Statistics Hypothesis Tests

Hypothesis Testing with respect to a single population Parameter.

Let’s say we make a statement with respect to a population parameter. A population parameter is nothing but a numerical characteristic of a population. This statement is a reason for a null hypothesis which is still an invalid proposed explanation and an alternative hypothesis available as another possible proposed explanation.

The proposed explanation - null hypothesis though invalid is put forward to be true. Depending on a given sample data,

we either

refuse to agree to the null hypothesis, which means that we come to a conclusion that null hypothesis is actually not true though it has been proposed to be true,

or

we accept the null hypothesis, which means that we come to a conclusion that null hypothesis is actually true and acceptable.

How can we come to such a conclusion either reject or do not reject the null hypothesis? We can come to a conclusion by doing a test and we call this test hypothesis test. It is actually calculations testing both hypothesis - null and its alternative. We do hypothesis tests to clarify null hypothesis is acceptable or it should be removed. Let’s see how to do hypothesis tests.

There are basically four types of hypothesis tests.

The first type is doing hypothesis tests about a population mean, a fixed average value of some sort of measured data about a population, with a stated value of sample standard deviation σ (sigma), the amount a single measurement differs from the mean. However, the tests can be done when the sample size is equal to or more than 30. However, when the sample size is small, then we do hypothesis tests when the population is almost but not completely normal and with no outliers. In this case, we must do verification on the normality of the data. To do verification, use normal probability plots. Use boxplot to check for outliers.

The second type is doing hypothesis tests about a population mean, a fixed average value of some sort of measured data about a population, with no stated value of sample standard deviation σ (sigma). We know nothing about σ which is the amount a single measurement differs from the mean. However, the tests can be done when the sample size is equal to or more than 30. However, when the sample size is small, then we do hypothesis tests when the population is almost but not completely normal and with no outliers. In this case, we must do verification on the normality of the data. To do verification, use normal probability plots. Use boxplot to check for outliers.

The third type is doing hypothesis tests with respect to a population proportion, p, which is finite or infinite collection of items considered in comparative relation to the whole population. To do this type of test,

we need a value,

from a mathematical computation which is the product of sample size - n, population proportion - p, the resultant value of the subtraction of population proportion from the whole which is 1 in numerical expression,

be greater than or  equal to 10.

In mathematical expression, we write it in this way: n * p * (1 - p) 10

p is the population proportion. It means p is a part considered in comparative relation to the whole population. (1 - p) is the rest of the part of the whole population without the part p. So, this whole expression n * p * (1 - p) represents a number which is produced by increasing the population size n by the amount of p and then the amount of (1 - p). To do the third type of hypothesis tests, it is required that this number from this operation n * p * (1 - p) must be greater than or equal to 10. If you get some number less than 10, do not perform third type of hypothesis testing.

If the number from this operation n * p * (1 - p) is greater than or equal to 10, then use the normal model to do the hypothesis tests.

The fourth test is doing hypothesis tests with respect to a population standard deviation or variance which is the quantity equal to the square of the sample standard deviation.

Make sure you know that sample standard deviation is σ (sigma), and the square of sample standard deviation σ (sigma) is population standard deviation which is called variance.

The population under consideration needs to be normally distributed for doing this type of hypothesis tests. The sample must be drawn from this population. Normality is required to do this type of hypothesis tests.

All four tests

Use classical methods or P-value approach to do all types of hypothesis tests. we like to use  P-value approach because it is easier to decide whether or not rejecting or accepting the null hypothesis. In P-value approach, you can always discard the null hypothesis if P-value is less than the level of significance α.

When we doing the hypothesis tests, sometimes we make mistakes. There are two types of mistakes we can make known as Type I and Type II errors.

Type I error is an error done when we reject the null hypothesis though we should not have because the null hypothesis is actually true in this case.

Type II error is an error done when we accept the null hypothesis when we should have rejected it because the null hypotheses is false or not true in this case.

The power of a hypothesis test is nothing but a value. This value is the probability of rejecting the null hypothesis when the alternative hypothesis is true. Keep in mind that, we do reject the null hypothesis because it is false when the alternative is calculated to be true. So, the power of a hypothesis test is the probability of rejecting the null hypothesis when the alternative hypothesis is true.

The closer the value of the true population mean is to the value of the hypothesized population mean mentioned in the null hypothesis, the lower the probability of rejecting the null hypothesis or accepting the alternative hypothesis, also known as the power of the test.

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Null Hypothesis

Alternative Hypothesis

False

True

Null Hypothesis

Alternative Hypothesis

True

False

Accept

Null Hypothesis

Reject Alternative Hypothesis

Accept

Alternative Hypothesis

Reject       Null Hypothesis

Probability of rejecting the null when alternative is true = Power of test

Rejecting the null when it is true= Type I Error

Accepting the null when it is false = Type II Error

Hypothesis testing Going into details.