IB Mathematics Applications & Interpretation Question Type 7: Performing a Hypothesis Test on Population Mean Using the Normal Approximation of a Poisson Distribution Exercises

An email server receives on average 2 spam messages per minute. In a test minute, 7 messages are recorded. At the 5% level, test if the spam rate has increased. Use a normal approximation.

With the same data ($n=1400$, observed faults $=14$), perform a two-sided test at $\alpha =0.05$ to check if the fault rate differs from $4$ per $200$. Use the normal approximation.

In the scenario of an improved machine inspecting 1400 products with 14 faults, calculate the one-sided $p$-value for testing $H_1:\lambda <28$ at the 5% significance level using the normal approximation.

A factory produces shampoo with $4$ faulty products on average for every $200$ products. The manager installs a better machine and inspects the next $1400$ products, finding $14$ faulty items. At a $5\%$ significance level, test whether there is evidence that the new machine has reduced the fault rate. Use the normal approximation to the Poisson distribution.

A hospital expects an average of $0.5$ emergencies per hour. In a $48$-hour period, $30$ emergencies occur. At the $5\%$ significance level, test the hypothesis that the emergency rate has increased, using a normal approximation.

A website logs on average $100$ visits per hour. Over a $10$-hour period, it logs $850$ visits. Test at the $5\%$ significance level (two-sided) whether the mean hourly visits differ from $100$, using a normal approximation to the Poisson distribution.

A machine produces bolts with an average of 3 defects per 1000. In a random sample of 3000 bolts, 5 defects are found. Use a normal approximation to perform a one-tailed test at the 5% significance level to determine whether there is evidence that the defect rate has decreased.

A lab records 60 radioactive counts in a minute, which are expected to be 50 on average. Using a two-sided test at $\alpha =0.10$ and approximating $\text{Poisson}(50)$ by a normal distribution, determine whether the count rate has changed.

A traffic flow study expects 20 cars per minute at a toll. In a sample of 50 minutes, 1300 cars pass. At $\alpha =0.05$ (two-sided), test whether the car flow rate differs significantly from the expectation using a normal approximation.

A process yields on average $10$ defects per $500$ items. After a change, a sample of $1500$ items produced $18$ defects. Test at the $\alpha =0.05$ significance level (one-tailed) whether the defect rate has decreased, using the normal approximation.

At a significance level of $\alpha =0.01$ (one-sided decrease), test if the call rate has decreased, using a normal approximation.

A factory originally has a mean of 4 faulty shampoos per 200. After installing a new line, 1000 shampoos are checked and 30 faults are found.

Test, at the 1% significance level, whether the fault rate has reduced. Use a normal approximation to the Poisson distribution.