This lecture presents some examples of Hypothesis testing, focusing on tests of hypothesis about the variance, that is, on using a sample to perform tests of hypothesis about the variance of an unknown distribution.
Table of contents
The higher the standard deviation the more variability or spread you have in your data. Standard deviation measures how much your entire data set differs from the mean. The larger your standard deviation, the more spread or variation in your data. Small standard deviations mean that most of your data is clustered around the mean. In the following graph, the mean is 84.47, the standard. Jul 20, 2019 Variance comes from models that are highly complex, employing a significant number of features. Typically models with high bias have low variance, and models with high variance have low bias. This is because the two come from opposite types of models.
Normal IID samples - Known mean
In this example we make the same assumptions we made in the example of set estimation of the variance entitled Normal IID samples - Known mean. The reader is strongly advised to read that example before reading this one.
The sample
The sample is made of independent draws from a normal distribution having known mean and unknown variance . Specifically, we observe realizations , ., of independent random variables, ., , all having a normal distribution with known mean and unknown variance . The sample is the -dimensional vector , which is a realization of the random vector.
The null hypothesis
We test the null hypothesis that the variance is equal to a specific value :
The alternative hypothesis
We assume that the parameter space is the set of strictly positive real numbers, i.e., . Therefore, the alternative hypothesis is
The test statistic
To construct a test statistic, we use the following point estimator of the variance:
The test statistic isThis test statistic is often called Chi-square statistic (also written as -statistic) and a test of hypothesis based on this statistic is called Chi-square test (also written as -test).
The critical region
Let and . We reject the null hypothesis if or if . In other words, the critical region isThus, the critical values of the test are and .
The power function
The power function of the test iswhere is a Chi-square random variable with degrees of freedom and the notation is used to indicate the fact that the probability of rejecting the null hypothesis is computed under the hypothesis that the true variance is equal to . Lion casino games.
The power function can be written aswhere we have definedAs demonstrated in the lecture entitled Point estimation of the variance, the estimator has a Gamma distribution with parameters and , given the assumptions on the sample we made above. Multiplying a Gamma random variable with parameters and by one obtains a Chi-square random variable with degrees of freedom. Therefore, the variable has a Chi-square distribution with degrees of freedom.
The size of the test
When evaluated at the point , the power function is equal to the probability of committing a Type I error, i.e., the probability of rejecting the null hypothesis when the null hypothesis is true. This probability is called the size of the test and it is equal to where is a Chi-square random variable with degrees of freedom (this is trivially obtained by substituting with in the formula for the power function found above).
Normal IID samples - Unknown mean
This example is similar to the previous one. The only difference is that we now relax the assumption that the mean of the distribution is known.
The sample
In this example, the sample is made of independent draws from a normal distribution having unknown mean and unknown variance . Specifically, we observe realizations , ., of independent random variables , ., , all having a normal distribution with unknown mean and unknown variance . The sample is the -dimensional vector , which is a realization of the random vector .
The null hypothesis
We test the null hypothesis that the variance is equal to a specific value :
The alternative hypothesis
We assume that the parameter space is the set of strictly positive real numbers, i.e., . Therefore, the alternative hypothesis is
The test statistic
We construct a test statistic by using the sample meanand either the unadjusted sample varianceor the adjusted sample variance
The test statistic isThis test statistic is often called Chi-square statistic (also written as -statistic) and a test of hypothesis based on this statistic is called Chi-square test (also written as -test).
The critical region
Wild spin casino. Let and . We reject the null hypothesis if or if . In other words, the critical region isThus, the critical values of the test are and .
The power function
The power function of the test iswhere the notation is used to indicate the fact that the probability of rejecting the null hypothesis is computed under the hypothesis that the true variance is equal to and has a Chi-square distribution with degrees of freedom.
The power function can be written aswhere we have definedGiven the assumptions on the sample we made above, the unadjusted sample variance has a Gamma distribution with parameters and (see Point estimation of the variance), so that the random variablehas a Chi-square distribution with degrees of freedom.
The size of the test
The size of the test is equal to where has a Chi-square distribution with degrees of freedom (this is trivially obtained by substituting with in the formula for the power function found above).
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Denote by the distribution function of a Chi-square random variable with degrees of freedom. Suppose you observe independent realizations of a normal random variable. What is the probability, expressed in terms of , that you will commit a Type I error if you run a Chi-square test of the null hypothesis that the variance is equal to , based on the observed realizations, and choosing and as the critical values?
The probability of committing a Type I error is equal to the size of the test:where has a Chi-square distribution with degrees of freedom. ButThus,If you wish, you can utilize some statistical software to compute the values of the distribution function. For example, with the MATLAB commands chi2cdf(65,39) and chi2cdf(15,39) we obtainAs a consequence, the size of the test is
Exercise 2
Make the same assumptions of the previous exercise and denote by the inverse of . Change the critical value in such a way that the size of the test becomes exactly equal to .
Replace with in the formula for the size of the test:You need to set in such a way that . In other words, you need to solvewhich is equivalent toProvided the right-hand side of the equation is positive, this is solved byIf you wish, you can compute numerically. From the previous exercise we know thatTherefore, we need to computeIn MATLAB, this is done with the command chi2inv(0.0444,39), which gives as a result
Exercise 3
Make the same assumptions of Exercise 1 above. If the unadjusted sample variance is equal to 0.9, is the null hypothesis rejected?
What Does Low Variance Mean Stats
In order to carry out the test, we need to compute the test statisticwhere is the sample size, is the value of the variance under the null hypothesis, and is the unadjusted sample variance.
Thus, the value of the test statistic isSince and , we have thatIn other words, the test statistic does not exceed the critical values of the test. As a consequence, the null hypothesis is not rejected. Paypal scratch cards.
How to cite
Please cite as:
Taboga, Marco (2017). 'Hypothesis tests about the variance', Lectures on probability theory and mathematical statistics, Third edition. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/hypothesis-testing-variance.
Variance measures how far a set of data is spread out. A variance of zero indicates that all of the data values are identical. All non-zero variances are positive. |
A small variance indicates that the data points tend to be very close to the mean, and to each other. A high variance indicates that the data points are very spread out from the mean, and from one another. Variance is the average of the squared distances from each point to the mean.
The process of finding the variance is very similar to finding the MAD, mean absolute deviation. The only difference is the squaring of the distances. Process: (1) Find the mean (average) of the set. (2) Subtract each data value from the mean to find its distance from the mean. (3) Square all distances. (4) Add all the squares of the distances. (4) Divide by the number of pieces of data (for population variance). |
Equation For Variance
One problem with the variance is that it does not have the same unit of measure as the original data. For example, original data containing lengths measured in feet has a variance measured in square feet.
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Standard deviation shows how much variation (dispersion, spread, scatter) from the mean exists. It represents a 'typical' deviation from the mean. It is a popular measure of variability because it returns to the original units of measure of the data set. |
A low standard deviation indicates that the data points tend to be very close to the mean. A high standard deviation indicates that the data points are spread out over a large range of values.
The standard deviation can be thought of as a 'standard' way of knowing what is normal (typical), what is very large, and what is very small in the data set.
Standard deviation is a popular measure of variability because it returns to the original units of measure of the data set. For example, original data containing lengths measured in feet has a standard deviation also measured in feet.
To compute standard deviation by hand: The standard deviation is simply the square root of the variance. This description is for computing population standard deviation. If sample standard deviation is needed, divide by n - 1 instead of n. Since standard deviation is the square root of the variance, we must first compute the variance. |
2. Subtract the mean from each data value and square each of these differences (the squared differences). |
3. Find the average of the squared differences (add them and divide by the count of the data values). This will be the variance. |
4. Take the square root. This will be the population standard deviation. Round the answer according to the directions in the problem. |
| Normal Curve |
A normal curve is a symmetric, bell-shaped curve. The center of the graph is the mean, and the height and width of the graph are determined by the standard deviation. When the standard deviation is small, the curve will be tall and narrow in spread. When the standard deviation is large, the curve will be short and wide in spread. The mean and median have the same value in a normal curve.
Normal Curve Empirical Rule: • 68% of the data lie within one standard deviation of the mean. • 95% of the data lies within two standard deviations of the mean. • 99.7% of the data lies within three standard deviations of the mean. Vegas joker casino. IQR for a normal curve is 1.34896 x standard deviation. |
