AI HL · 13.4
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Type I errors
13.4.19
Screenshot and video to come
Screenshot and video to come
A test of H₀: μ = 50 against H₁: μ > 50, with σ = 10. H₀ is rejected if the sample mean is above the critical value c. Drag the orange point to move c.
Type I error: H₀ is rejected when it is true. Type II error: H₀ is not rejected when it is false. When n changes, c moves so that P(Type I error) stays the same.
Drag c along the axis.
X ~ N(μ, 6.4²) and a sample of n = 12 is used to test H0: μ = 85 against H1: μ ≠ 85 at the 4% significance level. Find the probability of making a Type I error.
Have a go first. Then open the working one line at a time.
Type I error: reject H0 when H0 is true
The probability is found assuming H0 is true.
The sample mean has a continuous distribution, so the critical region can be chosen to have probability exactly 0.04 under H0.
For a continuous test, the significance level is the probability of rejecting a true H0.
P(Type I error) = 0.04
Two-tailed or one-tailed, it is the significance level: 0.02 + 0.02 here.
X ~ N(μ, 3.5²) and a sample of n = 9 is used to test H0: μ = 48 against H1: μ < 48. The critical region is \(\bar X < 46\). Find the probability of making a Type I error.
Have a go first. Then open the working one line at a time.
Under H0, X̄ ~ N(48, 3.5²/9), standard deviation = 3.5/√9 = 1.17
Type I error is rejecting H0 when it is true, so use the H0 mean.
P(Type I error) = P(X̄ < 46)
The critical region is the rejection region.
From the calculator: normal cumulative probability, mean 48, standard deviation 1.17: P(X̄ < 46) = 0.0432
Lower bound −∞ or a very small number.
X ~ B(12, p) and the hypotheses H0: p = 0.35, H1: p > 0.35 are tested. The critical region is X ≥ 8. Find the probability of making a Type I error.
Have a go first. Then open the working one line at a time.
Under H0, X ~ B(12, 0.35)
Use the H0 value of p.
P(Type I error) = P(X ≥ 8) = 1 − P(X ≤ 7)
For an upper tail, subtract the cumulative probability up to one below the boundary.
From the calculator: P(X ≤ 7) = 0.97449, so P(Type I error) = 0.0255
The value is the exact probability. It is not a significance level you chose.
X ~ Po(λ) and the hypotheses H0: λ = 6.2, H1: λ < 6.2 are tested. The critical region is X ≤ 2. Find the probability of making a Type I error.
Have a go first. Then open the working one line at a time.
Under H0, X ~ Po(6.2)
Use the H0 mean.
P(Type I error) = P(X ≤ 2)
Lower tail: no subtraction needed.
From the calculator: Poisson cumulative probability with mean 6.2: P(X ≤ 2) = 0.0536
This is the chance of wrongly rejecting H0.
A medicine is said to work for 40% of patients. A doctor thinks it works for fewer and tests this at the 5% significance level using a random sample of 20 patients. Find the critical region, and find the probability of making a Type I error.
Have a go first. Then open the working one line at a time.
H0: p = 0.4, H1: p < 0.4
p is the proportion of patients for whom the medicine works.
Under H0, X ~ B(20, 0.4). From the calculator: P(X ≤ 3) = 0.0160 ≤ 0.05 and P(X ≤ 4) = 0.0510 > 0.05
Critical region: X ≤ 3. Take the largest region with probability at most 5%.
P(Type I error) = P(X ≤ 3) = 0.0160
The 5% is the nominal level. The actual probability of a Type I error is lower because X takes whole values only.