A basketball player claims that the probability of scoring a free throw is greater than $0.40$. During $20$ independent free throws, the player scores $13$ times. The probability of scoring is assumed to remain constant.

Two independent groups of students use different revision methods. Their scores on the same test are summarized below. Scores in both populations are assumed to be normally distributed with equal unknown variances.

Method A: $n_A=14$, $\bar{x}_A=18.2$, $s_A=2.40$

Method B: $n_B=12$, $\bar{x}_B=20.1$, $s_B=2.10$

A teacher wishes to test whether the population mean scores for the two methods are different.

Eight participants complete a task before and after a training session. For each participant, the difference is defined as

$d=\text{score after training}-\text{score before training}$

The observed differences are $2,1,3,-1,4,2,0,3$. The differences are assumed to come from a normal population.

The number of faults recorded by a monitoring system in one hour is modelled by a Poisson distribution. The usual population mean is $2.5$ faults per hour. During eight independent hours after a software update, a total of $29$ faults is recorded. An engineer tests whether the update has increased the mean fault rate.

For a random sample of $10$ cities, a researcher records annual public transport expenditure per resident and the percentage of journeys made by public transport. The sample product moment correlation coefficient is $r=0.762$. The paired variables are assumed to follow a bivariate normal distribution.

A company tests

$H_0:\pi=0.20\qquad\text{against}\qquad H_1:\pi>0.20$

using a random sample of $25$ independent items. Let $X$ be the number of defective items. The significance level is $5\%$.

The lifetime, in hours, of a component is normally distributed with known population standard deviation $12$ hours. A test of

$H_0:\mu=100\qquad\text{against}\qquad H_1:\mu>100$

uses the mean lifetime $\bar X$ of a random sample of $36$ components. The significance level is $5\%$.

A museum records the payment method and age group of a random sample of 200 visitors. The observed frequencies are shown in the table. A $\chi^2$ test for independence is performed at the $5\%$ significance level.

A seed supplier claims that packets contain seeds from five varieties in the proportions $0.10$, $0.20$, $0.40$, $0.20$ and $0.10$. A random sample of 200 seeds gives observed frequencies $14$, $48$, $72$, $43$ and $23$.

A delivery company claims that unsuccessful deliveries occur equally often on each of the five working days. During 200 unsuccessful deliveries, the frequencies from Monday to Friday are $32$, $41$, $29$, $50$ and $48$.

A conservationist compares the mean time taken to identify a bird species using two field guides. Independent random samples give the following summaries.

Guide A: $n_A=10$, $\bar{x}_A=42.3$ seconds, $s_A=5.10$ seconds.

Guide B: $n_B=12$, $\bar{x}_B=38.1$ seconds, $s_B=4.60$ seconds.

Times in both populations are assumed normal with equal unknown variances.

Two independent laboratories measure the concentration of a mineral in samples from two water sources. The measurements are assumed to come from normal populations with equal unknown variances.

Source P: $n_P=15$, $\bar{x}_P=6.42$, $s_P=0.51$.

Source Q: $n_Q=13$, $\bar{x}_Q=6.05$, $s_Q=0.46$.

The mass $M$, in grams, of a product is claimed to follow $N(50,8^2)$. A sample of 300 products is grouped into the intervals $M<42$, $42\leq M<50$, $50\leq M<58$ and $M\geq58$. The observed frequencies are $37$, $112$, $111$ and $40$.

A school compares the mean waiting times at two canteens. The observed difference in sample means is $2.8$ minutes. Under the null hypothesis of equal population means, 5000 random reallocations of the recorded waiting times are performed. In 118 reallocations, the difference is at least $2.8$ minutes in the specified direction.

An online course provider records the device used and whether each of 190 randomly selected learners completed a course. The observed frequencies are shown in the table.

The breaking force of a type of cable is assumed to be normally distributed. The population standard deviation is unknown. A sample of 16 cables has mean breaking force $102.8\ \text{kN}$ and sample standard deviation $5.60\ \text{kN}$. A manufacturer claims that the population mean is $100\ \text{kN}$.

For a random sample of 12 farms, a researcher records annual rainfall and crop yield. The sample product moment correlation coefficient is $r=0.492$. The researcher claims that greater rainfall is associated with greater crop yield.

A bottling company fills containers with juice. Fill volume is normally distributed with population mean $\mu$ and known population standard deviation $8\ \text{ml}$. A random sample of $49$ containers is used to test

The significance level is $5\%$.

Ten musicians test a new breathing exercise. For each musician, the increase in the time for which a note can be sustained is calculated as

The sample of differences has mean $\bar d=1.80\ \text{s}$ and sample standard deviation $s_d=1.40\ \text{s}$. The differences are assumed to come from a normal population.

Two independent cooling systems are tested using separate electronic devices. Cooling times are assumed to be normally distributed with equal but unknown population variances. Summary statistics are shown for the two systems.

A marine scientist investigates the relationship between water temperature and the growth rate of an algae species at $14$ locations. The paired variables are assumed to follow a bivariate normal distribution. For all $14$ locations, the sample product moment correlation coefficient is $r=0.667$.

A national park classifies $200$ volunteers by age group and preferred conservation activity. The observed frequencies are shown in the table. A chi-squared test is used to investigate whether age group and preferred activity are independent.

A renewable-energy company predicts that maintenance requests occur in five categories with probabilities $0.10$, $0.20$, $0.30$, $0.25$ and $0.15$. A random sample of $200$ requests gives observed frequencies $12$, $48$, $71$, $43$ and $26$, respectively.

A language-learning application claims that a learner answers each vocabulary question correctly with probability $0.50$. A researcher tests whether an adaptive feature increases this probability. Each learner answers $20$ independent questions. One learner answers $15$ correctly. A simulation under the null hypothesis produces $10\,000$ samples, of which $218$ contain at least $15$ correct answers.

A digital randomizer is intended to produce each of six symbols with equal probability. In $120$ trials, the observed frequencies are $11$, $17$, $22$, $19$, $21$ and $30$.

A call centre models the number of complaints received in a $30$-minute interval by a Poisson distribution with mean $2$. During $100$ intervals, the observed numbers of complaints are grouped as $0$, $1$, $2$, $3$, and at least $4$, with observed frequencies $10$, $25$, $31$, $17$ and $17$.

The thickness of a glass panel is normally distributed with known population standard deviation $0.24\ \text{mm}$. A sample of 25 panels is used to test
$H_0:\mu=8.00\qquad\text{against}\qquad H_1:\mu<8.00$
at the $1\%$ significance level.