Statistics Paper 4 June 2010

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.

Find the mean of that Poisson distribution.

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.

Find the value of h.

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.

Write down the mode.

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.

Find the lower quartile.

A local bookshop reported that 11% of books sold in a year are romance novels. The bookshop sells 316 books on a given day, and the number of those that are romance novels is to be modelled by a normal approximation to the binomial.

Find the standard deviation of that normal distribution.

A car is never kept in a garage at night. The probability that a night is wet in summer is 0,80. On the morning following a wet night the probability that the car does not start is 0,25. On the morning following a dry night this probability is 0,04. The starter performance is independent of each morning.

Find the probability that the car does not start on a given morning in summer.

A car is never kept in a garage at night. The probability that a night is wet in summer is 0,80. On the morning following a wet night the probability that the car does not start is 0,25. On the morning following a dry night this probability is 0,04. The starter performance is independent of each morning.

Find the probability that the night was wet, given that the car did not start.

A random variable X has a binomial distribution, $X\sim Bin(n,\,p)$, with $E(X)=\frac{3}{2}$ and $Var(X)=\frac{9}{8}$.

Find the value of p.

A random variable X has a binomial distribution, $X\sim Bin(n,\,p)$, with $E(X)=\frac{3}{2}$ and $Var(X)=\frac{9}{8}$.

Find the value of n.

A random variable X has a binomial distribution, $X\sim Bin(n,\,p)$, with $E(X)=\frac{3}{2}$ and $Var(X)=\frac{9}{8}$.

Find $P(X\ge 3)$.

The following table shows the observed frequency distribution of the number of matches per box in a random sample of 100 boxes.

Calculate the unbiased estimate of the mean number of matches per box.

The following table shows the observed frequency distribution of the number of matches per box in a random sample of 100 boxes.

Calculate the unbiased estimate of the variance of the number of matches per box.

The following table shows the observed frequency distribution of the number of matches per box in a random sample of 100 boxes.

Determine a 95% confidence interval for the mean number of matches per box.

A Harare commuter records the duration of the delays of her commuter train, in minutes and seconds, on twenty mornings: 1:15, 12:17, 5:49, 2:09, 8:54, 6:22, 2:43, 2:39, 6:24, 4:21, 3:13, 0:28, 1:47, 10:37, 1:25, 3:10, 4:58, 3:16, 5:28, 1:32.

The delays are grouped into 2-minute intervals, each interval taking the delays from its lower end up to but not including its upper end. State the frequency of the interval from 2 to 4 minutes.

A Harare commuter records the duration of the delays of her commuter train, in minutes and seconds, on twenty mornings: 1:15, 12:17, 5:49, 2:09, 8:54, 6:22, 2:43, 2:39, 6:24, 4:21, 3:13, 0:28, 1:47, 10:37, 1:25, 3:10, 4:58, 3:16, 5:28, 1:32.

The delays are grouped into 2-minute intervals, each interval taking the delays from its lower end up to but not including its upper end. State the interval with the highest frequency.

A Harare commuter records the duration of the delays of her commuter train, in minutes and seconds, on twenty mornings: 1:15, 12:17, 5:49, 2:09, 8:54, 6:22, 2:43, 2:39, 6:24, 4:21, 3:13, 0:28, 1:47, 10:37, 1:25, 3:10, 4:58, 3:16, 5:28, 1:32.

A stem and leaf plot is drawn using the number of minutes as the stem and the number of seconds as the leaf. State how many leaves appear on the stem 1.

A Harare commuter records the duration of the delays of her commuter train, in minutes and seconds, on twenty mornings: 1:15, 12:17, 5:49, 2:09, 8:54, 6:22, 2:43, 2:39, 6:24, 4:21, 3:13, 0:28, 1:47, 10:37, 1:25, 3:10, 4:58, 3:16, 5:28, 1:32.

A railway spokesman claims that 80% of trains leave within 5 minutes of the scheduled time of departure. Using these twenty mornings, calculate the percentage of trains that left within 5 minutes.

John bought a packet of 15 tomato seeds for his garden. The probability of any one seed not germinating is 0.01, independently of the others.

Find the probability that every one of the 15 seeds germinates.

John bought a packet of 15 tomato seeds for his garden. The probability of any one seed not germinating is 0.01, independently of the others.

Find the probability that he gets at most 13 tomato plants from his seeds.

In a competition, the title holder has a probability of 0.97 of hitting the target on each of 200 shots, independently. The number of shots he misses is to be modelled by a Poisson approximation to the binomial.

Find the mean of that Poisson distribution.

In a competition, the title holder has a probability of 0.97 of hitting the target on each of 200 shots, independently. The number of shots he misses is to be modelled by a Poisson approximation to the binomial.

Use that Poisson approximation to find the probability that he hits the target at least 190 times in the 200 shots.

During a practice session a gunman fires at a target from the same place until he succeeds. Independently for each attempt, the probability of hitting the target is 0,8. X is the number of attempts up to and including his first success.

Find the probability that he will need at least 3 attempts to his first success.

During a practice session a gunman fires at a target from the same place until he succeeds. Independently for each attempt, the probability of hitting the target is 0,8. X is the number of attempts up to and including his first success.

Find the variance of X.

During a practice session a gunman fires at a target from the same place until he succeeds. Independently for each attempt, the probability of hitting the target is 0,8. X is the number of attempts up to and including his first success.

Y is the time in minutes he will take over his practice session, where Y = 3X + 2. Find the mean of Y.

During a practice session a gunman fires at a target from the same place until he succeeds. Independently for each attempt, the probability of hitting the target is 0,8. X is the number of attempts up to and including his first success.

Y is the time in minutes he will take over his practice session, where Y = 3X + 2. Find the variance of Y.

A very small proportion of a country's adult population has some knowledge of a particular foreign language. A random sample of adults is taken, and the number in the sample with some knowledge of the language is modelled by a Poisson distribution with mean $\lambda$. The sample size is chosen so that the probability of obtaining at least one adult having some knowledge of the language is 0.98.

Find the value of $\lambda$.

For a particular foreign language, only a very small proportion r % of a country's adult population have some knowledge of it. Using a Poisson approximation, the number n of adults that must be selected at random so that the probability of obtaining at least one adult having some knowledge of the language is 0.98 is given by $n=\dfrac{391.2}{r}$.

Find n for the case where r = 1.2.

In a country, a proportion $r=\frac{1}{3}$ per cent of the adult population have some knowledge of a particular foreign language. A random sample of $n=917$ adults is taken, and the number in the sample with some knowledge of the language is modelled by a Poisson distribution.

Find the probability that precisely 5 adults in the sample will have some knowledge of the language.

Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.

A chi-squared test is carried out at the 5% level of significance to determine whether the grades obtained are independent of the school entered.

Calculate the expected frequency of School X pupils obtaining Grade C.

Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.

A chi-squared test is carried out at the 5% level of significance to determine whether the grades obtained are independent of the school entered.

State the number of degrees of freedom for the test.

Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.

A chi-squared test is carried out at the 5% level of significance to determine whether the grades obtained are independent of the school entered.

Calculate the value of the chi-squared test statistic.

Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.

A chi-squared test is carried out at the 5% level of significance to determine whether the grades obtained are independent of the school entered.

State the critical value of the test statistic at the 5% level of significance.

Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.

A chi-squared test is carried out at the 5% level of significance to determine whether the grades obtained are independent of the school entered.

State the conclusion of the test.

The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.

The regression line of goods vehicles registered on goods carried is $X=a+bY$.

Find the value of b.

The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.

The regression line of goods vehicles registered on goods carried is $X=a+bY$.

Find the value of a.

The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.

The regression line of goods vehicles registered on goods carried is $X=827+9,75Y$. Use this equation to estimate the number of goods vehicles registered, in thousands, if the amount of goods carried is 120 million tonnes.

The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.

Calculate the product moment correlation coefficient between goods carried and the number of goods vehicles registered.

The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.

The product moment correlation coefficient between goods carried and the number of goods vehicles registered is 0,708. Comment on the relationship between the two quantities.