Testing a Poisson mean Problem set B One question at a time

For each test, X ~ Po(λ). Find the p-value and state the conclusion.

1. (a) H0: λ = 15, H1: λ < 15, significance level 5%, observed value x = 10.

   • Under H0, X ~ Po(15).
   • p-value = P(X ≤ 10) = 0.118
   • 0.118 > 0.05, so do not reject H0. Insufficient evidence.

2. (b) H0: λ = 6.4, H1: λ < 6.4, significance level 5%, observed value x = 2.

   • Under H0, X ~ Po(6.4).
   • p-value = P(X ≤ 2) = 0.0463
   • 0.0463 < 0.05, so reject H0. Sufficient evidence.

For each test, X ~ Po(λ). Find the p-value and state the conclusion.

1. (a) H0: λ = 3.5, H1: λ > 3.5, significance level 5%, observed value x = 8.

   • Under H0, X ~ Po(3.5).
   • p-value = P(X ≥ 8) = 1 − P(X ≤ 7) = 0.0267
   • 0.0267 < 0.05, so reject H0. Sufficient evidence.

2. (b) H0: λ = 14, H1: λ > 14, significance level 10%, observed value x = 20.

   • Under H0, X ~ Po(14).
   • p-value = P(X ≥ 20) = 1 − P(X ≤ 19) = 0.0765
   • 0.0765 < 0.1, so reject H0. Sufficient evidence.

Find the critical region for each test.

1. (a) X ~ Po(λ), H0: λ = 18, H1: λ < 18, significance level 5%.

   • Under H0, X ~ Po(18).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≤ 11) = 0.0549 > 0.05
   • P(X ≤ 10) = 0.0304 < 0.05
   • Critical region: X ≤ 10

2. (b) X ~ Po(λ), H0: λ = 6.5, H1: λ < 6.5, significance level 10%.

   • Under H0, X ~ Po(6.5).
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≤ 3) = 0.112 > 0.1
   • P(X ≤ 2) = 0.0430 < 0.1
   • Critical region: X ≤ 2

Find the critical region for each test.

1. (a) X ~ Po(λ), H0: λ = 15, H1: λ > 15, significance level 5%.

   • Under H0, X ~ Po(15).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≥ 22) = 0.0531 > 0.05
   • P(X ≥ 23) = 0.0327 < 0.05
   • Critical region: X ≥ 23

2. (b) X ~ Po(λ), H0: λ = 0.8, H1: λ > 0.8, significance level 2%.

   • Under H0, X ~ Po(0.8).
   • From the calculator, cumulative probabilities near the 2% mark.
   • P(X ≥ 3) = 0.0474 > 0.02
   • P(X ≥ 4) = 0.00908 < 0.02
   • Critical region: X ≥ 4

A machine has averaged 9 faults per week. After a service, the engineer thinks the average rate has decreased. Over the next 4 weeks there are 25 faults. Test his belief at the 5% significance level.

1. (a) Write down hypotheses, defining λ and X.

   • The rate is 9 per week. For 4 weeks, λ = 9 × 4 = 36.
   • Let X be the number of faults in 4 weeks. X ~ Po(λ).
   • H0: λ = 36, H1: λ < 36, where λ is the mean number of faults per 4 weeks.

2. (b) Find the p-value.

   • Under H0, X ~ Po(36).
   • p-value = P(X ≤ 25) = 0.0345

3. (c) State the conclusion in context.

   • 0.0345 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean rate of faults has decreased.

A toll gate has averaged 2.5 motorbikes per hour. After a new road opens, the operator thinks the average rate has increased. In a 4-hour period 15 motorbikes pass. Test his belief at the 10% significance level.

1. (a) Write down hypotheses, defining λ and X.

   • The rate is 2.5 per hour. For 4 hours, λ = 2.5 × 4 = 10.
   • Let X be the number of motorbikes in 4 hours. X ~ Po(λ).
   • H0: λ = 10, H1: λ > 10, where λ is the mean number of motorbikes per 4 hours.

2. (b) Find the p-value.

   • Under H0, X ~ Po(10).
   • p-value = P(X ≥ 15) = 1 − P(X ≤ 14) = 0.0835

3. (c) State the conclusion in context.

   • 0.0835 < 0.1, so reject H0.
   • There is sufficient evidence, at the 10% significance level, that the mean rate of motorbikes has increased.

A bakery sells an average of 16 sourdough loaves per hour in the morning. After a rival bakery opens next door, the owner thinks the average rate has decreased. He counts the loaves sold in one morning hour. Use a 5% significance level.

1. (a) Define X and write down hypotheses.

   • Let X be the number of loaves in one hour. λ is the mean number per hour.
   • H0: λ = 16, H1: λ < 16

2. (b) Find the critical region.

   • Under H0, X ~ Po(16).
   • P(X ≤ 10) = 0.0774 > 0.05
   • P(X ≤ 9) = 0.0433 < 0.05
   • Critical region: X ≤ 9

3. (c) There are 10 loaves. State the conclusion.

   • 10 is not in the critical region, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the mean number of loaves sold per hour has decreased.

Find the critical region for each binomial test.

1. (a) X ~ B(90, p), H0: p = 0.12, H1: p > 0.12, significance level 5%.

   • Under H0, X ~ B(90, 0.12).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≥ 16) = 0.0692 > 0.05
   • P(X ≥ 17) = 0.0383 < 0.05
   • Critical region: X ≥ 17

2. (b) X ~ B(50, p), H0: p = 0.3, H1: p > 0.3, significance level 10%.

   • Under H0, X ~ B(50, 0.3).
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≥ 19) = 0.141 > 0.1
   • P(X ≥ 20) = 0.0848 < 0.1
   • Critical region: X ≥ 20

A quarry has averaged 0.9 rockfalls per week. A geologist thinks the average rate has increased after heavy rain. He counts the rockfalls over a 10-week period and tests at the 5% significance level.

1. (a) State two conditions that must hold for the Poisson distribution to be a suitable model.

   • The events occur independently of each other.
   • They occur at a constant average rate, one at a time, so the rate does not change over the period.

2. (b) Write down hypotheses and find the critical region for the number of rockfalls in 10 weeks.

   • 0.9 × 10 = 9 per 10 weeks. Under H0, X ~ Po(9).
   • H0: λ = 9, H1: λ > 9
   • P(X ≥ 14) = 0.0739 > 0.05
   • P(X ≥ 15) = 0.0415 < 0.05
   • Critical region: X ≥ 15

3. (c) There are 16 rockfalls in the 10 weeks. State the conclusion in context.

   • 16 is in the critical region, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean rate of rockfalls has increased.