Hypothesis Testing on Amit Rajan

Hypothesis Testing (Part 6)

Mon, 26 Nov 2018 07:26:51 +0530

### Tests for Variances of Normal Populations :

Let $X_1, X_2, ..., X_n$ be a simple random sample from a normal population given as $N(\mu, \sigma^2)$. The sample variance $s^2$ is given as:

\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (X_i - \overline{X})^2\]

Then the test statistic $\frac{(n-1)s^2}{\sigma_0^2}$ follows a chi-square distribution with $n-1$ degrees of freedom. The null hypothesis can take any of the form:

\[H_0: \sigma^2 \leq \sigma_0^2;\ \sigma^2 = \sigma_0^2;\ \sigma^2 \geq \sigma_0^2\]

Example: To check the reliability of a scale in a butcher shop, a test weight known to weigh 400 grams was weighed 16 times. For the scale to be considered reliable, the variance of repeated measurements must be less than 1. The sample variance of the 16 measured weights was $s^2$ = 0.81. Assume that the measured weights are independent and follow a normal distribution. Can we conclude that the population variance of the measurements is less than 1?

Hypothesis Testing (Part 5)

Sun, 25 Nov 2018 17:06:08 +0530

### Tests with Categorical Data :

A generalization of the Bernoulli trial is the multinomial trial, which is an experiment that can rsult in any one of the $k$ outcomes where $k \geq 2$. Let the probabilities of the $k$ outcomes be denoted by $p_1, p_2, p_3, ..., p_k$ and the prespecified values be $p _{01}, p _{02}, ..., p _{0k}$. We need to conduct a test to chek whether the probabilities are equal to the prespecified values or not. The null hypothesis will be

Hypothesis Testing (Part 4)

Sun, 25 Nov 2018 11:36:18 +0530

### Distribution-Free Tests :

The one assumption for the Student’s t tests performed above was the fact that the samples should come from the normal distribution. In distribution-free tests, this restriction is relaxed, i.e. the samnples are not required to come from any specific distribution. Distribution-free tests are sometimes called as nonparametric tests. Mainly, there are two types of distribution-free tests: Wilcoxon signed-rank test(test for population mean) and Wilcoxon rank-sum test / Mann-Whitney test (analogous to the two-sample t test).

Hypothesis Testing (Part 3)

Sat, 24 Nov 2018 01:08:17 +0530

### Tests for the Difference Between Two Means (Large-Sample) :

The basic idea to conduct the hypothesis test for difference between two means is to find the distribution for the difference of two means and test whether it is equal to 0 or not. Here is an example.

Example: Suppose that a production manager for a manufacturer of industrial machinery is concerned that ball bearings produced in environments with low ambient temperatures may have smaller diameters than those produced under higher temperatures. To investigate this concern, she samples 120 ball bearings that were manufactured early in the morning, before the shop was fully heated, and finds their mean diameter to be 5.068 mm and their standard deviation to be 0.011 mm. She independently samples 65 ball bearings manufactured during the afternoon and finds their mean diameter to be 5.072 mm and their standard deviation to be 0.007 mm. Can she conclude that ball bearings manufactured in the morning have smaller diameters, on average, than ball bearings manufactured in the afternoon?

Hypothesis Testing (Part 2)

Fri, 23 Nov 2018 19:38:29 +0530

### Tests for a Population Proportion :

The hypothesis testing for population proportion can be conducted in a similar manner. Here are some examples to depict it.

Example: A supplier of semiconductor wafers claims that of all the wafers he supplies, no more than 10% are defective. A sample of 400 wafers is tested, and 50 of them, or 12.5%, are defective. Can we conclude that the claim is false?

Hypothesis Testing (Part 1)

Thu, 22 Nov 2018 09:17:41 +0530

Hypothesis Testing is a method by which we can test an assumption made for a population parameter. For example, the statement $\mu > 10$ is an assumption or hypothesis about the population mean $\mu$. To check the validity of this hypothesis, we must conduct a hypothesis test. Hypothesis tests are closely related to confidence intervals.

Prior to performing a hypothesis test, we need to formulate the null and alternate hypothesis. Null hypothesis states that the effect indicated by the sample is only due to chance or random variation. The alternate hypothesis indicates that the effect indicated by the sample is real and it accurately represents the whole population. For example, if we want to test that whether the population mean $\mu > 10$ by analyzing the data from a sample, the null and alternate hypothesis will be:

Hypothesis testing

Mon, 29 Oct 2018 02:01:07 +0100

### Introduction :

Hypothesis testing is a procedure that is used to determine that whether a made statistical statement (known as hypothesis) is a reasonable one and should not be rejected, or is unreasonable and should be rejected. Hypothesis testing setup is initialized by formulating a Null Hypothesis ($H_0$), which is the hypothesis associated with a contradiction to the theory that one would like to prove and Alternate Hypothesis ($H_A$), which is the hypothesis associated with the theory that one would like to prove. Then an appropriate test statistic and level of significance is chosen.