject H Correct Type II error       The probability of a Type I error is designated by the Greek letter alpha (α) and is called the Type I error rate; the probability of a Type II error (the Type II error rate) is designated by the Greek letter beta (ß) . A Type II error is only an error in the sense that an opportunity to reject the null hypothesis correctly was lost. It is not an error in the sense that an incorrect conclusion was drawn since no conclusion is drawn when the null hypothesis is not rejected.      A Type I error, on the other hand, is an error in every sense of the word. A conclusion is drawn that the null hypothesis is false when, in fact, it is true. Therefore, Type I errors are generally considered more serious than Type II errors. The probability of a Type I error (α) is called the significance level and is set by the experimenter. There is a tradeoff between Type I and Type II errors. The more an experimenter protects himself or herself against Type I errors by choosing a low level, the greater the chance of a Type II error. Requiring very strong evidence to reject the null hypothesis makes it very unlikely that a true null hypothesis will be rejected. However, it increases the chance that a false null hypothesis will not be rejected, thus lowering power. The Type I error rate is almost always set at .05 or at .01, the latter being more conservative since it requires stronger evidence to reject the null hypothesis at the .01 level then at the .05 level.       A type I error occurs when one rejects the null hypothesis when it is true. The probability of a type I error is the level of significance of the test of hypothesis, and is denoted by *alpha*. Usually a one-tailed test hypothesis is is used when one talks about type I error. Examples:     If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, and men with cholesterol levels over 225 are diagnosed as not healthy, what is the probability of a type one error?  z=(225-180)/20=2.25; the corresponding tail area is .0122, which is the probability of a type I error.     If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, at what level (in excess of 180) should men be diagnosed as not healthy if you want the probability of a type one error to be 2%?  2% in the tail corresponds to a z-score of 2.05; 2.05 × 20 = 41; 180 + 41 = 221.      A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true.       The probability of a type II error is denoted by *beta*. One cannot evaluate the probability of a type II error when the alternative hypothesis is of the form µ > 180, but often the alternative hypothesis is a competing hypothesis of the form: the mean of the alternative population is 300 with a standard deviation of 30, in which case one can calculate the probability of a type II error. Examples:      If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, but only men with a cholesterol level over 225 are diagnosed as predisposed to heart disease, what is the probability of a type II error (the null hypothesis is that a person is not predisposed to heart disease).  z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, which is the probability of a type II error (*beta*).       If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, above what cholesterol level should you diagnose men as predisposed to heart disease if you want the probability of a type II error to be 1%? (The null hypothesis is that a person is not predisposed to heart disease.)  1% in the tail corresponds to a z-score of 2.33 (or -2.33); -2.33 × 30 = -70; 300 - 70 = 230.

#### Surviving the Dissertation Process

 Posted on February 28, 2016 at 9:59 PM comments (83)

#### Effect Size and Sample Size Calculations

 Posted on February 28, 2016 at 9:18 PM comments (100)