t-Test Made Easy: Reject or Accept?

A t-test is a statistical test used to compare the means of two groups.  It determines if there's a significant difference between the means, considering the variability within each group.  

SAMPLE t-TEST PROBLEM

A teacher believes their students perform better than the national average on a standardized test. The national average score is 75. The teacher's class of 20 students has an average score of 80, with a standard deviation of 8.  Is the teacher's class's average score significantly higher than the national average? (Assume a significance level of α = 0.05 and a one-tailed test.) 

Here's how to solve the one-sample t-test problem:

 

Step 1: State the Hypotheses

Ho: μ ≤ 75

- The teacher's class average is not significantly different from the national average 

 

Ha: μ > 75

- The teacher's class average is significantly higher than the national average 

Step 2: Determine the Significance Level

The significance level (alpha) is given as α = 0.05.

 

Step 3: Calculate the t-statistic

 

We'll use the formula for a one-sample t-test:

Where:

x is the sample mean = 80

μ is the population mean = 75

s is the sample standard deviation = 8

n is the sample size = 20

 

Step 4: Determine the Degrees of Freedom

Degrees of freedom (df) 

= n - 1 

= 20 - 1 

= 19

Step 5: Find the Critical Value

Using a t-table or statistical software with df = 19 and α = 0.05 for a one-tailed test, the critical t-value is approximately 1.729.

Step 6: Make a Decision

 

The calculated t-statistic 2.80 is greater than the critical t-value 1.729. Therefore, we reject the null hypothesis.

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Very tiring right? Now, what if there's an easier way in computing for a t-Test problem? 

Introducing: t-Test Calculator

First, how does it work? 

With the same sample problem;

Step 1: Enter the hypothesized mean

Step 2: Enter the sample mean

Step 3: Enter the sample size

Step 4: Enter the sample standard deviation 

Step 5: After completing all the needed data, click calculate

Step 6: Analyze

In the calculator, all data needed for analyzation (whether to reject or accept the null hypothesis) are already given. In the picture, we can see that the test statistic and critical value are automatically given. 

Answer: The calculated t-statistic 2.80 is greater than the critical t-value 1.729. Therefore, we reject the null hypothesis. 

YOUR TURN! 

Using the calculator, try to answer these problems and share your answers in the comment section:

Problem 1:

A manufacturer claims that their light bulbs have an average lifespan of 1000 hours. A consumer group tests a sample of 25 bulbs and finds an average lifespan of 980 hours with a standard deviation of 50 hours. Is there sufficient evidence to reject the manufacturer's claim at a significance level of α = 0.05? (Assume a two-tailed test).

 

Problem 2:

A psychologist is testing a new therapy designed to reduce anxiety levels. The average anxiety score for the general population is 50. A sample of 15 patients undergoing the new therapy has an average anxiety score of 45 with a standard deviation of 6. Is the new therapy effective in reducing anxiety at a significance level of α = 0.05? (Assume a one-tailed test).

 

Problem 3:

A company produces cans of soda with a stated volume of 355 ml. A quality control inspector randomly selects 30 cans and measures their volumes. The average volume is 353 ml with a standard deviation of 2 ml. Is there evidence to suggest that the cans are being underfilled at a significance level of α = 0.05? (Assume a one-tailed test).