State the null and alternative hypotheses for a test that the proportion of defective items produced by a machine is greater than 5%.
\( H_0: p=0.05 \), \( H_1: p>0.05 \) (one-tailed).
Hypothesis Testing
10 exam-style questions on one- and two-tailed tests for binomial and normal distributions. Tap Show answer for full working.
3 Easy4 Medium3 Hard
A coin is tossed 20 times and 14 heads are observed. Test at the 5% level whether the coin is biased towards heads. Use \( X\sim B(20,0.5) \).
\( H_0: p=0.5 \), \( H_1: p>0.5 \). \( P(X\ge14)=\sum_{k=14}^{20}\dbinom{20}{k}(0.5)^{20}\approx0.0577>0.05 \). Do not reject \( H_0 \).
Define a Type I error and a Type II error in the context of hypothesis testing.
Type I: rejecting \( H_0 \) when it is true. Type II: accepting \( H_0 \) when it is false.
A gardener claims that 70% of seeds germinate. In a trial of 15 seeds, only 7 germinate. Test at the 5% significance level whether the germination rate is lower than claimed.
\( H_0: p=0.7 \), \( H_1: p<0.7 \). \( X\sim B(15,0.7) \). \( P(X\le7)\approx0.0500 \) (borderline). Critical region typically \( X\le7 \) or check exact tables; conclusion depends on exact critical value — often reject or borderline at 5%.
The mean mass of a product is claimed to be 250 g. A sample of 40 items has mean 247 g. Assuming the population SD is 8 g, test at the 5% level whether the mean is less than 250 g.
\( H_0: \mu=250 \), \( H_1: \mu<250 \). \( Z=\dfrac{247-250}{8/\sqrt{40}}=\dfrac{-3}{1.265}\approx-2.37 \). Critical value \( -1.645 \). Reject \( H_0 \).
Find the critical region for a two-tailed test of \( H_0: p=0.4 \) against \( H_1: p eq0.4 \) based on \( X\sim B(20,0.4) \) at approximately the 5% level.
Find smallest \( a \) and largest \( b \) so tails each ≈0.025. Typically critical region \( X\le3 \) or \( X\ge13 \) (check cumulative probabilities).
A normal population has known variance \( \sigma^2=25 \). A sample of size 16 has mean 52. Test \( H_0: \mu=50 \) against \( H_1: \mu eq50 \) at the 5% level.
\( Z=\dfrac{52-50}{5/\sqrt{16}}=\dfrac{2}{1.25}=1.6 \). Critical values \( \pm1.96 \). Do not reject \( H_0 \).
In a binomial test, \( X\sim B(25,p) \), \( H_0: p=0.5 \), \( H_1: p>0.5 \). The critical region is \( X\ge17 \). Find the significance level of the test and the probability of a Type I error.
Significance level \( =\mathrm{P}(X\ge17\mid p=0.5)\approx0.0539 \) (≈5.4%). This is also \( \mathrm{P}(\text{Type I error}) \).
A company claims average battery life is 100 hours with SD 10 hours. A sample of 25 batteries has mean 96 hours. Test at 1% whether the mean life is less than claimed. State your conclusion clearly.
\( H_0: \mu=100 \), \( H_1: \mu<100 \). \( Z=\dfrac{96-100}{10/\sqrt{25}}=-2 \). Critical value at 1% is \( -2.326 \). \( -2>-2.326 \), so do not reject \( H_0 \) at 1%.
A two-tailed test is required for \( H_0: p=0.3 \) using \( X\sim B(30,0.3) \) at the 10% level. Determine a suitable critical region and the actual significance level.
Aim for each tail ≈0.05. Critical region approximately \( X\le4 \) or \( X\ge14 \). Actual significance level \( =P(X\le4)+P(X\ge14) \) under \( p=0.3 \) (evaluate with calculator/tables ≈0.09–0.11 depending on exact cut-offs).