Hypothesis test for the mean of a Poisson distribution Quiz

1. A test at the $5\%$ level has $\mathrm{H}_0:\lambda=4$ and $\mathrm{H}_1:\lambda>4$. Which probability is compared with $0.05$ when $x=9$ is observed with $X\sim\mathrm{Po}(4)$?

• A. $\mathrm{P}(X\leq9)$
• B. $\mathrm{P}(X\geq9)$
• C. $\mathrm{P}(X\leq8)$
• D. $\mathrm{P}(X=9)$

2. Cars pass a checkpoint at a mean rate of $2$ per minute. What is the distribution of the number passing in $5$ minutes under $\mathrm{H}_0:\lambda=2$?

• A. $\mathrm{Po}(2)$
• B. $\mathrm{Po}(5)$
• C. $\mathrm{Po}(10)$
• D. $\mathrm{Po}(7)$

3. Which pair of hypotheses is correct for 'the mean number of faults per metre has changed from $0.5$'?

• A. $\mathrm{H}_0:\lambda=0.5,\ \mathrm{H}_1:\lambda\neq0.5$
• B. $\mathrm{H}_0:\lambda\neq0.5,\ \mathrm{H}_1:\lambda=0.5$
• C. $\mathrm{H}_0:\lambda=0.5,\ \mathrm{H}_1:\lambda>0.5$
• D. $\mathrm{H}_0:X=0.5,\ \mathrm{H}_1:X\neq0.5$

4. For $X\sim\mathrm{Po}(8)$ and $\mathrm{H}_1:\lambda\neq8$ at the $10\%$ level, $\mathrm{P}(X\leq3)=0.0424$ and $\mathrm{P}(X\leq2)=0.0138$. What is the lower tail of the critical region?

• A. $X\leq2$
• B. $X\leq3$
• C. $X\leq4$
• D. $X\leq1$

5. A result is not significant at the $5\%$ level. What is the best conclusion?

• A. The mean rate has not changed.
• B. $\mathrm{H}_0$ is proved true.
• C. There is insufficient evidence that the mean rate has changed.
• D. The mean rate has definitely increased.