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Table of Contents

 


 

Comparing a Population Mean to a Sample Mean (T-test)

 

Used to compare a mean from a random sample to the mean (mu) of a population.  It is especially useful to compare a mean from a random sample to an established data source such as census data to ensure you have an unbiased sample.  Examples include comparing mean age and education levels of survey respondents to known values in the population.

 

Problem: Compare the mean age of incoming students to the known mean age for all previous incoming students. A random sample of 30 incoming college freshmen revealed the following statistics: mean age 19.5 years, standard deviation 1 year. The college database shows the mean age for previous incoming students was 18.

Assumptions

Interval/ratio level data

Random sampling

Normal distribution in population

 

State the Hypothesis

Ho: There is no significant difference between the mean age of past college students and the mean age of current incoming college students.

Ha: There is a significant difference between the mean age of past college students and the mean age of current incoming college students.

 

Set the Rejection Criteria

Significance level .05 alpha, 2-tailed test

Degrees of Freedom = n-1 or 29

Critical value from t-distribution = 2.045

Compute the Test Statistic

Standard error of the sample mean

       

 

Test statistic

      

  

 

Decide Results of Null Hypothesis

Given that the test statistic (8.197) exceeds the critical value (2.045), the null hypothesis is rejected in favor of the alternative. There is a statistically significant difference between the mean age of the current class of incoming students and the mean age of freshman students from past years. In other words, this year's freshman class is on average older than freshmen from prior years.

If the results had not been significant, the null hypothesis would not have been rejected. This would be interpreted as the following: There is insufficient evidence to conclude there is a statistically significant difference in the ages of current and past freshman students.

Software Output Example


Google

 

 


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