Testing a Poisson mean Problem set A Worked solutions

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

1. (a) H0: λ = 12, H1: λ < 12, significance level 5%, observed value x = 8.

   • Under H0, X ~ Po(12).
   • p-value = P(X ≤ 8) = 0.155
   • 0.155 > 0.05, so do not reject H0. Insufficient evidence.

2. (b) H0: λ = 7.5, H1: λ < 7.5, significance level 5%, observed value x = 3.

   • Under H0, X ~ Po(7.5).
   • p-value = P(X ≤ 3) = 0.0591
   • 0.0591 > 0.05, so do not reject H0. Insufficient evidence.

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

1. (a) H0: λ = 4, H1: λ > 4, significance level 5%, observed value x = 9.

   • Under H0, X ~ Po(4).
   • p-value = P(X ≥ 9) = 1 − P(X ≤ 8) = 0.0214
   • 0.0214 < 0.05, so reject H0. Sufficient evidence.

2. (b) H0: λ = 10, H1: λ > 10, significance level 10%, observed value x = 15.

   • Under H0, X ~ Po(10).
   • p-value = P(X ≥ 15) = 1 − P(X ≤ 14) = 0.0835
   • 0.0835 < 0.1, so reject H0. Sufficient evidence.

Find the critical region for each test.

1. (a) X ~ Po(λ), H0: λ = 12, H1: λ < 12, significance level 5%.

   • Under H0, X ~ Po(12).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≤ 7) = 0.0895 > 0.05
   • P(X ≤ 6) = 0.0458 < 0.05
   • Critical region: X ≤ 6

2. (b) X ~ Po(λ), H0: λ = 8.2, H1: λ < 8.2, significance level 10%.

   • Under H0, X ~ Po(8.2).
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≤ 5) = 0.174 > 0.1
   • P(X ≤ 4) = 0.0887 < 0.1
   • Critical region: X ≤ 4

Find the critical region for each test.

1. (a) X ~ Po(λ), H0: λ = 7.5, H1: λ > 7.5, significance level 5%.

   • Under H0, X ~ Po(7.5).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≥ 12) = 0.0792 > 0.05
   • P(X ≥ 13) = 0.0427 < 0.05
   • Critical region: X ≥ 13

2. (b) X ~ Po(λ), H0: λ = 1.8, H1: λ > 1.8, significance level 2%.

   • Under H0, X ~ Po(1.8).
   • From the calculator, cumulative probabilities near the 2% mark.
   • P(X ≥ 5) = 0.0364 > 0.02
   • P(X ≥ 6) = 0.0104 < 0.02
   • Critical region: X ≥ 6

A customer service desk has averaged 4.2 complaints per day. After staff training, the manager thinks the average rate has decreased. Over the next 5 days there are 14 complaints. Test her belief at the 5% significance level.

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

   • The rate is 4.2 per day. For 5 days, λ = 4.2 × 5 = 21.
   • Let X be the number of complaints in 5 days. X ~ Po(λ).
   • H0: λ = 21, H1: λ < 21, where λ is the mean number of complaints per 5 days.

2. (b) Find the p-value.

   • Under H0, X ~ Po(21).
   • p-value = P(X ≤ 14) = 0.0716

3. (c) State the conclusion in context.

   • 0.0716 > 0.05, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the mean rate of complaints has decreased.

A hospital emergency department has averaged 3.2 arrivals with sports injuries per hour on Saturdays. After a tournament starts nearby, a nurse thinks the average rate has increased. In a 2-hour period there are 11 such arrivals. Test her belief at the 5% significance level.

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

   • The rate is 3.2 per hour. For 2 hours, λ = 3.2 × 2 = 6.4.
   • Let X be the number of arrivals in 2 hours. X ~ Po(λ).
   • H0: λ = 6.4, H1: λ > 6.4, where λ is the mean number of arrivals per 2 hours.

2. (b) Find the p-value.

   • Under H0, X ~ Po(6.4).
   • p-value = P(X ≥ 11) = 1 − P(X ≤ 10) = 0.0614

3. (c) State the conclusion in context.

   • 0.0614 > 0.05, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the mean rate of arrivals has increased.

An astronomer sees an average of 11 meteors per night from her garden. After a new street lamp is fitted, she thinks the average rate has decreased. She counts the meteors on one night. Use a 5% significance level.

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

   • Let X be the number of meteors in one night. λ is the mean number per night.
   • H0: λ = 11, H1: λ < 11

2. (b) Find the critical region.

   • Under H0, X ~ Po(11).
   • P(X ≤ 6) = 0.0786 > 0.05
   • P(X ≤ 5) = 0.0375 < 0.05
   • Critical region: X ≤ 5

3. (c) There are 4 meteors. State the conclusion.

   • 4 is in the critical region, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean number of meteors per night has decreased.

Find the critical region for each binomial test.

1. (a) X ~ B(60, p), H0: p = 0.15, H1: p > 0.15, significance level 5%.

   • Under H0, X ~ B(60, 0.15).
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≥ 14) = 0.0578 > 0.05
   • P(X ≥ 15) = 0.0291 < 0.05
   • Critical region: X ≥ 15

2. (b) X ~ B(40, p), H0: p = 0.2, H1: p > 0.2, significance level 10%.

   • Under H0, X ~ B(40, 0.2).
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≥ 11) = 0.161 > 0.1
   • P(X ≥ 12) = 0.0875 < 0.1
   • Critical region: X ≥ 12

A helpline has averaged 2.4 calls per hour. The manager thinks the average rate has increased since a new advertisement. She counts the calls over a 5-hour 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 calls in 5 hours.

   • 2.4 × 5 = 12 per 5 hours. Under H0, X ~ Po(12).
   • H0: λ = 12, H1: λ > 12
   • P(X ≥ 18) = 0.0630 > 0.05
   • P(X ≥ 19) = 0.0374 < 0.05
   • Critical region: X ≥ 19

3. (c) There are 19 calls in the 5 hours. State the conclusion in context.

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