A researcher repeatedly takes random samples of the same size from a population and calculates a $95.5\%$ confidence interval for the population mean from each sample.

The masses, in kilograms, of six independently selected food containers are denoted by $X_1, X_2, \dots, X_6$. Each mass is normally distributed with mean $2.4$ and standard deviation $0.3$. The total shipping mass is

where $0.8$ kg is the mass of the shipping box.

The daily amount spent by a visitor at a market is strongly right-skewed, with mean €$18$ and standard deviation €$12$. A random sample of $49$ visitors is selected and $\bar X$ is their mean daily expenditure.

The lifetime of a type of battery is normally distributed with mean $75$ hours and standard deviation $12$ hours. A random sample of $n$ batteries is selected.

The standard deviation of the diameter of components produced by a factory is known to be $7.2$ mm. A random sample will be used to calculate a $99.5\%$ confidence interval for the population mean diameter.

A random sample of $36$ observations is taken from a normal population with known standard deviation $\sigma$. A $90.5\%$ confidence interval for the population mean is

Two independent random samples of customer ratings are taken for brands A and B. The ratings for each brand are assumed to come from a normal population.

For brand A, $n=25$, $\bar x=82$ and $s=10$.

For brand B, $n=30$, $\bar x=78$ and $s=9$.

A university investigates student satisfaction with its online library. The proposed questionnaire includes the item: “How much do you agree that the convenient and excellent online library has improved your studies?” Responses are recorded on a scale from 1 to 5. A random sample of 36 students gives a mean response of 3.42 and a sample standard deviation of 0.88.

A gift hamper contains three independently selected bags of coffee and two independently selected boxes of tea. The mass $X$, in grams, of a coffee bag is normally distributed with mean $480$ and standard deviation $12$. The mass $Y$, in grams, of a tea box is normally distributed with mean $320$ and standard deviation $8$. The empty hamper has mass $50$ grams. Let $H$ be the total mass of a completed hamper.

The amount paid for an individual home-insurance claim is strongly right-skewed, with population mean €240 and population standard deviation €300. Claim amounts are independent. An insurer studies the mean amount paid for a random sample of 100 claims.

The temperature at which a type of sensor activates is assumed to be normally distributed. For a random sample of 15 sensors, the mean activation temperature is $6.84^\circ\text{C}$ and the sample standard deviation is $1.12^\circ\text{C}$. The population standard deviation is unknown.

A streaming platform asks 400 active subscribers to estimate the number of hours of television watched per week. The sample mean is $7.8$ hours. For active subscribers, the population standard deviation is known to be $2.4$ hours. A researcher wants to use the results to estimate mean weekly television viewing for all adults in the country.

A computer repeatedly selects independent random samples of the same size from a population with fixed mean $\mu$. For each sample it constructs an $80\%$ confidence interval for $\mu$. A total of 250 intervals are constructed, of which 207 contain $\mu$.

The time spent on three independent stages of a delivery process is represented by $A$, $B$ and $C$, measured in minutes. The distributions are $A\sim N(18,16)$, $B\sim N(12,9)$ and $C\sim N(8,4)$. A fixed administration time of 5 minutes is added. The total delivery time is $T=A+B+C+5$.

A filling machine dispenses coffee into jars. The mass dispensed is normally distributed with unknown mean $\mu$ and known standard deviation $12.8\ \text{g}$. A random sample of $64$ jars has mean mass $503.2\ \text{g}$.

An archaeologist measures the depths, in centimetres, of a particular layer at $14$ randomly selected locations. The depths are assumed to come from a normal population. The sample has mean $127.4\ \text{cm}$ and standard deviation $8.6\ \text{cm}$.

A school wants to estimate the mean number of hours of sleep obtained by its senior students on a school night. It sends a questionnaire only to students attending an optional early-morning study club. One item asks, “How many hours of healthy sleep do you normally get?” A total of $120$ students respond, with mean $6.42$ hours and sample standard deviation $1.35$ hours. Assume the sleep times in the intended population are normally distributed.

The time between successive calls to a help centre has a strongly right-skewed distribution with mean $10$ minutes and standard deviation $10$ minutes. A simulation repeatedly selects independent samples and records their means. Three simulations use sample sizes $4$, $36$ and $100$. Their sampling distributions are labelled $A$, $B$ and $C$, but the labels do not state the sample sizes.

A music-streaming company uses responses posted through its application to estimate the daily listening time of all people aged $16$ to $25$ in a country. Its questionnaire asks, “How many minutes do you spend enjoying our excellent music service each day?” For $100$ respondents, the mean is $74$ minutes. The population standard deviation of reported listening time is taken to be $30$ minutes.

Two independent greenhouses use different lighting systems. Plant heights are assumed normally distributed in each greenhouse. For system $A$, $n=20$, $\bar x=34.2\ \text{cm}$ and $s=3.5\ \text{cm}$. For system $B$, $n=22$, $\bar x=36.0\ \text{cm}$ and $s=4.1\ \text{cm}$.

The duration of a sea turtle's nesting visit is strongly right-skewed. Historical research indicates that the population standard deviation is $20$ minutes, but the current population mean is unknown. A random sample of $64$ current visits has mean duration $56$ minutes.

The concentration of a chemical in bottles of cleaning solution has known population standard deviation $3.6\ \text{g L}^{-1}$. A random sample of 64 bottles has mean concentration $18.7\ \text{g L}^{-1}$.

A language school designs two forms, A and B, of a vocabulary test. Both forms are completed by the same 10 students. The correlation between the two sets of scores is $0.93$. Let $D$ be a student's score on form A minus the student's score on form B. The differences are assumed to come from a normal population and have sample mean $4.8$ and sample standard deviation $3.1$.

The breaking strength of a type of climbing cord is assumed to be normally distributed. A random sample of 20 cords has mean breaking strength $52.4\ \text{kN}$ and sample standard deviation $6.5\ \text{kN}$. An engineer constructs the confidence interval $(49.6,55.2)\ \text{kN}$ for the population mean.

A finite population contains 120 measurements from a non-normal distribution with mean 70 and standard deviation 16. A simulation selects independent random samples with replacement and records the sample mean. Simulation A uses samples of size 16, while simulation B uses samples of size 64.

Two independent farms grow the same variety of tomato using different fertilizers. Tomato masses at each farm are assumed to be normally distributed. Farm A samples 16 tomatoes and obtains $\bar x_A=102\ \text{g}$ and $s_A=12\ \text{g}$. Farm B samples 20 tomatoes and obtains $\bar x_B=96\ \text{g}$ and $s_B=10\ \text{g}$.

A physiotherapist records the time, in seconds, taken by each of 12 patients to complete a mobility task before and after a treatment. For each patient, the reduction in completion time is calculated as the time before treatment minus the time after treatment. The reductions are assumed to come from a normal population. Assume that the 12 patient reductions form an independent random sample from this population. They have sample mean $1.85$ seconds and sample standard deviation $2.40$ seconds.

The amount paid for an individual household insurance claim is strongly right-skewed, with mean $2400$ euros and standard deviation $1800$ euros. Independent claims are selected at random. Let $\bar X$ be the mean amount paid for a sample of $81$ claims.

A gift package contains three independently selected packets of tea and two independently selected packets of biscuits. The mass $X$, in grams, of a tea packet is normally distributed with mean $18$ and standard deviation $2$. The mass $Y$, in grams, of a biscuit packet is normally distributed with mean $12$ and standard deviation $1.5$. The box has a fixed mass of $5$ grams. The total mass is
$T=X_1+X_2+X_3+Y_1+Y_2+5$

A biologist tests whether seed-germination times follow a proposed distribution. Six adjacent time categories have observed frequencies $4$, $5$, $11$, $12$, $20$ and $18$, and expected frequencies $2.6$, $4.9$, $9.5$, $14.0$, $22.0$ and $17.0$, respectively. Two parameters of the proposed distribution were estimated from the sample.

A random sample of $25$ measurements is taken from a normal population with unknown mean and standard deviation. A $95\%$ confidence interval for the population mean is
$41.7<\mu<48.3$

An island has $8$ marked ecological plots. A researcher studies samples of $3$ distinct plots selected without replacement. Each sample is chosen by simple random sampling, so each of the $\binom{8}{3}=56$ unordered samples is equally likely. She proposes listing every possible unordered sample to obtain the exact sampling distribution of the sample mean. A colleague instead proposes generating $5000$ random samples by computer.

The output, in kilograms, from a machine during one production cycle is normally distributed. Under setting $X$, the output has mean $50$ and standard deviation $6$. Under setting $Y$, it has mean $53$ and standard deviation $8$. Independent samples of $9$ cycles under $X$ and $16$ cycles under $Y$ are taken. Let
$D=\bar Y-\bar X$

A university calculates an admissions index from an aptitude score $A$ and an interview score $B$ using
$S=0.4A+0.6B$
For the applicant population, $A\sim N(70,8^2)$ and $B\sim N(60,10^2)$. Initially, assume that $A$ and $B$ are independent.