TwoSample Tests
What is a ttest?
 A ttest is used to compare the means of two normally distributed populations
 In the exam the population variance will always be unknown
What assumptions are needed for the ttest?
 The underlying distribution for each variable must be normal
 In the exam you will need to assume the variance for the two groups are equal
 You will need to use the pooled twosample ttest
What are the steps for a pooled twosample ttest?
 STEP 1: Write the hypotheses
 H_{0 }: μ_{x}_{ }= μ_{y}
 Where μ_{x}_{ }and μ_{y }are the population means
 Make sure you make it clear which mean corresponds to each population
 In words this means the two population means are equal
 H_{1 }: μ_{x}_{ }< μ_{y} or H_{1 }: μ_{x}_{ }> μ_{y }or H_{1 }: μ_{x}_{ }≠ μ_{y}
 The alternative hypothesis will depend on what is being tested (see sections for onetailed and twotailed tests)
 H_{0 }: μ_{x}_{ }= μ_{y}
 STEP 2: Enter the data into your GDC
 Enter two lists of data – one for each sample
 Choose the pooled option
 Your GDC will then give you the pvalue
 STEP 3: Decide whether there is evidence to reject the null hypothesis
 Compare the pvalue with the given significance level
 If pvalue < significance level then reject H_{0}
 If pvalue > significance level then accept H_{0}
 Compare the pvalue with the given significance level
 STEP 4: Write your conclusion
 If you reject H_{0}
 There is sufficient evidence to suggest that the population mean of X is bigger than/smaller than/different to the population mean of Y
 This will depend on the alternative hypothesis
 If you accept H_{0}
 There is insufficient evidence to suggest that the population mean of X is bigger than/small than/different to the population mean of Y
 Therefore this suggests that the population means are equal
 If you reject H_{0}
Onetailed Tests
How do I perform a onetailed ttest?
 A onetailed test is used to test one of the two following cases:
 The population mean of X is bigger than the population mean of Y
 The alternative hypothesis will be: H_{1 }: μ_{x}_{ }> μ_{y}
 Look out for words such as increase, bigger, higher, etc
 The population mean of X is smaller than the population mean of Y
 The alternative hypothesis will be: H_{1 }: μ_{x}_{ }< μ_{y}
 Look out for words such as decrease, smaller, lower, etc
 The population mean of X is bigger than the population mean of Y
 If you reject the null hypothesis then
 This suggests that the population mean of X is bigger than the population mean of Y
 If the alternative hypothesis is H_{1 }: μ_{x}_{ }> μ_{y}
 This suggests that the population mean of X is smaller than the population mean of Y
 If the alternative hypothesis is H_{1 }: μ_{x}_{ }< μ_{y}
 This suggests that the population mean of X is bigger than the population mean of Y
Worked Example
The times (in minutes) for children and adults to complete a puzzle are recorded below.
Children 
3.1 
2.7 
3.5 
3.1 
2.9 
3.2 
3.0 
2.9 


Adults 
3.1 
3.6 
3.5 
3.6 
2.9 
3.6 
3.4 
3.6 
3.7 
3.0 
The creator of the puzzle claims children are generally faster at solving the puzzle than adults. A ttest is to be performed at a 1% significance level.
Twotailed Tests
How do I perform a twotailed ttest?
 A twotailed test is used to test the following case:
 The population mean of X is different to the population mean of Y
 The alternative hypothesis will be: H_{1 }: μ_{x}_{ }≠ μ_{y}
 Look out for words such as change, different, not the same, etc
 The population mean of X is different to the population mean of Y
 If you reject the null hypothesis then
 This suggests that the population mean of X is different to the population mean of Y
 You can not state which one is bigger as you were not testing for that
 All you can conclude is that there is evidence that the means are not equal
 To test whether a specific one is bigger you would need to use a onetailed test
Worked Example
In a school all students must study either French or Spanish as well as maths. 18 students in a maths class complete a test and their scores are recorded along with which language they study.
Studies French 
61 
82 
77 
80 
99 
69 
75 
71 
81 
Studies Spanish 
74 
79 
83 
66 
95 
79 
82 
81 
85 
The maths teacher wants to investigate whether the scores are different between the students studying each language. A ttest is to be performed at a 10% significance level.