Table of contents | |
T-test Formula | |
What Is the T-test Formula? | |
One-Sample T-Test Formula | |
Independent Sample T-Test | |
Paired Samples T-Test | |
Examples Using 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.
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.
where
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
According to the t-test formula,
Applying the known values in the t-test formula, we get
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
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?
According to the t-test formula, we know that
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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