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AI HL · 13.4

Hypothesis Testing

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13.4.11

Critical values and critical regions for a test of a mean

13.4.11

Critical values and critical regions for a test of a mean

Target: I can say what a critical region and a critical value are. (+4 more)

By the end of this section

Key points

What a critical region is

  • The critical region is the set of sample means that would make you reject H0. The values on its boundary are the critical values.
  • If H0 is true, the probability that the sample mean falls in the critical region equals the significance level.
  • You will not be asked for a critical region for a t-test. Critical regions are for the z-test, where σ is known.

The form depends on H1

  • H1: μ < μ0 gives a lower region, \(\bar X < c\), with probability α in the tail.
  • H1: μ > μ0 gives an upper region, \(\bar X > c\), with probability α in the tail.
  • H1: μ ≠ μ0 gives two regions, \(\bar X < c_1\) and \(\bar X > c_2\), with probability α/2 in each tail.

Finding the critical value

  • Under H0, \(\bar X \sim N(\mu_0, \sigma^2/n)\). The standard deviation of the sample mean is \(\sigma/\sqrt{n}\), not σ.
  • Use the inverse normal with that mean and standard deviation. The area is α for a lower region, 1 − α for an upper region, and α/2 and 1 − α/2 for two tails.
  • Check the answer: a lower critical value is below μ0, an upper one is above μ0.
  • Reject H0 if the sample mean is in the critical region. This gives the same decision as comparing a p-value with α.

On the calculator

Inverse normal for a critical value

  • Choose the inverse normal. Enter the area to the left of the value you want, the mean μ0 and the standard deviation \(\sigma/\sqrt{n}\).
  • For a lower tail the area is α. For an upper tail the area is 1 − α. For two tails use α/2 and 1 − α/2.

Screenshot and video to come

Explore: the p-value and the critical region

The p-value is an area under the curve of the test statistic, if H₀ is true. Choose the test and H₁, then drag the orange point to the value of the statistic. The gold area is α, over the critical region.

Significance level, α
Critical value
p-value

Drag the point with the mouse or a finger, or select it (Tab) and use the arrow keys. Type a value for an exact one.

Drag the point to choose the test statistic.

  1. Drag the statistic further from the middle. What happens to the p-value? Why is a statistic far out strong evidence against H₀?
  2. Press "Move to the critical value". What is the p-value? Compare it with α.
  3. Change H₁ from > to ≠ and keep the statistic. Why does the p-value double? Where is the other half of the area?
  4. In t, make the degrees of freedom large. Which curve does the t curve become? Compare the critical values of t and z.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A lower critical region

A bakery's loaves have masses that are normally distributed with standard deviation 8 g. The mean is meant to be 400 g. After a new oven is fitted, the mean mass of a sample of 25 loaves is used to test, at the 5% significance level, whether the mean mass has decreased. Find the critical region.

Working
  1. Hypotheses

    H0: μ = 400, H1: μ < 400

    A decrease: the critical region is in the lower tail.

  2. Distribution

    X̄ ~ N(400, 8²/25), standard deviation = 8/√25 = 1.6

    The sample mean is less spread out than one loaf. Divide σ by √n.

  3. Critical value

    From the calculator: inverse normal, area 0.05, mean 400, standard deviation 1.6: c = 397

    5% of sample means lie below this value if H0 is true.

  4. Critical region

    X̄ < 397

    A sample mean below this value gives sufficient evidence that the mean has decreased.

Video walkthrough coming
An upper critical region

The battery life of a phone is normally distributed with standard deviation 1.2 hours and mean 14.0 hours. After a software update, the mean battery life of a sample of 40 phones is used to test, at the 1% significance level, whether the mean has increased. Find the critical region.

Working
  1. Hypotheses

    H0: μ = 14.0, H1: μ > 14.0

    An increase: the critical region is in the upper tail.

  2. Distribution

    X̄ ~ N(14.0, 1.2²/40), standard deviation = 1.2/√40 = 0.190

    Use σ/√n.

  3. Critical value

    From the calculator: inverse normal, area 0.99, mean 14.0, standard deviation 0.190: c = 14.4

    The area to the left of c is 0.99, so 1% lies above it.

  4. Critical region

    X̄ > 14.4

    The critical value is above 14.0, as it must be for an upper tail.

Video walkthrough coming
A two-tailed critical region

A machine cuts metal rods to a mean length of 48.0 cm. The lengths are normally distributed with standard deviation 0.9 cm. The mean length of a sample of 9 rods is used to test, at the 10% significance level, whether the mean length has changed. Find the critical region.

Working
  1. Hypotheses

    H0: μ = 48.0, H1: μ ≠ 48.0

    A change in either direction: two tails.

  2. Distribution

    X̄ ~ N(48.0, 0.9²/9), standard deviation = 0.9/√9 = 0.300

    Use σ/√n.

  3. Split the level

    Each tail has probability 0.10 ÷ 2 = 0.05

    Halve the significance level, one half for each tail.

  4. Critical values

    From the calculator: inverse normal, areas 0.05 and 0.95, mean 48.0, standard deviation 0.300: c1 = 47.5, c2 = 48.5

    The two values are the same distance from 48.0.

  5. Critical region

    X̄ < 47.5 and X̄ > 48.5

    Reject H0 if the sample mean is in either region.

Video walkthrough coming
Using the critical region

The rod-cutting machine above is tested at the 10% significance level. The sample mean of 9 rods is 48.7 cm. Is there sufficient evidence that the mean length has changed?

Working
  1. Critical region

    X̄ < 47.5 or X̄ > 48.5

    From the example above.

  2. Compare

    48.7 > 48.5

    The sample mean is in the upper part of the critical region.

  3. Conclusion

    Reject H0. There is sufficient evidence, at the 10% significance level, that the mean length has changed.

    The sample mean is above 48.0, so the evidence points to an increase.

Video walkthrough coming