IB Mathematics Applications & Interpretation Question Type 5: Performing a Hypothesis Test on Single Proportion Using the Binomial Distribution Exercises

A production process has an average of 3 defects per 100 items. After an update to the process, a sample of 1000 items is found to yield 25 defects. Use a normal approximation to the Poisson distribution to test, at the 5% significance level, whether there has been a decrease in the mean number of defects.

It is claimed that 50% of users click an ad within 10 seconds. A sample of 220 users yields 95 clicks. At 5% significance, test if the click rate is less than claimed.

A hospital records an average of 5 infection cases per month. Following the introduction of a new hygiene protocol, the number of infection cases in the next month is 2.

At the 5% level of significance, perform a Poisson test to determine whether there is evidence that the infection rate has decreased.

A website claims that 55% of its users make a purchase on their first visit. A researcher surveys 190 users and finds 45 purchases. At the 5% significance level, test whether the true purchase rate is less than 55%.

A factory averages 4 faulty items per 200 produced. After changes, 1,400 items are checked and 14 faults are found. At the 5% significance level, test for improvement using a normal approximation to the Poisson distribution.

A poll indicates 70% support for a policy. In a sample of 200 people, 127 support it. At the 5% significance level, test whether the support differs from 70%.

A clinic records on average 5 infections per month. Over 12 months after an intervention, 50 infections occur. At 5% significance, test for a reduction using a normal approximation to the Poisson distribution.

Traffic studies indicate that accidents on a specific stretch of road occur at an average rate of 2 per day. Following a safety campaign, 10 accidents are recorded over a period of 14 days.

Test, at the 5% significance level, the hypothesis that the safety campaign has succeeded in reducing the accident rate. State your hypotheses clearly.

A shampoo factory produces on average 4 faulty bottles per 200 bottles. After installing a new machine, 400 bottles are inspected and 2 are found to be faulty.

At the 5% significance level, test whether the fault rate has decreased. Use a Poisson model.

A city averages 2 road accidents per day. Over the past year (365 days), 680 accidents occurred.

At the 5% significance level, test whether the daily accident rate has decreased using the normal approximation to the Poisson distribution.

A company asserts that $60\%$ of its visitors subscribe on their first visit. In a sample of $150$ visitors, $80$ subscribe. At the $10\%$ significance level, test whether the true subscription rate is less than $60\%$.

A process has historically 3 defects per 100 items. After improvement, 300 items are checked and 6 defects are found. At the 1% significance level, test for a reduction using a Poisson model.