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# Calculating critical values of t and z using their formulas

Critical value (t-value) is a statistics topic used for testing the hypothesis. The hypothesis is the statement used widely in statistics for testing the expectation or prediction of the sample values. Critical values are usually states for telling the upper and lower bounds of the confidence interval while testing in statistics.

Critical values tell the region for the hypothesis whether the null hypothesis is true or false. If the null hypothesis is false the alternative hypothesis must be true.

## Critical value formula

The first way to find the critical value is by using its general formula. This formula is written as.

t = (x – µ) / (s / √n)

In the above formula, all the terms are for the sample such as x is used for the mean of the sample, µ is for the population mean, s for the standard deviation of the given sample, and n is for the size of the given sample.

The other way to calculate the critical value or t value is by using a table known as the t table. For this method, you must have a confidence interval like 0.05, 0.02, 0.5, etc. and along with a confidence interval, you must have a degree of freedom. For this method, you have to check the values and get your required result from the table.

For example, if the significance level is 0.005 and the degree of freedom is 43, check it by using the table yourself and verify the result by using t table calculator. T table method is a very important method to find the exact and accurate result of any problem in just a few seconds.

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As you just have to check the value of the confidence interval in the row and the value of the degree of freedom in the column and take that value where both intersects.

## How to calculate the t value?

Let’s understand this concept by taking some examples.

### Example 1

#### A dairy products company wants to improve the sale of its product. The earlier sales data of products showed the average sale of 49 salesmen was Rs 500 per transaction. After training, the studied data showed an average sale of Rs 800 per transaction. If the standard deviation is Rs 210, find the t value?

Solution:

Step 1: Write the general critical value formula.
t = (x – µ) / (s / √n)

Step 2: Identify values from given problem.
Sample Mean = x = Rs 800
Population mean = µ = Rs 500
Standard deviation = s = Rs 210
Size of sample = n = 49

Step 3: Put the values in the general formula of critical value.
t = (x – µ) / (s / √n)
t = (800 – 500) / (210/ √49)
t = 300 / (210/7)
= 300 / 30
t = 10
t = 10 is the degree of freedom for critical value, now check the degree of freedom at a significance level at 0.05 in t table.

The critical value is 2.7638, so we can conclude that the absolute value is more than the critical value, which implies 6>2.7638

Hence, the training given by the company to the employees boosts the sales.

### Example 2

#### Calculate the t value, if the sample size is 10 and the significance level is 0.05.

Solution

Step 1: Identify the confidence interval and sample size from the given problem.
Size of sample = n = 10
Confidence interval = α = 0.05

Step 2: Calculate the degree of freedom by using sample size, as subtracting one from the sample size will give the degree of freedom.
Degree of freedom = df = n – 1 = 10 – 1 = 9

Step 3: Use the t table and find the t value by watching rows and columns.
t value = 1.8331

### Example 3

#### Find critical value if the sample size is 21 and the significance level is 0.005.

Solution:

Step 1: Identify the confidence interval and sample size from the given problem.
Size of sample = n = 21
Confidence interval = α = 0.005

Step 2: Calculate the degree of freedom by using sample size, as subtracting one from the sample size will give the degree of freedom.
Degree of freedom = df = n – 1 = 21 – 1 = 20

Step 3: Use the t table and find the t value by watching rows and columns.
t value = 2.8454

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## Application of critical value

For cutting the defined regions in which test statistics are unlikely to lie, critical values are essentially used for this purpose. In statistics, critical values are widely used for the testing of hypotheses. A point on the distribution of the test of the hypothesis that compared with the test statistic to find whether to accept or reject the null hypothesis, that point is said to be the critical value.

If the absolute value, that value which is away from zero, of the test statistics is larger or more than the t value or critical value, you can announce that the statistical significance and reject the null hypothesis, and accept the alternative hypothesis.
For example, if the significance level is 0.01 and the degree of freedom is 10, then the critical value or t value for this test must be 2.7638. If we have to find the change in fats from 1981 to 1991 and the absolute value of this form of test is given as 4.65, then we can conclude that the absolute value is more than the critical value or t value, in this case, we must reject the null hypothesis and announce that there is non-zero change on the average in fats from 1981 to 1991.

In simple words, the critical value is that value in the test statistic which tells us the upper and lower bounds of a confidence interval or tells us statistical significance in the given test. Confidence interval and degree of freedom are very essential for this test.

## Summary

Now you can see that this topic is not difficult. The critical value is usually used for the calculation of a hypothesis whether the null hypothesis is true or false. There are two methods to calculate critical value one is by using formula and the other is by using t table.