Hypothesis testing

Section FS1-4

2 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

Open the printable pack

FS1-4.1

Extend ideas of hypothesis tests to test for the mean of a Poisson distribution.

For a Poisson-mean test, state $H_0$ and $H_1$ in terms of the population parameter $\lambda$ or $\mu$ , not the observed count.
Under $H_0$ , combine independent observation periods into one Poisson total and use the direction of $H_1$ to choose the tail.
A critical region is the most extreme attainable tail whose probability under $H_0$ does not exceed the significance level; its probability is the actual size.
A common error is to say that $H_0$ is proved when it is not rejected; the conclusion should state that there is insufficient evidence for $H_1$ .

Tier 1 · Easy

2 marks

ORIGINAL

A manager claims that the mean number of alerts per shift is $4.2$ . State hypotheses to test whether the mean has increased, defining your parameter.

Tier 2 · Standard

7 marks

ORIGINAL

Under $H_0$ , defects occur at a mean rate of $2.5$ per hour. Eight independent hours give a total of $28$ defects. Test at a $5\%$ significance level whether the mean rate has increased. State the critical region and its actual size.

Tier 3 · Hard

8 marks

ORIGINAL

A Poisson model has mean $1.8$ events per batch. Twelve independent batches produce $14$ events in total. Test at a $5\%$ significance level whether the mean per batch has decreased. Give the critical region, its size and the $p$ -value.

FS1-4.2

Extend hypothesis testing to test for the parameter p of a geometric distribution.

For a geometric-parameter test, hypotheses are stated in terms of the success probability $p$ .
A larger $p$ tends to produce an earlier first success, so small trial numbers support $H_1:p>p_0$ and large trial numbers support $H_1:p<p_0$ .
The sum of $r$ independent geometric variables with the same $p$ counts the trial of the $r$ th success and has a negative binomial distribution.
A common error is to reverse the tail: long waiting times are evidence for a smaller, not larger, success probability.

Tier 1 · Easy

3 marks

ORIGINAL

A geometric model uses $p=0.35$ . State hypotheses for testing whether the probability of success has fallen, and state which tail of the waiting time is critical.

Tier 2 · Standard

7 marks

ORIGINAL

Six independent geometric observations count trials to first success. Test $H_0:p=0.3$ against $H_1:p>0.3$ at the $5\%$ level of significance using their sum $S$ . Find the critical region and its size, then state the conclusion when $S=9$ .

Tier 3 · Hard

8 marks

ORIGINAL

Five independent geometric observations have common parameter $p$ . Use their sum $S$ to test $H_0:p=0.4$ against $H_1:p<0.4$ at the $5\%$ level. Determine the critical region and its size. Find the $p$ -value and conclusion when $S=19$ .