Hypothesis Testing for Mean (One Sample) (DP IB Applications & Interpretation (AI)): Revision Note

One-Sample z-tests

What is a one-sample z-test?

• A one-sample z-test is used to test the mean (μ) of a normally distributed population

  • You use a z-test when the population variance (σ²) is known

• The mean of a sample of size n is calculated  and a normal distribution is used to test the test statistic

•  can be used as the test statistic

  • In this case you would use the distribution 

    • Remember when using this distribution that the standard deviation is 

•  can be used as the test statistic

  • In this case you would use the distribution 

    • This is a more old-fashioned approach but your GDC still might tell you the z-value when you do the test

    • You will not need to use this method in the exam as your GDC should be capable of doing the other method

What are the steps for performing a one-sample z-test on my GDC?

• STEP 1: Write the hypotheses

  • H₀ : μ = μ₀

    • Clearly state that μ represents the population mean

    • μ₀  is the assumed population mean

• For a one-tailed test H₁ : μ < μ₀ or H₁ : μ > μ₀

• For a two-tailed test: H₁ : μ ≠ μ₀

  • The alternative hypothesis will depend on what is being tested

• STEP 2: Enter the data into your GDC and choose the one-sample z-test

  • If you have the raw data

    • Enter the data as a list

    • Enter the value of σ

  • If you have summary statistics

    • Enter the values of , σ and n

  • Your GDC will give you the p-value

• STEP 3: Decide whether there is evidence to reject the null hypothesis

  • If the p-value < significance level then reject H₀

• STEP 4: Write your conclusion

  • If you reject H₀­ then there is evidence to suggest that...

    • The mean has decreased (for H₁ : μ < μ₀)

    • The mean has increased (for H₁ : μ > μ₀)

    • The mean has changed (for H₁ : μ ≠ μ₀)

  • If you accept H­₀ then there is insufficient evidence to reject the null hypothesis which suggests that...

    • The mean has not decreased (for H₁ : μ < μ₀)

    • The mean has not increased (for H₁ : μ > μ₀)

    • The mean has not changed (for H₁ : μ ≠ μ₀)

How do I find the p-value for a one-sample z-test using a normal distribution?

• The p-value is determined by the test statistic 

• For H₁ : μ < μ₀ the p-value is 

• For H₁ : μ > μ₀ the p-value is 

• For H₁ : μ ≠ μ₀ the p-value is 

  • If then this can be calculated easier by 

  • If then this can be calculated easier by 

How do I find the critical value and critical region for a one-sample z-test?

• The critical region is determined by the significance level α%

  • For H₁ : μ < μ₀ the critical region is  where

  • For H₁ : μ > μ₀ the critical region is  where 

  • For H₁ : μ ≠ μ₀ the critical regions are  and where 

• The critical value(s) can be found using the inverse normal distribution function

  • When rounding the critical value(s) you should choose:

    • The lower bound for the inequalities 

    • The upper bound for the inequalities 

  • This is so that the probability does not exceed the significance level

One-Sample t-tests

What is a one-sample t-test?

• A one-sample t-test is used to test the mean (μ) of a normally distributed population

  • You use a t-test when the population variance (σ²) is unknown

  • You need to use the unbiased estimate for the population variance ()

• The mean of a sample of size n is calculated  and a t-distribution is used to test it

  • t-distributions are similar to normal distributions

    • As the sample size gets larger the t-distribution tends towards the standard normal distribution

• You won’t be expected to find the critical value

  • The p-value can be found using the test function on your GDC

What are the steps for performing a one-sample t-test on my GDC?

• STEP 1: Write the hypotheses

  • H₀ : μ = μ₀

    • Clearly state that μ represents the population mean

    • μ₀  is the assumed population mean

• For a one-tailed test H₁ : μ < μ₀ or H₁ : μ > μ₀

• For a two-tailed test: H₁ : μ ≠ μ₀

  • The alternative hypothesis will depend on what is being tested

• STEP 2: Enter the data into your GDC and choose the one-sample t-test

  • If you have the raw data

    • Enter the data as a list

  • If you have summary statistics

    • Enter the values of , s_n-1 (sometimes written as s_x on a GDC)   and n

  • Your GDC will give you the p-value

• STEP 3: Decide whether there is evidence to reject the null hypothesis

  • If the p-value < significance level then reject H₀

• STEP 4: Write your conclusion

  • If you reject H₀­ then there is evidence to suggest that...

    • The mean has decreased (for H₁ : μ < μ₀)

    • The mean has increased (for H₁ : μ > μ₀)

    • The mean has changed (for H₁ : μ ≠ μ₀)

  • If you accept H­₀ then there is insufficient evidence to reject the null hypothesis which suggests that...

    • The mean has not decreased (for H₁ : μ < μ₀)

    • The mean has not increased (for H₁ : μ > μ₀)

    • The mean has not changed (for H₁ : μ ≠ μ₀)