Hypothesis Testing for Mean (One Sample) (DP IB Applications & Interpretation (AI): HL): 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, $\overline{x}$, of a sample of size n is calculated, and a normal distribution is used to test the test statistic

• $\overline{x}$ can be used as the test statistic

  • In this case you would use the distribution $\overline{X}~\mathrm{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$

    • Remember when using this distribution that the standard deviation is $\frac{\sigma }{\sqrt{n}}$

• $z=\frac{\overline{x}-\mu }{{\displaystyle \frac{\sigma }{\sqrt{n}}}}$ can be also used as the test statistic

  • In this case you would use the distribution $Z~\mathrm{N}(0,1^2)$

    • 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 $\overline{x}$, 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 $\overline{x}$

  • For H₁ : μ < μ₀ the p-value is $\mathrm{P}\left(\overline{X}<\overline{x}|\mu ={\mu }_0\right)$

  • For H₁ : μ > μ₀ the p-value is $\mathrm{P}\left(\overline{X}>\overline{x}|\mu ={\mu }_0\right)$

  • For H₁ : μ ≠ μ₀ the p-value is $\mathrm{P}\left(\left|\overline{X}-{\mu }_0\right|>\left|x-{\mu }_0\right||\mu ={\mu }_0\right)$

    • If $\overline{x}<{\mu }_0$ then this can be calculated more easily by $2\times \mathrm{P}\left(\overline{X}<\overline{x}|\mu ={\mu }_0\right)$

    • If $\overline{x}>{\mu }_0$ then this can be calculated more easily by $2\times \mathrm{P}\left(\overline{X}>\overline{x}|\mu ={\mu }_0\right)$

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 $\overline{X}<c$ where $\mathrm{P}\left(\overline{X}<c|\mu ={\mu }_0\right)=\alpha \%$

  • For H₁ : μ > μ₀ the critical region is $\overline{X}>c$ where $\mathrm{P}\left(\overline{X}>c|\mu ={\mu }_0\right)=\alpha \%$

  • For H₁ : μ ≠ μ₀ the critical regions are $\overline{X}<c_1$ and $\overline{X}>c_2$ where $\mathrm{P}\left(\overline{X}<c_1|\mu ={\mu }_0\right)=\mathrm{P}\left(\overline{X}>c_2|\mu ={\mu }_0\right)=\frac{1}{2}\alpha \%$

• 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 $\overline{X}<c$

    • The upper bound for the inequalities $\overline{X}>c$

  • 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 ($s_{n-1}^2$)

• The mean, $\overline{x}$, 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 $\overline{x}$, 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)