Statistics Paper 4 June 2013

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

Age is a continuous quantity, so the class boundaries are 0 to 1, 1 to 2.5, 2.5 to 4.5, 4.5 to 9.5 and 9.5 to 14.5. Calculate the mean age, in years, of the children who suffered from the disease.

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

In a histogram of these data the bars are drawn to frequency density, since the classes are of unequal width. Taking the boundaries of the 5 - 9 class as 4.5 to 9.5, find the frequency density of that class.

During the 2010 World Cup in a certain city, the probability that there was electricity on any particular day was $\dfrac{1}{3}$. In the case that there was no electricity, a generator would be switched on. Independently, the probability that John watched a soccer match being screened live was $\dfrac{1}{4}$.

Given that John watched a soccer match, find the probability that there was no electricity. Give your answer as a fraction in its lowest terms.

During the 2010 World Cup in a certain city, the probability that there was electricity on any particular day was $\dfrac{1}{3}$. In the case that there was no electricity, a generator would be switched on. Independently, the probability that John watched a soccer match being screened live was $\dfrac{1}{4}$.

Find the probability that on a particular day there was no electricity and John watched a soccer match.

An ice-cream vendor records his daily takings ($x$) over a period of 30 days. The results are summarised by $\sum x = 900$ and $\sum x^{2} = 34\ 000$.

Find the unbiased estimate of the population mean.

An ice-cream vendor records his daily takings ($x$) over a period of 30 days. The results are summarised by $\sum x = 900$ and $\sum x^{2} = 34\ 000$.

Find the unbiased estimate of the population variance.

An ice-cream vendor records his daily takings ($x$) over a period of 30 days. The results are summarised by $\sum x = 900$ and $\sum x^{2} = 34\ 000$.

The unbiased estimate of the population variance is 241.38. Assuming the daily takings are normally distributed and using $z = 1.96$, calculate the upper limit of the 95% confidence interval for the mean amount he receives.

The probability that a learner driver passes the driving test at the Vehicle Inspection Department is $\dfrac{1}{4}$, and the attempts are independent of one another. A learner counts the number of attempts made up to and including the one on which the test is passed.

State the statistical distribution which best models the number of attempts.

The probability that a learner driver passes the driving test at the Vehicle Inspection Department is $\dfrac{1}{4}$, and the attempts are independent of one another. A learner counts the number of attempts made up to and including the one on which the test is passed.

Find the mean number of attempts.

The probability that a learner driver passes the driving test at the Vehicle Inspection Department is $\dfrac{1}{4}$, and the attempts are independent of one another. A learner counts the number of attempts made up to and including the one on which the test is passed.

Find the variance of the number of attempts.

The probability that a learner driver passes the driving test at the Vehicle Inspection Department is $\dfrac{1}{4}$, and the attempts are independent of one another. A learner counts the number of attempts made up to and including the one on which the test is passed.

Find the smallest value of $n$ for which the probability that the learner needs only $n$ or fewer attempts is at least 0.7.

A continuous random variable X has probability density function

$$f(x)=\begin{cases}\dfrac{1}{2} & 0 \le x < 0.5\\[4pt] \dfrac{1}{5}(3-x), & 0.5 \le x \le 3\\[4pt] 0, & \text{otherwise.}\end{cases}$$

Which of these describes the graph of $f(x)$?

A continuous random variable X has probability density function

$$f(x)=\begin{cases}\dfrac{1}{2} & 0 \le x < 0.5\\[4pt] \dfrac{1}{5}(3-x), & 0.5 \le x \le 3\\[4pt] 0, & \text{otherwise.}\end{cases}$$

Find the median of X.

A continuous random variable X has probability density function

$$f(x)=\begin{cases}\dfrac{1}{2} & 0 \le x < 0.5\\[4pt] \dfrac{1}{5}(3-x), & 0.5 \le x \le 3\\[4pt] 0, & \text{otherwise.}\end{cases}$$

Evaluate $P(x < 1.2)$.

A machine on a production line is set to make components with a mean diameter of 2 cm. A random sample of 10 components had their diameters, in cm, measured:

2.17 1.93 2.02 1.97 2.00 2.01 2.02 1.89 1.99 2.01

Calculate the mean diameter of the sample, in cm.

A machine on a production line is set to make components with a mean diameter of 2 cm. A random sample of 10 components had their diameters, in cm, measured:

2.17 1.93 2.02 1.97 2.00 2.01 2.02 1.89 1.99 2.01

The ten readings give $\sum x = 20.01$ and $\sum x^{2} = 40.0879$. Calculate the unbiased estimate of the population standard deviation, in cm.

A machine on a production line is set to make components with a mean diameter of 2 cm. A random sample of 10 components had their diameters, in cm, measured:

2.17 1.93 2.02 1.97 2.00 2.01 2.02 1.89 1.99 2.01

The mean diameter is to be tested against the set value of 2 cm at the 5% level of significance, with the spread estimated from the sample itself. State the positive critical value of the test statistic.

A machine on a production line is set to make components with a mean diameter of 2 cm. A random sample of 10 components had their diameters, in cm, measured:

2.17 1.93 2.02 1.97 2.00 2.01 2.02 1.89 1.99 2.01

For these data the test statistic works out at 0.043 and the two-tailed 5% critical values on 9 degrees of freedom are $\pm 2.262$. What is the conclusion of the test?

A political party claims that it commands 60% of the voters. To test this, a random sample of 300 potential voters were asked which party they would vote for, and 160 confirmed that they would vote for that party. The claim is tested at the 10% level of significance.

Calculate the sample proportion of voters who confirmed that they would vote for the party.

A political party claims that it commands 60% of the voters. To test this, a random sample of 300 potential voters were asked which party they would vote for, and 160 confirmed that they would vote for that party. The claim is tested at the 10% level of significance.

Taking $H_{0}: p = 0.6$ against $H_{1}: p < 0.6$, and the sample proportion as $\dfrac{160}{300}$, calculate the value of the test statistic z.

A political party claims that it commands 60% of the voters. To test this, a random sample of 300 potential voters were asked which party they would vote for, and 160 confirmed that they would vote for that party. The claim is tested at the 10% level of significance.

The test statistic works out at $z = -2.357$ and the 10% one-tailed critical value is $-1.282$. What is the conclusion?

Mangoes are boxed into cartons, each containing 500 mangoes. The probability that any one mango is rotten is 0.002, independently of the others. A buyer returns any carton that contains 4 or more rotten mangoes.

Find the expected number of rotten mangoes per carton.

Mangoes are boxed into cartons, each containing 500 mangoes. The probability that any one mango is rotten is 0.002, independently of the others. A buyer returns any carton that contains 4 or more rotten mangoes.

State the most appropriate statistical distribution for modelling the number of rotten mangoes in a carton, with the reason for it.

Mangoes are boxed into cartons, each containing 500 mangoes. The probability that any one mango is rotten is 0.002, independently of the others. A buyer returns any carton that contains 4 or more rotten mangoes.

Two cartons of mangoes are chosen at random. Find the expected number of rotten mangoes in the two cartons together.

Mangoes are boxed into cartons, each containing 500 mangoes. The probability that any one mango is rotten is 0.002, independently of the others. A buyer returns any carton that contains 4 or more rotten mangoes.

Find the probability that a carton of mangoes is not returned.

Mangoes are boxed into cartons, each containing 500 mangoes. The probability that any one mango is rotten is 0.002, independently of the others. A buyer returns any carton that contains 4 or more rotten mangoes.

Two cartons of mangoes are chosen at random. Find the probability that between them they contain at least three rotten mangoes.

The table shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

Assuming there is no association between ownership and social class, calculate the expected number of executive households that own a satellite dish.

The table shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

State the number of degrees of freedom for a chi-squared test of association on this table.

The table shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

The expected frequencies are 15.3 and 9.7 for Executive, 19.0 and 11.98 for Managerial, and 57.66 and 36.3 for Working. Calculate the value of the chi-squared test statistic.

The table shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

The test is carried out at the 5% level of significance on 2 degrees of freedom. State the critical value of the chi-squared statistic.

The table shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

The test statistic works out at 2.79 and the 5% critical value on 2 degrees of freedom is 5.991. What is the conclusion of the test?

Oranges are sold in small pockets whose weights are normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. The probability that a randomly chosen pocket weighs more than 3.2 kg is 0.1, and the probability that a randomly chosen pocket weighs less than 2.4 kg is 0.2.

Find $\sigma$, in kg.

Oranges are sold in small pockets whose weights are normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. The probability that a randomly chosen pocket weighs more than 3.2 kg is 0.1, and the probability that a randomly chosen pocket weighs less than 2.4 kg is 0.2.

Find $\mu$, in kg.

Oranges are sold in small pockets whose weights are normally distributed with mean 2.7 kg and standard deviation 0.38 kg. Potatoes are sold in small pockets whose weights are normally distributed with mean 5 kg and standard deviation 0.6 kg. All the pockets are chosen independently of one another.

Find the mean of the combined weight of two orange pockets and one potato pocket, in kg.

Oranges are sold in small pockets whose weights are normally distributed with mean 2.7 kg and standard deviation 0.38 kg. Potatoes are sold in small pockets whose weights are normally distributed with mean 5 kg and standard deviation 0.6 kg. All the pockets are chosen independently of one another.

Find the probability that two randomly chosen orange pockets together weigh less than one randomly chosen pocket of potatoes.

Oranges are sold in small pockets whose weights are normally distributed with mean 2.7 kg and standard deviation 0.38 kg. Potatoes are sold in small pockets whose weights are normally distributed with mean 5 kg and standard deviation 0.6 kg. All the pockets are chosen independently of one another.

Find the probability that two randomly chosen orange pockets and one randomly chosen pocket of potatoes weigh more than 11 kg altogether.

The manager of a clothing shop surveyed ten women, recording each woman's age X in years and her annual expenditure on clothes Y in dollars.

Calculate the mean annual expenditure on clothes, in dollars.

For ten women, the age X in years and the annual expenditure on clothes Y in dollars gave $n = 10$, $\sum x = 320$, $\sum y = 2225$, $\sum x^{2} = 11\,342$, $\sum y^{2} = 545\,425$ and $\sum xy = 64\,430$.

Calculate the gradient of the regression line of Y on X.

For ten women, the age X in years and the annual expenditure on clothes Y in dollars gave $n = 10$, $\sum x = 320$, $\sum y = 2225$, $\sum x^{2} = 11\,342$, $\sum y^{2} = 545\,425$ and $\sum xy = 64\,430$.

The gradient of the regression line of Y on X is $-6.143$, and $\bar{x}=32$, $\bar{y}=222.5$. Which is the equation of that line?

For ten women, the age X in years and the annual expenditure on clothes Y in dollars gave $n = 10$, $\sum x = 320$, $\sum y = 2225$, $\sum x^{2} = 11\,342$, $\sum y^{2} = 545\,425$ and $\sum xy = 64\,430$.

The regression line of Y on X is $Y = -6.143X + 419.08$. Use it to estimate the amount, in dollars, likely to be spent on clothes by a 40 year old woman.

For ten women, the age X in years and the annual expenditure on clothes Y in dollars gave $n = 10$, $\sum x = 320$, $\sum y = 2225$, $\sum x^{2} = 11\,342$, $\sum y^{2} = 545\,425$ and $\sum xy = 64\,430$.

Calculate the product-moment correlation coefficient between age and annual expenditure.