T-Test | Applied Mathematics for Class 12 - Commerce PDF Download

T-test Formula

The t-test formula helps us to compare the average values of two data sets and determine if they belong to the same population or are they different. The t-score is compared with the critical value obtained from the t-table. The large t-score indicates that the groups are different and a small t-score indicates that the groups are similar.

What Is the T-test Formula?


The t-test formula is applied to the sample population. The t-test formula depends on the mean, variance, and standard deviation of the data being compared. There are 3 types of t-tests that could be performed on the n number of samples collected.

  • One-sample test,
  • Independent sample t-test and
  • Paired samples t-test

The critical value is obtained from the t-table looking for the degree of freedom(df = n-1) and the corresponding α value(usually 0.05 or 0.1). If the t-test obtained statistically > CV then the initial hypothesis is wrong and we conclude that the results are significantly different.

One-Sample T-Test Formula


For comparing the mean of a population T-Test | Applied Mathematics for Class 12 - Commerce from n samples, with a specified theoretical mean μ, we use a one-sample t-test.
T-Test | Applied Mathematics for Class 12 - Commerce
where σ/√n is the standard error
w here,
`x' bar is the mean of the sample,
N is the assumed mean,
σ is the standard deviation
and n is the number of observations

Independent Sample T-Test


Students t-test is used to compare the mean of two groups of samples. It helps evaluate if the means of the two sets of data are statistically significantly different from each other.
T-Test | Applied Mathematics for Class 12 - Commerce
T-Test | Applied Mathematics for Class 12 - Commerce

where

  • t = Student's t-test
  • x1 = mean of first group
  • x2 = mean of second group
  • s1 = standard deviation of group 1
  • s2 = standard deviation of group 1
  • n1 = number of observations in group 1
  • n2 = number of observations in group 2

Paired Samples T-Test


Whenever two distributions of the variables are highly correlated, they could be pre and post test results from the same people. In such cases, we use the paired samples t-test.
T-Test | Applied Mathematics for Class 12 - Commerce
where
t = Student's t-test
x1 − x2 = Difference mean of the pairs
s = standard deviation
n = sample size

Examples Using t-test Formula


Example 1: Calculate a t-test for the following data of the number of times people prefer coffee or tea in five time intervals.
T-Test | Applied Mathematics for Class 12 - Commerce

 let x1 be the sample of data that prefers coffee and x2 be the sample of data that prefers tea.
let us find the mean, variance and the SD
T-Test | Applied Mathematics for Class 12 - CommerceT-Test | Applied Mathematics for Class 12 - Commerce
According to the t-test formula,
T-Test | Applied Mathematics for Class 12 - Commerce
Applying the known values in the t-test formula, we get
T-Test | Applied Mathematics for Class 12 - Commerce
t = 0.47


Example 2: A company wants to improve its sales. The previous sales data indicated that the average sale of 25 salesmen was $50 per transaction. After training, the recent data showed an average sale of $80 per transaction. If the standard deviation is $15, find the t-score. Has the training provided improved the sales?

H0 accepted hypothesis:the population mean = the claimed value ⇒ μ = μ0
H0 alternate hypothesis: the population mean not equal to the claimed value ⇒ μ ≠ μ0
t-test formula for independent test is T-Test | Applied Mathematics for Class 12 - Commerce
Mean sale = 80, μ = 50, s = 15 and n = 25
substituting the values, we get t = (80-50)/(15/√25)
t = (30 ×5)/10 = 10
looking at the t-table we find 10 > 1.711 . (I.e. CV for α = 0.05).
∴ the accepted hypothesis is not true. Thus we conclude that the training boosted the sales.


Example 3: A pre-test and post-test conducted during a survey to find the study hours of Patrick on weekends. Calculate the t-score and determine (for α = 0.25) if the pre-test and post-test surveys are significantly different?
T-Test | Applied Mathematics for Class 12 - Commerce

According to the t-test formula, we know that
T-Test | Applied Mathematics for Class 12 - Commerce
Σ(X-Y)= -3 = 3
s = Σ(X-Y)2/(n-1) = 52/1 = 25
t = 3/(25/2) = 6/25 = 0.24
here degree of freedom is n-1 = 2-1 =1 and the corresponidng critical value in the t-table for α = 0.25, is 1.
t < CV.
Therefore the scores are not significantly different.

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