## Critical Value Calculator

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## What is Critical Value

Any point on a line that divides the graph into two equal parts is considered to have a critical value, to put it simply. Depending on the area in which the value falls, the null hypothesis is either rejected or accepted. One of the two divisions created by the critical value is known as the rejection region. The null hypothesis would not be accepted if the test value were to be found in the rejection region.

## Critical Value Formula

The critical value is the point on a test statistic distribution where it is decided whether to reject or not to reject the null hypothesis. The critical value formula depends on the kind of test being run and the level of significance (alpha) picked.

For Example, the critical value formula for a two-tailed z-test with a normal distribution is:

Critical Value (CV) = z(alpha/2)

where "alpha" denotes the level of significance and "z" denotes the standard normal distribution (usually 0.05).

The formula for the critical value in a two-tailed t-test is:

CV = t(alpha/2,df)

where "df" stands for degrees of freedom, "alpha" for level of significance, and "t" stands for the t-distribution.

The formula for a one-tailed test varies depending on which side of the distribution you want to test.

It's crucial to remember that while these formulas provide the critical value for a specific level of significance, the critical value for a specific probability level can also be determined by using the inverse of the cumulative distribution function of the test statistic.

## How to Calculate Critical Values: A Step-by-Step Guide

Calculating critical values is an important part of statistical analysis, and understanding the process can help you make better decisions based on data. In this guide, we'll walk you through the steps for calculating critical values.

### Step 1: Determine the Type of Hypothesis Test

The first step is to determine whether you are conducting a one-tailed or two-tailed hypothesis test. In a one-tailed test, the null hypothesis is that there is no effect or a specific direction of effect (i.e., "greater than" or "less than"). In a two-tailed test, the null hypothesis is that there is no effect, without specifying the direction of the effect.

### Step 2: Choose the Level of Significance

The level of significance, denoted by α (alpha), is the probability of rejecting the null hypothesis when it is true. Common levels of significance are 0.05 (5%) and 0.01 (1%), but the specific value depends on the researcher's preference and the context of the study.

### Step 3: Determine the Degrees of Freedom

The degrees of freedom, denoted by df, represent the number of independent pieces of information in the sample that can vary.
The formula for degrees of freedom depends on the type of test and the sample size.

For a one-tailed test with a sample size of `n, df = n - 1.`

For example, if you have a sample size of `n = 20,` the degrees of freedom for a one-tailed test would be ` df = 20 - 1 = 19.`

For a two-tailed test with a sample size of ` n, df = n - 2.`

For example, if you have a sample size of n = 30, the degrees of freedom for a two-tailed test would be ` df = 30 - 2 = 28.`

### Step 4: Look Up the Critical Value

Once you know the type of test, level of significance, and degrees of freedom, you can find the critical value from a statistical table. The critical value is the minimum value of the test statistic that will lead to the rejection of the null hypothesis.

For example, suppose you are conducting a one-tailed test with a level of significance ` α = 0.05` and degrees of freedom `df = 19.` From a t-distribution table, the critical value is `1.734.`

For a two-tailed test, you need to find the critical value for both tails. The critical values are typically denoted by ` tα/2` and `-tα/2`

For example, if ` α = 0.01 ` and `df = 28,` the critical values from a t-distribution table are `-2.763` and `2.763,` respectively.

### Step 5: Calculate the Test Statistic

Calculate the test statistic using the sample data and the null hypothesis. The test statistic is the value used to determine whether to reject or fail to reject the null hypothesis.

For example, suppose you are testing the null hypothesis that the mean weight of a certain population is 50 kg, and your sample mean is 55 kg with a sample standard deviation of 10 kg. The test statistic for a one-tailed test is calculated as:

t = (sample mean - null hypothesis) / (sample standard deviation/sqrt (sample size)) ` = (55 - 50) / (10 / sqrt(20)) = 3.162`

Since the test statistic of 3.162 is greater than the critical value of 1.734, we reject the null hypothesis at the 0.05 level of significance. This means that we have evidence to support the alternative hypothesis that the mean weight of the population is greater than 50 kg.

## Common confidence levels and their critical values

In statistical inference, confidence levels are used to quantify the uncertainty associated with estimating a population parameter based on a sample. A confidence level is the probability that a statistical interval, such as a confidence interval, contains the true population parameter. Common confidence levels used in practice include 90%, 95%, and 99%.

The critical values for a given confidence level depend on the distribution of the test statistic and the degrees of freedom. Let's overview some examples of common confidence levels and their corresponding critical values for different distributions.

### Normal Distribution

For a normal distribution, the critical values for different confidence levels are given by the z-score. The z-score is the number of standard deviations a value is away from the mean. The critical values for different confidence levels are:

90% confidence level: `z = 1.645`

95% confidence level: ` z = 1.96`

99% confidence level: `z = 2.576`

Example: If you want to construct a 95% confidence interval for the population mean based on a sample from a normal distribution, you would use the critical value of `z = 1.96`

### Student's t-Distribution

For small sample sizes or when the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. The critical values for different confidence levels for the t-distribution depend on the degrees of freedom. Some common values are:

90% confidence level: `t(df) = 1.645`

95% confidence level: `t(df) = 1.96`

99% confidence level: `t(df) = 2.576`

Example: If you want to construct a 90% confidence interval for the population mean based on a sample from a t-distribution with 10 degrees of freedom, you would use the critical value of ` t(10) = 1.645`

### Chi-Squared Distribution

The chi-squared distribution is used for hypothesis tests and confidence intervals involving the variance of a normally distributed population. The critical values for different confidence levels depend on the degrees of freedom. Some common values are:

90% confidence level: `χ²(df) = 14.68`

95% confidence level: `χ²(df) = 16.92`

99% confidence level: `χ²(df) = 23.59`

Example: If you want to construct a 99% confidence interval for the population variance based on a sample from a normal distribution, you would use the critical value of ` χ2(n-1) = 23.59,` where n is the sample size.

## Types of Critical Values

Critical values are specific values that are used to determine whether to reject or fail to reject the null hypothesis. There are different types of critical values used in different statistical tests, and understanding these types can help in making accurate statistical inferences. Below are some types of critical values:

### One-tailed critical values

One-tailed critical values are used in hypothesis testing when the alternative hypothesis is directional. To put it another way, the test is made to see if the sample mean differs significantly from the population mean.

### Two-tailed critical values

Two-tailed critical values are used in hypothesis testing when the alternative hypothesis is non-directional. Two-tailed critical values are located in the middle of the distribution and correspond to a specified level of significance split between the two tails.

### Upper-tailed critical values

Upper-tailed critical values are used in hypothesis testing when the test is designed to determine if the sample mean is significantly greater than the population mean. Upper-tailed critical values are located at the extreme right end of the distribution and correspond to a specified level of significance.

### Lower-tailed critical values

Lower-tailed critical values are used in hypothesis testing when the test is designed to determine if the sample mean is significantly less than the population means. A certain level of significance is corresponding to lower-tailed critical values, which are situated at the extreme left end of the distribution.

## Critical Value of Z

The critical value of z is a term used in statistics to indicate the value of the standard normal distribution that corresponds to a particular level of significance or alpha (α). The standard normal distribution is a continuous probability distribution that is often used to model random variables that are approximately normal.

As part of a statistical procedure called hypothesis testing, which determines whether there is enough data to support or disprove a null hypothesis, the critical value of z is frequently used. The null hypothesis is a statement that assumes that there is no significant difference between two or more populations or sets of data. For example, suppose you want to perform a two-tailed hypothesis test with a level of significance of 0.05 (i.e., α = 0.05).

The critical value of z at this level of significance is ±1.96. It means that if the test statistic falls outside of the range of -1.96 to 1.96, we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.

## Assistance offered by this critical value calculator

The assistance offered by this critical value calculator includes the following:
• Calculating critical values for various statistical distributions
• Customizable inputs
• User-friendly interface
• Results in multiple formats

With customizable inputs, users can specify the significance level, degrees of freedom, and other parameters necessary to obtain accurate results. The user-friendly interface of our calculator makes it easy to enter inputs and view results.