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 normal approximation.

A machine produces bolts with an average of 3 defects per 1000. Over 3000 bolts, 5 defects are found. Test at 5% (one‐sided decrease) whether defect rate has reduced. Use 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 lab records 60 radioactive counts in a minute, expected to be 50 on average. Using a two‐sided test at α=0.10 and approximating Poisson(50) by normal, determine whether the count rate has changed.

A hospital expects an average of 0.5 emergencies per hour. In 48 hours, 30 emergencies occur. At 5% significance (one‐sided increase), test if the emergency rate has increased, using normal approximation.

A process yields on average 10 defects per 500 items. After a change, 1500 items produced 18 defects. Test at α=0.05 (one‐sided) whether the defect rate has decreased, using the normal approximation.

In the scenario of the improved machine inspecting 1400 products with 14 faults, calculate the one‐sided p‐value for testing H₁: λ<28 at 5% significance using the normal approximation.

A call center averages 12 calls per hour. During an 8‐hour trial, 90 calls are received. At α=0.01 (one‐sided decrease), test if call rate has decreased, using normal approximation.

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

A website logs on average 100 visits per hour. Over 10 hours, it logs 850 visits. Test at 5% significance (two‐sided) whether the mean hourly visits differ from 100, using normal approximation to Poisson.

A traffic flow study expects 20 cars per minute at a toll. In a sample of 50 minutes, 1300 cars pass. At α=0.05 (two‐sided), test if flow rate differs from expectation using 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 1% significance (one‐sided decrease) whether the fault rate has reduced. Approximate using the normal approximation to Poisson.