Central limit theorem + Linear combination of multiple independent NORMAL random variables: notes and practice questions

A beverage company uses an automated machine to fill plastic bottles with water. The actual volume of water, $V$, in a randomly chosen bottle is normally distributed with a mean of 500 mL and a standard deviation of 15 mL.

In a quality control check, if a randomly chosen bottle contains less than 485 mL of water, the company fails the check.

Find the probability that the company fails this quality control check.

A quality assurance manager suggests that a fairer check would be to pass if the mean volume of eight randomly chosen bottles is greater than 485 mL.

Find the probability of passing the quality control check if the manager's suggestion is followed.

A beverage company produces bottles of orange juice. The volume of juice in each bottle, in mL, can be modelled by a normal distribution with a mean of 1005 mL and a standard deviation of 12 mL.

Find the probability that a randomly selected bottle contains less than its labelled volume of 1000 mL.

Find the upper quartile of the volumes of the bottles.

The bottles are packed into cartons, with 8 bottles in each carton. The volumes of juice in the bottles are independent of each other.

Find the probability that the total volume of juice in a carton exceeds 8050 mL.

The length of a newly manufactured "Micro-Resistor" ($R$) is normally distributed with a mean of $12.0$ mm and a standard deviation of $0.5$ mm. A circuit board requires five such resistors to be placed in series. Assume the lengths of the resistors are independent.

(a) Calculate the probability that the total length of these five resistors exceeds $62.0$ mm.

The time taken to assemble a "Control Unit" ($C$) is normally distributed with a mean of $45.0$ minutes and a standard deviation of $2.0$ minutes. The time taken to test a "Sensor Module" ($S$) is normally distributed with a mean of $14.0$ minutes and a standard deviation of $1.0$ minute. Assume the assembly time and testing time are independent.

(b) Determine the probability that the assembly time of a randomly selected Control Unit is less than three times the testing time of a randomly selected Sensor Module.

(a) A logistics company handles two types of packages: small (S) and large (L). The weight of each type of package follows a normal distribution with parameters as shown in this table:

Type of Package

Mean weight (kg)

Standard deviation (kg)

S (Small)

$1.5$

$0.2$

L (Large)

$5.0$

$0.5$

One package of each type is selected at random. Find the probability that the large package weighs less than four times the weight of the small package.

(b) One large package and three small packages are selected at random. Find the probability that the large package weighs more than the total weight of the three small packages.

A factory produces light bulbs. The probability that a randomly selected light bulb is defective is $p = 0.15$. For a single light bulb, let the discrete random variable $X$ equal $1$ if the bulb is defective, and $0$ if it is not defective.

(a) For the random variable $X$, write down

(i) the mean

(ii) the variance.

A quality control manager inspects a random sample of $n = 120$ light bulbs. Let the sample mean $\bar{X} = \frac{1}{n} \sum_{i=1}^{120} X_i$ be the proportion of defective bulbs in the sample. The Central Limit Theorem states that the distribution of $\bar{X}$ can be approximated by a normal distribution for a sufficiently large $n$.

(b) Write down the

(i) mean

(ii) variance

of $\bar{X}$, when approximated by a normal distribution.

(c) Use this normal approximation to estimate $P(\bar{X} > 0.18)$.

(d) Hence, write down $P\left(\sum_{i=1}^{120} X_i > k\right)$ for an appropriate value of $k$, when using this normal approximation.

(e) Let $T = \sum_{i=1}^{120} X_i$ be the total number of defective light bulbs in the sample.

(i) State the true distribution that $T$ satisfies.

(ii) Find the exact value of $P(T > 21.5)$, which can be construed as the probability that more than $21$ bulbs are defective.

A factory produces specialized electronic components. The total "quality score" of a component, $T$, is a combination of scores from three independent inspection stages:

• Stage 1: Automated visual inspection. The score $X$ from this stage has an expectation of $2.5$ and a standard deviation of $0.7$.

• Stage 2: Manual functional test. A batch of $8$ critical functions are tested, and the score $Y$ is the number of functions that pass. Each function has a $0.35$ probability of passing, independently.

• Stage 3: Environmental stress test. The score $Z$ from this stage, representing the number of successful stress cycles, follows a Poisson distribution with a mean of $4.2$.

The overall quality score for a component is given by $T = X + 2Y + Z$.

Calculate the expected value and variance of the total quality score $T$.

Given that the distribution of $T$ can be approximated by a Normal distribution, find the probability that a randomly selected component has a total quality score between $10$ and $14$ (inclusive of $10$, exclusive of $14$).

The factory manager wants to ensure that the mean total quality score of a sample of $n$ components is within $0.8$ units of the true mean, with a probability of at least $0.95$. Find the minimum sample size $n$ required.

(a) State the Central Limit Theorem, including the conditions under which it applies.

(b) A logistics company is investigating the average commute time for its employees. It is known that the commute times have a population variance of $\sigma^2 = 9$ minutes$^2$. A random sample of $64$ employees reported their daily commute times, and the sample mean was $\bar{x} = 40$ minutes.

Calculate the $95\%$ confidence interval for the population mean commute time, $\mu$.

A company manufactures specialized medical devices. Each device consists of a main circuit board and a protective casing. The weight of the circuit board, $C$, is normally distributed with a mean of $120$ g and a standard deviation of $5$ g. The weight of the protective casing, $P$, is normally distributed with a mean of $30$ g and a standard deviation of $2$ g. The weights of the circuit board and the casing are independent.

Find the probability that a randomly chosen complete device has a total weight of less than $145$ g.

A batch of $10$ such devices is to be packed into a container. The container has a maximum weight capacity of $1510$ g. The weight of each device is independent.

Find the probability that the total weight of the $10$ devices is greater than the capacity of the container.

The company sources a critical microchip component from two different suppliers, Supplier X and Supplier Y. An engineer claims that microchips from Supplier Y have a lower response time than those from Supplier X. To test this claim, a random sample is taken from each supplier.

The eight microchips in the sample from Supplier X have response times, in milliseconds (ms), of:

$19.65, 19.65, 19.79, 20.75, 20.97, 21.15, 22.28, 22.37$.

Find the

(i) mean response time for the sample from Supplier X.

Find the

(ii) unbiased estimate of the population variance for the sample from Supplier X.

The seven microchips in the sample from Supplier Y have a mean response time of $19.5$ ms and an unbiased estimate of the population standard deviation ($s_{n-1}$) of $1.05$ ms.

Perform a suitable test, at the $5\%$ significance level, to test the engineer's claim that microchips from Supplier Y have a lower response time than those from Supplier X. You may assume the response times of microchips from each supplier are normally distributed with equal population variance.

A company manufactures steel beams. The total length of these beams, $L$, is normally distributed with a mean of $400$ cm and a standard deviation of $1.2$ cm. For installation purposes, engineers often consider the 'effective half-length' of the beam, which is half of its total length.

(a) Find the distribution of the effective half-length of the beams.

These beams are designed to fit into pre-fabricated slots. The length of the slots, $S$, is also normally distributed with a mean of $201.5$ cm and a standard deviation of $0.45$ cm. Assume that the length of a beam and the length of a slot are independent.

(b) Calculate the probability that the clearance (the difference between the slot length and the beam's effective half-length, $S-H$) is less than $0.5$ cm.

A customer service department receives contacts from two independent sources: phone calls and emails. The number of phone calls received per hour can be modelled by a Poisson distribution with a mean of $5.5$. The number of emails received per hour can be modelled by a Poisson distribution with a mean of $3.8$.

(a)(i) Find the probability that the department receives at most $20$ phone calls in a $3$-hour period.

(a)(ii) Find the probability that the department receives a total of more than $12$ contacts (phone calls and emails combined) in one hour.

Each phone call takes an average of $4$ minutes to handle, and each email takes an average of $7$ minutes to handle. Let $M$ be the total time (in minutes) spent handling contacts in one hour.

(b)(i) Find $E(M)$.

(b)(ii) Find $Var(M)$.

(c) State one reason why the distribution of $M$ cannot be a Poisson distribution.

(d) The customer service department operates for an $8$-hour shift. Use the Central Limit Theorem to find the probability that the mean time spent per hour handling contacts during this shift is greater than $45$ minutes.

Tech Solutions Inc. is a customer support company. They are analyzing their call center efficiency. Over a long period, they collect data on the number of calls, $C$, already in the queue when a new customer's call arrives. The probability distribution of $C$ is shown in the following table.

Find the probability that there are at least two calls in the queue when a new customer's call arrives.

Find E($C$).

The time in seconds, $T$, taken to resolve a single customer's issue can be modelled by the normal distribution $T \sim \text{N}(130, 25^2)$.

The company's management estimates that the expected total time a new customer will wait before their issue is resolved can be found by calculating E($C$) $\times$ E($T$).

Find the value of E($C$) $\times$ E($T$).

The company considers a service time to be 'long' if it takes more than three minutes to resolve a single customer's issue.

Using the distribution of $T$ given above, find the probability that it takes more than three minutes to resolve a randomly selected customer's issue.

Find the probability it takes more than four minutes in total to resolve two randomly selected customers' issues. You may assume all service times are independent of all other service times.

The company assumes that when a new customer's call arrives, the person at the front of the queue has only just reached a support agent. They also assume that if there are three or more customers already in the queue, the new customer will definitely wait more than three minutes before being served.

Using these assumptions and the probabilities for $C$ given in the table above,

find the probability a customer just arriving at the call center will wait more than three minutes before being served.

The company has a policy to employ more staff if the probability that a customer has to wait more than three minutes before being served is greater than $0.35$.

Hence state whether Tech Solutions Inc. will decide to employ more staff.

The scores of students on a standardized mathematics test are modelled by a normal distribution with a mean of $68$ marks and a standard deviation of $12$ marks.

If $15$ students are chosen independently and at random, find the probability that more than $5$ of them scored above $80$ marks.

A study group consists of $10$ students chosen independently and at random. Write down the probability distribution that models the total score of these $10$ students.

The total score of $10$ students in a study group exceeds $s$ marks on less than $2.5\%$ of all occasions. Find the value of $s$.

A tutoring center claims that $35\%$ of students who take the standardized mathematics test achieve a distinction. A rival tutor believes that this is an underestimation of the true proportion. The rival tutor collects data from $180$ of their clients, finding that $72$ of them achieved a distinction.

The rival tutor decides to perform a test using the binomial distribution on their data for the population proportion, $p$, that achieve a distinction.

(i) State two assumptions that the rival tutor makes in order to conduct the test.

(ii) Write down the null and the alternative hypotheses for the rival tutor's test, in terms of $p$.

(iii) Using the data from the rival tutor's sample, perform the test at a $5\%$ significance level to determine whether the rival tutor's belief is reasonable.

The battery life, in hours, of a particular smartphone model, $L$, can be modelled by a normal distribution with a mean of 24 hours and a standard deviation of 2 hours.

(a) Find the probability that a randomly selected smartphone has a battery life greater than 27 hours.

Two smartphones are selected at random and independently of each other.

(b) (i) Find the probability that both smartphones have a battery life greater than 27 hours.

(b) (ii) Find the probability that their total battery life is greater than 52 hours.

A software update is released which is claimed to improve battery life. The manufacturer decides to take a random sample of 20 smartphones to test this claim at the 1% significance level, assuming the standard deviation of the battery life has not changed.

(c) Write down the null and alternative hypotheses for the test.

(d) Find the critical region for this test.

Unknown to the manufacturer, the software update has resulted in all smartphones having a 5% longer battery life than the original model.

(e) Find the mean and standard deviation of the battery life for smartphones with the update.

(f) Find the probability of a Type II error in the manufacturer’s test.

(a) The length of an aluminum segment for a custom telescope tube is modelled by a normal distribution with mean $220$ cm and standard deviation $7$ cm.

Find the probability that a randomly chosen aluminum segment is less than $210$ cm in length.

(b) Find the endpoints of the interval, symmetric about the mean, such that $90\%$ of the aluminum segments have a length that lies in this interval.

(c) The total length of the telescope tube assembly, $T$, is formed by two independent aluminum segments ($A_1, A_2$) and one central optical housing ($H$). The length of the optical housing is modelled by a normal distribution with mean $120$ cm and standard deviation $5$ cm, and is independent of the lengths of the aluminum segments.

(i) Calculate the mean and standard deviation of $T$.

(ii) Hence find $P(545 < T < 570)$.

(d) Due to a design change, the total length of the telescope tube assembly is now normally distributed with a mean of $565$ cm and a standard deviation of $11$ cm. The model for the length of an aluminum segment does not change.

(i) Use the results for the sum of independent random variables to find a revised value for the mean of the optical housing length.

(ii) Find the standard deviation of the revised optical housing length.

(e) Under this revised model, $85\%$ of the optical housings have a length less than $k$ cm. Calculate the value of $k$.

A factory produces two types of bottled drinks, "Energize" and "Refresh". The volumes of liquid, in ml, are modelled as independent normal distributions.

The volume of an "Energize" bottle, $A$, is modelled as $A \sim N(500, 10^2)$.
The volume of a "Refresh" bottle, $B$, is modelled as $B \sim N(330, 8^2)$.

(a) Find the probability that a randomly chosen "Energize" bottle contains between 495 ml and 510 ml of liquid.

(b) Find the probability that the volume of a randomly chosen "Energize" bottle is more than 1.5 times the volume of a randomly chosen "Refresh" bottle.

(c) Three "Energize" bottles and two "Refresh" bottles are emptied into a large jug with a capacity of 2.2 litres (2200 ml). Find the probability that the total volume of liquid overflows the jug.