A Level Statistics Practice Questions

A group of twenty people played a game. The frequency distribution of their scores is: score 1 with frequency 2, score 2 with frequency 5, score 4 with frequency 7, and score $x$ with frequency 6. If the mean score is 5, what is the value of $x$?

A group of twenty people played a game with scores 1, 2, 4 and 10, having frequencies 2, 5, 7 and 6 respectively (mean score 5). What is the variance of this distribution?

A head of school can contact parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively, using only one method. The probability that parents receive the message is 0.6 if e-mail is used, 0.8 if letter is used and 1 if cellphone is used. What is the probability that the parents receive the message?

A head of school contacts parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively. The probability parents receive the message is 0.6 for e-mail, 0.8 for letter and 1 for cellphone, so the overall probability of receiving the message is 0.82. Given that the parents received the message, what is the probability that it was sent by e-mail?

Which of the following is not one of the four traditional components of a time series?

Under which conditions can the Poisson distribution be used as an approximation to the Binomial distribution?

Potato seeds are packed in packets of 200 seeds, with on average 2% of the seeds in a packet rotten. A packet is substandard if it contains 5 or more rotten seeds. Using a Poisson approximation with mean $\lambda=200\times0.02=4$, what is the probability that a packet is substandard, correct to 3 decimal places?

A random sample of 80 pockets of manure gives $\sum x=79.53$ and $\sum x^2=100.4621$ (in kg). Testing $H_0:\mu=1.10$ against $H_1:\mu<1.10$, what is the value of the test statistic $t$, to 2 decimal places?

A test of $H_0:\mu=1.10$ kg against $H_1:\mu<1.10$ kg for the mass of manure pockets gives a test statistic of $t=-1.82$ with 79 degrees of freedom, compared against a one-tailed 5% critical value of $-1.664$. What is the correct conclusion?

Six sales representatives made the following numbers of calls to potential customers: 7, 6, 8, 6, 1, 2, with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. What is the product moment correlation coefficient between number of calls and sales turnover, to 2 decimal places?

Six sales representatives made calls (7, 6, 8, 6, 1, 2) with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. Using the least squares regression equation of turnover on number of calls, what is the gradient (slope) of the regression line, to 3 decimal places?

A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. What is the probability that the student is early (not late) for school?

A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. Given that the student arrives early on a particular day, what is the probability, as a fraction in lowest terms, that the student travelled on foot?

The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the median time, in minutes?

The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the interquartile range (Q3 minus Q1) of these times, in minutes?

The times to the nearest minute taken by 12 students to complete a task are 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44, giving Q1 = 42, median = 44.5, and Q3 = 46.5 minutes (IQR = 4.5). The upper fence for outliers is Q3 + 1.5×IQR = 53.25 minutes. What can be concluded about the value 55 and the overall shape of the distribution?

According to the Central Limit Theorem, for a sufficiently large sample size $n$ drawn from any population with mean $\mu$ and variance $\sigma^2$, the distribution of the sample mean $\bar{X}$ is approximately

120 bags of mealie-meal have mean mass 750g and standard deviation 3.9g. What is the standard error of the mean, correct to 3 decimal places?

120 bags of mealie-meal have mean mass 750g and standard deviation 3.9g. Using a 95% confidence interval with $z = 1.96$, what is the upper bound of the confidence interval for the mean mass, in grams, correct to 2 decimal places?

An ice cream vendor's sales are 40% chocolate, 35% vanilla and 25% peach. The probability of a cone sale is 0.80 for chocolate, 0.60 for vanilla and 0.40 for peach. What is the overall probability that a randomly selected sale is by cone?

The volume of dish washing liquid in a satchet is normally distributed with mean 400 ml and standard deviation 45 ml. The volume of fabric softener in a bottle is normally distributed with mean 650 ml and standard deviation 50 ml, independently of the satchets. Find the probability that the total volume of 4 randomly chosen satchets is less than the total volume of 2 randomly chosen bottles.

A machine packs washing powder into packets with mean weight 2 kg. The weights are normally distributed, and 10% of packets weigh less than 1.95 kg. Find the standard deviation of the packet weights, correct to 3 decimal places (in kg).

A machine packs washing powder into packets that are normally distributed with mean 2 kg and standard deviation 0.039 kg. Find the proportion of packets that weigh more than 2.10 kg.

A continuous random variable X has probability density function $f(x) = \dfrac{3}{8}x^2$ for $0 \leq x \leq 2$. Find the value of the cumulative distribution function $F(1)$.

Which of the following is an advantage of using telephone interviews as a way of collecting data?

When selecting a number of objects from a set, which condition makes the selection a permutation rather than a combination?

What is the formula, in terms of n and r, for the number of permutations of r objects chosen from n distinct objects?

Whenever there is a power-cut, a school is equally likely (probability 1/3 each) to switch on one of its 3 generators A, B or C. The independent probabilities of a breakdown are 0.2 for A, 0.3 for B and 0.25 for C. For a randomly chosen day with a power-cut, what is the probability that there was a generator breakdown?

Using the same power-cut scenario (each generator equally likely to be switched on with probability 1/3; independent breakdown probabilities 0.2 for A, 0.3 for B, 0.25 for C), given that there was a generator breakdown, what is the probability that it was generator C?

An unbiased die with faces marked 1, 2, 2, 3, 3, 3 is rolled twice. If X is the total score on the two rolls, what is P(X = 4), as a fraction out of 36?

For a data set with D: 0, 5, 10, 15, 20, 25, 30, 35 and corresponding M: 90, 82, 56, 68, 58, 46, 30, 20, what is the product moment correlation coefficient r, to 3 decimal places?

For a data set with D: 0, 5, 10, 15, 20, 25, 30, 35 and corresponding M: 90, 82, 56, 68, 58, 46, 30, 20, the product moment correlation coefficient works out to r is approximately -0.957. Which statement best describes this correlation?

A random sample of 100 observations from a population gave $\sum(x - 50) = 123.5$. What is the unbiased estimate of the population mean $\mu$, to 3 decimal places?

A company receives on average 6 orders per day, following a Poisson distribution. What is the probability that no more than 2 orders will be received on a given day, to 3 decimal places?

Which statement correctly distinguishes a statistic from a parameter?

A sample of 10 toothpick lengths (cm): 4.99, 4.96, 5.00, 4.98, 5.01, 4.95, 4.96, 4.97, 4.99, 4.97 gives a sample mean of 4.978 cm and standard deviation of about 0.107 cm. Testing H0: mu = 5 against H1: mu is not equal to 5 at the 1% significance level gives a t-statistic of about -0.649, compared with a critical value of t(9) = 3.250. What is the conclusion?

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$, $f(x) = k(x-8)$ for $3 \le x \le 8$, and $f(x) = 0$ otherwise, where $k$ is a constant. Since the total area under $f$ must equal 1, what is the value of $k$?

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$ and $f(x) = -\dfrac{1}{20}(x-8)$ for $3 \le x \le 8$ (zero otherwise). What is $E(X)$, correct to 2 decimal places?

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$ and $f(x) = -\dfrac{1}{20}(x-8)$ for $3 \le x \le 8$ (zero otherwise). What is the median of $X$, correct to 2 decimal places?

A method of collecting statistical data involves obtaining information from every single member of a population, rather than from a sample of it. What is this method of data collection called?

In a survey of 100 athletes at a marathon, the amount of water taken, in litres, was recorded as: 0-0.5 litres by 8 athletes, 0.5-1.0 litres by 20 athletes, 1.0-1.5 litres by 29 athletes, 1.5-2.0 litres by 22 athletes, and 2.0-2.5 litres by 21 athletes. Using the midpoint of each class, what is the estimated mean amount of water taken, in litres?

In a survey of 100 athletes at a marathon, the amount of water taken, in litres, was recorded as: 0-0.5 litres by 8 athletes, 0.5-1.0 litres by 20 athletes, 1.0-1.5 litres by 29 athletes, 1.5-2.0 litres by 22 athletes, and 2.0-2.5 litres by 21 athletes. Using the midpoint of each class and a mean of 1.39 litres, what is the estimated standard deviation of the amount of water taken, correct to 3 decimal places?

The volume of dish washing liquid in satchets is normally distributed with mean 400 ml and standard deviation 45 ml, independent of the volume of fabric softener in bottles, which is normally distributed with mean 650 ml and standard deviation 50 ml. Let W be the total volume of 2 randomly chosen bottles of fabric softener minus the total volume of 4 randomly chosen satchets of dish washing liquid. What is the standard deviation of W, in ml (to 1 decimal place)?

A new detergent is made by mixing the contents of 1 satchet of dish washing liquid, mean volume 400 ml, with 2 bottles of fabric softener, mean volume 650 ml per bottle. What is the expected total volume of the new detergent, in ml?

A machine packs washing powder into packets with mean weight 2 kg. The weights are normally distributed, and 10% of packets weigh less than 1,95 kg. What is the standard deviation of the packet weights, in kg (to 3 decimal places)?

Packets of washing powder have weights normally distributed with mean 2 kg and standard deviation 0,039 kg. What proportion of packets weigh more than 2,10 kg? Give your answer to 3 decimal places or as a percentage.

A continuous random variable X has probability density function f(x) = (3/8)x^2 for 0 <= x <= 2. What is the cumulative distribution function F(x) for 0 <= x <= 2?

A continuous random variable X has cumulative distribution function F(x) = x^3/8 for 0 <= x <= 2. What is the median m of X? Give your answer to 3 decimal places.

A continuous random variable $X$ has probability density function $f(x) = 0.1x + k$ for $4 \leq x \leq 6$, $f(x) = 0.3$ for $6 \leq x \leq 8$, and $f(x) = 0$ otherwise. Find the value of the constant $k$.

A continuous random variable $X$ has probability density function $f(x) = 0.1x - 0.3$ for $4 \leq x \leq 6$, $f(x) = 0.3$ for $6 \leq x \leq 8$, and $f(x) = 0$ otherwise. Find $P(5 \leq X \leq 7)$.

Examination marks are displayed as below, with 1 alongside a leaf 3 standing for the mark 13 %.

The marks obtained by candidates in a mathematics examination are shown below, where a stem of 1 with a leaf of 3 means 13 %.

A manufacturing plant uses three machines, A, B and C, in its production process. The total daily output contributions of machines A, B and C are 40%, 45% and 15% respectively. It is known that 4% of the tins produced by A are defective, 3% of those produced by B are defective, and 1% of those produced by C are defective. What is the probability that one tin chosen at random from the day's production is defective?

A manufacturing plant uses three machines, A, B and C, contributing 40%, 45% and 15% of daily output respectively, with defect rates of 4%, 3% and 1% for A, B and C respectively. Given that a randomly chosen tin is defective, what is the probability, to 3 decimal places, that it came from machine B?

A factory produces two types of nut and bolt with normally distributed masses. Type A bolts have mean mass 20.5 g and Type A nuts have mean mass 5 g; Type B bolts have mean mass 20 g and Type B nuts have mean mass 4.7 g. Each bolt is fitted with two nuts. What is the mean, in grams, of (total mass of a Type A bolt-and-nuts unit) minus (total mass of a Type B bolt-and-nuts unit)?

A factory produces two types of nut and bolt with independent, normally distributed masses, each with standard deviation 0.2 g. Type A bolts have mean 20.5 g and Type A nuts have mean 5 g; Type B bolts have mean 20 g and Type B nuts have mean 4.7 g. Each bolt is fitted with two nuts. What is the probability that the total mass of a Type A bolt-and-nuts unit is greater than the total mass of a Type B bolt-and-nuts unit?

Weekly supply to car dealer B follows a Poisson distribution with mean $\frac{2}{3}$ cars. What is the probability that, in a given week, fewer than three cars are supplied to B?

Weekly supply to car dealer A follows a Poisson distribution with mean $\frac{4}{3}$ cars, and weekly supply to car dealer B independently follows a Poisson distribution with mean $\frac{2}{3}$ cars. What is the probability that, in a given week, fewer than three cars in total are supplied to A and B combined?

A social scientist recorded ARD rate ($X\%$) and divorce rate ($Y\%$) over 6 periods, with $n=6$, $\sum X=684$, $\sum Y=64$, $\sum X^2=92312$, $\sum Y^2=926$, $\sum XY=9106$. What is the product moment correlation coefficient, to 3 decimal places?

A social scientist recorded ARD rate ($X\%$) and divorce rate ($Y\%$) over 6 periods, with $n=6$, $\sum X=684$, $\sum Y=64$, $\sum X^2=92312$, $\sum Y^2=926$, $\sum XY=9106$. What is the gradient of the regression line of $y$ upon $x$, to 3 decimal places?

A driver records the times (minutes) taken to complete a 50 km lap over 30 laps: Time 23 (frequency 3), 24 (7), 25 (8), 26 (6), 27 (3), 28 (2), 29 (1). What is the sample mean lap time, to 1 decimal place?

A sample of 30 lap times has mean 25.3 minutes and standard deviation 1.218 minutes. Testing $H_0:\mu=25$ against $H_1:\mu>25$, what is the value of the test statistic?

The diameters of 25 steel rods have a sample mean of 0.980 cm and a standard deviation of 0.015 cm. Assuming the diameters are normally distributed with this standard deviation, find the 99% confidence interval for the population mean diameter.

The masses (g) of 24 sweets, read off a stem-and-leaf diagram and placed in ascending order, are: 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45. Find the median mass.

Using the same 24 sweet masses (g) 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45, find the mode.

A sweet with mass greater than 1.2 g is classified as large. From the 24 masses (g) 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45, calculate the mean mass of the large sweets.

A stem-and-leaf diagram of pocket money (in dollars) received by a group of girls is given below, with key $3|30 = \$3.30$:

Using the same pocket money data (20 girls, values in cents: 50,50,50,75,100,100,100,150,175,200,200,200,250,250,300,325,330,375,450,550, mean \$2.14), what is the standard deviation, to the nearest cent?

A school's lower sixth intake is 55% from its own O-level pupils and 45% from other schools. Of those who did O-level elsewhere, 90% pass A-level; of those who did O-level at the school, 70% pass. A recent A-level graduate is selected at random. What is the probability that this pupil passed A-level?

Using the same school (55% own O-level pupils, 45% from elsewhere; pass rates 70% and 90% respectively, overall pass probability 0.79), given that a graduate passed A-level, what is the probability that they did O-level outside the school?

An experiment has only two possible outcomes. The first outcome occurs with probability $p$ and the second outcome occurs with probability $p^2$. Find the value of $p$, correct to 3 decimal places.

A survey in a city shows that the probability a person is in favour of capital punishment is 0.55 and the probability a person is against it is 0.45. If two people are selected at random, find the probability that at least one of them favours capital punishment.

The continuous random variable $X$ has probability density function $f(x)=\frac{1}{28}(6x-4)$ for $2\le x\le4$ (and 0 otherwise). Find $E(X)$, giving your answer as a fraction or correct to 3 decimal places.

The table below shows the masses of 200 students measured to the nearest kg.

A number X is randomly selected from the interval $(-\pi,\ \pi)$, so that X is uniformly distributed over that interval. Which of these is the cumulative distribution function of X?

A discrete random variable X has $P(X=x)=\dfrac{x-1}{16}$ for $x=2,3,4,5$ and $P(X=x)=\dfrac{9-x}{16}$ for $x=6,7,8$. Calculate $E(X)$.

A discrete random variable Y follows a geometric distribution with variance 12. Find the probability of success on a single trial.

A roulette wheel contains 38 numbers of which 18 are red, 18 are black and 2 are green. When the wheel is spun it is equally likely to land on any of the 38 numbers, and successive spins are independent. The wheel is spun twice. Find the probability that the ball lands on red both times, giving your answer correct to 4 decimal places.

A group of 50 children raised money for charity. The amount raised by each child, to the nearest dollar ($), is recorded in the table.

A random variable X has E(X) = 10 and Var(X) = 9. Find E(Y) where Y = 2X - 3.

In a particular survey involving 7076 households, 0.0248% were in favour of amending city council by-laws. The number in favour is to be modelled by a Poisson approximation to the binomial.

X is a continuous random variable with probability density function $f(x)$. The density is the curve $y=x^{2}$ for $0\le x\le 1$, followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.

X is a continuous random variable with probability density function $f(x)$. The density is the curve $y=x^{2}$ for $0\le x\le 1$, followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.

A random variable W has a geometric distribution, $W \sim Geo(p)$, counting the number of trials up to and including the first success. Given that $Var(W) = 30$, find the value of p.

The table shows the number of children below the age of 15 known to have suffered from measles in 2009 in a certain village.

A bag contains 24 counters of which 6 are red, 8 are green and 10 are yellow. Three counters are taken from the bag at random without replacement.

The diameters of washers produced by a machine follow a Normal distribution with standard deviation 0.1 mm and unknown mean $\mu$. The mean is to be set so that the probability that a diameter exceeds 2.0 mm is 0.03.

X is a discrete random variable with a geometric distribution, $X \sim Geo(0.4)$, counting the number of trials up to and including the first success.