Statistics Paper 4 November 2012

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

State the smallest possible amount which may have been raised by one child.

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

Calculate the mean amount raised.

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

Calculate the standard deviation of the amounts raised.

A discrete random variable X takes the values 0, 1 and 2 with probabilities $P_0$, $P_1$ and $P_2$ respectively. It is given that $E(X)=1,2$ and $\operatorname{Var}(X)=0,36$.

Find $E(X^{2})$.

A discrete random variable X takes the values 0, 1 and 2 with probabilities $P_0$, $P_1$ and $P_2$ respectively. It is given that $E(X)=1,2$ and $\operatorname{Var}(X)=0,36$.

Find the value of $P_2$.

A discrete random variable X takes the values 0, 1 and 2 with probabilities $P_0$, $P_1$ and $P_2$ respectively. It is given that $E(X)=1,2$ and $\operatorname{Var}(X)=0,36$.

Find the value of $P_0$.

A student wishes to be enrolled at one of the three Zimbabwean universities BUSE, NUST and MSU. It is equally likely that the student will apply to any one of these three universities. The probability that the student will be admitted at BUSE is 60 %, while the probabilities that the student will be admitted at NUST and at MSU are 45 % and 35 % respectively.

Find the probability that the student applies to BUSE and is admitted there.

A student wishes to be enrolled at one of the three Zimbabwean universities BUSE, NUST and MSU. It is equally likely that the student will apply to any one of these three universities. The probability that the student will be admitted at BUSE is 60 %, while the probabilities that the student will be admitted at NUST and at MSU are 45 % and 35 % respectively.

Find the probability that the student will not be admitted at any one of the universities.

A student wishes to be enrolled at one of the three Zimbabwean universities BUSE, NUST and MSU. It is equally likely that the student will apply to any one of these three universities. The probability that the student will be admitted at BUSE is 60 %, while the probabilities that the student will be admitted at NUST and at MSU are 45 % and 35 % respectively.

Find the probability that the student applied to NUST, given that the student was not admitted into any one of the universities.

The masses of passengers on a flight are normally distributed with mean $\mu$ and standard deviation $\sigma$. 1,7 % of the passengers had masses greater than 75 kg and 98 % had masses more than 45 kg.

Find the value of $\sigma$.

The masses of passengers on a flight are normally distributed with mean $\mu$ and standard deviation $\sigma$. 1,7 % of the passengers had masses greater than 75 kg and 98 % had masses more than 45 kg.

Find the value of $\mu$.

A continuous random variable X has probability density function

$f(x)=\begin{cases} ax^{2} & 0\le x\le 1 \\ \frac{1}{4}(7-3x) & 1\le x\le \frac{7}{3} \\ 0 & \text{otherwise}\end{cases}$

where $a$ is a constant. Find the value of $a$.

A continuous random variable X has probability density function

$f(x)=\begin{cases} ax^{2} & 0\le x\le 1 \\ \frac{1}{4}(7-3x) & 1\le x\le \frac{7}{3} \\ 0 & \text{otherwise}\end{cases}$

where $a$ is a constant, and it may be assumed that $a=1$.

Find the mean of X.

A continuous random variable X has probability density function

$f(x)=\begin{cases} ax^{2} & 0\le x\le 1 \\ \frac{1}{4}(7-3x) & 1\le x\le \frac{7}{3} \\ 0 & \text{otherwise}\end{cases}$

where $a$ is a constant, and it may be assumed that $a=1$.

Find the median of X.

20 % of the employees of an organisation joined a Housing Scheme.

From a random sample of 8 employees of the organisation, find the probability that only 2 employees joined the scheme.

20 % of the employees of an organisation joined a Housing Scheme.

From a random sample of 8 employees of the organisation, find the probability that less than 3 employees joined the scheme.

20 % of the employees of an organisation joined a Housing Scheme.

A random sample of 100 employees of the organisation is taken. Find the variance of the number of employees in the sample who joined the scheme.

20 % of the employees of an organisation joined a Housing Scheme.

A larger random sample of 100 employees of the organisation was taken. Calculate the probability that less than 25 employees joined the scheme.

A transport company has 2 vans available for daily hire to ferry vegetables to a market. The daily demand for these vans may be assumed to have a Poisson distribution with mean 2.

Find the probability that no van is demanded on a particular day.

A transport company has 2 vans available for daily hire to ferry vegetables to a market. The daily demand for these vans may be assumed to have a Poisson distribution with mean 2.

Find the probability that exactly 2 vans are demanded on a particular day.

A transport company has 2 vans available for daily hire to ferry vegetables to a market. The daily demand for these vans may be assumed to have a Poisson distribution with mean 2.

Calculate the probability that on a particular day the transporter is not able to meet the demand for these vans.

A transport company has 2 vans available for daily hire to ferry vegetables to a market. The daily demand for these vans may be assumed to have a Poisson distribution with mean 2.

Calculate the probability that exactly 4 hirings will occur in two consecutive days.

A truck is carrying boxes of bathing soap and washing powder. The mass of a box of bathing soap is normally distributed with mean 65 kg and variance 8, and the mass of a box of washing powder is normally distributed with mean 50 kg and variance 7. All the boxes are independent of one another.

Find the probability that a box of bathing soap weighs less than 62 kg.

A truck is carrying boxes of bathing soap and washing powder. The mass of a box of bathing soap is normally distributed with mean 65 kg and variance 8, and the mass of a box of washing powder is normally distributed with mean 50 kg and variance 7. All the boxes are independent of one another.

Find the standard deviation of the total mass of 2 randomly chosen boxes of bathing soap.

A truck is carrying boxes of bathing soap and washing powder. The mass of a box of bathing soap is normally distributed with mean 65 kg and variance 8, and the mass of a box of washing powder is normally distributed with mean 50 kg and variance 7. All the boxes are independent of one another.

Find the probability that 2 randomly chosen boxes of bathing soap weigh a total of more than 140 kg.

A truck is carrying boxes of bathing soap and washing powder. The mass of a box of bathing soap is normally distributed with mean 65 kg and variance 8, and the mass of a box of washing powder is normally distributed with mean 50 kg and variance 7. All the boxes are independent of one another.

Find the probability that the mass of a box of bathing soap is greater than $\frac{4}{3}$ of the mass of a box of washing powder.

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

Calculate the expected frequency for employees with 'A' level qualifications and low earnings.

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

Calculate the expected frequency for employees with a degree and low earnings.

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

State the number of degrees of freedom for this test.

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

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

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

Calculate the value of the chi-squared test statistic.

The table shows a random sample of 4 140 employees classified by educational qualifications and by earnings.

A chi-squared test is carried out, at the 5 % level of significance, of whether there is any association between qualification and earnings.

The calculated value of the test statistic is 45,2 and the critical value at the 5 % level is 9,488. What is the correct conclusion?

The percentage score, x, was recorded for 250 'O' level Mathematics students and the data is summarised by $\sum(x-80)=700$ and $\sum(x-80)^{2}=25\,142$. The population mean and variance of X are denoted by $\mu$ and $\sigma^{2}$ respectively.

Calculate the sample mean score.

The percentage score, x, was recorded for 250 'O' level Mathematics students and the data is summarised by $\sum(x-80)=700$ and $\sum(x-80)^{2}=25\,142$. The population mean and variance of X are denoted by $\mu$ and $\sigma^{2}$ respectively.

Show that the unbiased estimate of $\sigma^{2}$ is 93,1, correct to one decimal place, by calculating its value.

The percentage score, x, was recorded for 250 'O' level Mathematics students and the data is summarised by $\sum(x-80)=700$ and $\sum(x-80)^{2}=25\,142$. The population mean and variance of X are denoted by $\mu$ and $\sigma^{2}$ respectively.

The unbiased estimate of $\sigma^{2}$ is 93,1. Calculate the value of the test statistic for testing $H_0$: $\mu = 85$ against $H_1$: $\mu < 85$.

The percentage score, x, was recorded for 250 'O' level Mathematics students and the data is summarised by $\sum(x-80)=700$ and $\sum(x-80)^{2}=25\,142$. The population mean and variance of X are denoted by $\mu$ and $\sigma^{2}$ respectively.

The test statistic for $H_0$: $\mu = 85$ against $H_1$: $\mu < 85$ is $z = -3,61$, and the test is carried out at the 5 % level of significance. What is the correct conclusion?

The percentage score, x, was recorded for 250 'O' level Mathematics students and the data is summarised by $\sum(x-80)=700$ and $\sum(x-80)^{2}=25\,142$. The population mean and variance of X are denoted by $\mu$ and $\sigma^{2}$ respectively.

The sample mean is 82,8 and the unbiased estimate of $\sigma^{2}$ is 93,1. Calculate a symmetric 95 % confidence interval for the mean score.

A farmer supplies potatoes daily to a hypermarket. The masses of potatoes supplied and the fuel consumed by the delivery truck per trip were recorded for eight such trips.

The equation of the regression line of Y on X is of the form $Y=a+bX$. Find the value of the constant $b$.

A farmer supplies potatoes daily to a hypermarket. The masses of potatoes supplied and the fuel consumed by the delivery truck per trip were recorded for eight such trips.

The equation of the regression line of Y on X is of the form $Y=a+bX$, where $b=0,907$. Find the value of the constant $a$.

A farmer supplies potatoes daily to a hypermarket. The masses of potatoes supplied and the fuel consumed by the delivery truck per trip were recorded for eight such trips.

The regression line of Y on X is $Y=9,62+0,907X$, where X is the mass in tonnes and Y is the volume of fuel in litres. Use it to estimate the maximum mass of potatoes that may be transported by 14 litres of fuel.

A farmer supplies potatoes daily to a hypermarket. The masses of potatoes supplied and the fuel consumed by the delivery truck per trip were recorded for eight such trips.

Find the product moment correlation coefficient between the mass of potatoes and the volume of fuel consumed.

A farmer supplies potatoes daily to a hypermarket. The masses of potatoes supplied and the fuel consumed by the delivery truck per trip were recorded for eight such trips.

The product moment correlation coefficient for these data is 0,988. What does this say about the relationship between fuel consumption and the mass of potatoes?