# Critical value (case of T-score)

## 1.1 Definition

Critical Value is the measurement or gauge a researcher uses to calculate the Margin of Error within a set of data and Critical values can tell you what probability any particular data point will have. It sets a threshold to make you know the limits of your data observations.

## 1.2 Applicability of Critical Value-t score Approach

The computation of the Critical Value of small samples is similar to the Z-statistic approach. This Critical Value applies where the Sample Size is Small such as a sample size of strictly less than 30 observations (remember student t-statistic where n <30).

## 1.3 Role of Critical Value

• It represents a range of characteristics more correctly. That is, critical value provides the thresholds within which we can define a certain characteristic such as a particular factor.
• Gives an insight into the characteristics of the sample size you are evaluating. That is, it describes whether the characteristic being hypothesized influence the dependent or outcome variable in a statistically significant manner or not.
• It provides guidelines on disproving hypothesis when you test them. This is because the concern of the researcher is to find out whether the changes witnessed on the dependent variable is due to chance or due to the proposed predictor, the critical value provides the thresholds to adhere to.
• Used in calculating the Margin of Error. Margin of error is calculated by considering the size of the critical value.

## 1.4 Step-by-Step Procedure of Computing Critical Value (CV)

This procedure involves computation of first the alpha value and second Critical Probability as indicated in the following three steps;

STEP 1: Computation of Alpha Value (α)

When testing Null Hypothesis using t-score or t-statistics, critical value is the decision rule tool in guiding whether to accept or reject the null hypothesis. Then, before establishing the value of t you first compute the Alpha value (significant level). The three key alpha values determined are alright. However, the significance level which is commonly utilized in t-test is (0.05). This significance level portrays the t-distribution with ten (i.e., 10) degrees of freedom.

NOTE: That, Confidence Level is the probability of a statistical parameter also being true for the population you measure (expressed in a percent value). For example, if a researcher has collected data from a sample of a certain specific size (i.e., the criteria) so as to analyze it pertaining some attribute of the population characteristic such as average height of University students and feels that he/she is 95% confident that the sample selected represents the true average height of University students, then at 95%, he/she would compute the Alpha Value as follows; Specific Interpretation;

There is a 97.5% probability that with the sample size selected (criteria used), the sample statistic is the same as the population parameter being studied. i.e. the sample statistic is true for the entire population.

STEP 3: Computation of Critical Value

The critical value being established in this step is referred to as Computed Critical Value (don’t confuse this value with the theoretical value).

If sample size (n) is less than 30 (i.e., n <30), then you use NB1: That, the computed t Score is always compared with the theoretical or Table Critical Value which is gotten from the t-distribution Table.

NB2: That, the values for the connotations used in the formula are either given in a question in an examination set up or from actual data collected. The t score is further discussed under types of critical values as indicated below

## Types of critical values under t-score approach

To determine the Critical Value of t-Statistic, degrees of freedom (df) are used. But SE is the Standard Error

Conditions

Unlike in the case of Z-test, the t-test approach is attached to some conditions as follows;

One;

If the population standard deviation (δ) is known, then use In other words, since the sample standard deviation is computable, then use that statistic to establish the standard error value.

Two;

Degree of freedom (df) = Sample Size-1

The researcher needs to know precisely the degrees of freedom of the sample he/she is using. Degree of freedom is the number of values of observation that can vary given the maximum size of the sample size. The formula is;

Df= n-1

Where;

Df is degrees of freedom

n is the sample size

Example

If sample size=21

Df=21-1

=20

Based on the researcher’s postulation or theory (unproved supposition), critical value is used in carrying out three types of Null Hypothesis test, namely;

• Two Tailed Test Critical Values
• Right One-Tailed Test Critical Value
• Left One Tailed Test Critical Value

### 2.1 Two Tailed Test Critical Value-t Score

The Critical Value of a two tailed test of a Null Hypothesis exists where by the t-distribution curve has two equal critical regions on its both the left - and the right -hand side. NB1: That Two Tailed Test Critical Value is the value that is correspondently used when performing Two Tailed Null Hypothesis Test.

### 2.2 Right One-Tailed Test Critical Value-t Score

The Critical Value of right one-tailed test of a Null Hypothesis exists where by the t-distribution curve has one critical region on the right-hand side as the name suggests.

The corresponding Critical Value for the right-hand critical region is (α). Therefore, the t-distribution curve is graphically presented by the critical region being on the left-hand side as portrayed in Figure 1.2; NB2: That Right Tailed Test Critical Value is the value that is correspondently used when performing Right Tailed Null Hypothesis Test.

### 2.3 Left One-Tailed Test Critical Value-t Score

The Critical Value of left one-tailed test of a Null Hypothesis exists where by the t-distribution curve has one critical region on the Left-hand side as the name suggests.

The Critical Value of left one-tailed test of a Null Hypothesis exists where by the t-distribution curve has one critical region on the left-hand side as the name suggests. This case is the opposite of the right one tail test.

The corresponding Critical Value for the left-hand critical region is (α). Therefore, the t-distribution curve is graphically presented by the critical region being on the left-hand side as portrayed in Figure 1.3; NB3: That Right Tailed Test Critical Value is the value that is correspondently used when performing Right Tailed Null Hypothesis Test. About the Author - Dr Geoffrey Mbuva(PhD-Finance) is a lecturer of Finance and Accountancy at Kenyatta University, Kenya. He is an enthusiast of teaching and making accounting & research tutorials for his readers.