Statistics Paper 4 June 2014

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.

Which calculation gives the probability that exactly 2 of the three counters taken are green?

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 probability that exactly 2 of the counters taken are green is $\dfrac{56}{253}$.

Given that 2 of the counters are green, find the probability that the first counter taken is red.

The probability that a boy hits a target is 0.8. Shots are independent of each other, and during each practice period the boy fires shots until he hits the target.

Find the mean number of shots fired per practice period.

The probability that a boy hits a target is 0.8. Shots are independent of each other, and during each practice period the boy fires shots until he hits the target.

Find the standard deviation of the number of shots fired per practice period.

The probability that a boy hits a target is 0.8. Shots are independent of each other, and during each practice period the boy fires shots until he hits the target.

Find the probability that the boy will need to take at least five shots to hit the target.

A coin is tossed 5 times and at least 4 heads are obtained. A binomial test is used at the 5% level of significance to test whether the coin is biased towards heads, with $H_{0}: p = 0.5$ against $H_{1}: p > 0.5$, so that under $H_{0}$ the number of heads X has the distribution $B(5,\ 0.5)$.

Find $P(X \ge 4)$, the probability of the observed result or a more extreme one under $H_{0}$.

A coin is tossed 5 times and at least 4 heads are obtained. A binomial test is used at the 5% level of significance to test whether the coin is biased towards heads, with $H_{0}: p = 0.5$ against $H_{1}: p > 0.5$, so that under $H_{0}$ the number of heads X has the distribution $B(5,\ 0.5)$. The tail probability works out as $P(X \ge 4) = 0.1875$.

What is the conclusion of the test?

The following are television prices in dollars taken in 40 different shops.

Find the median price, in dollars.

The following are television prices in dollars taken in 40 different shops.

Find the lower quartile of the prices, in dollars.

The following are television prices in dollars taken in 40 different shops.

Find the upper quartile of the prices, in dollars.

The following are television prices in dollars taken in 40 different shops.

The lower quartile of these prices is 140 dollars and the upper quartile is 330 dollars. Find the interquartile range, in dollars.

The probability that a seed grown under specified conditions will germinate and produce a plant is 0.8. A number n of seeds are planted under these conditions and X is the number of them that germinate, so that $X \sim B(n,\ 0.8)$.

Which normal distribution is the suitable approximation to X for large n?

The probability that a seed grown under specified conditions will germinate and produce a plant is 0.8. A number n of seeds are planted under these conditions and X is the number of them that germinate, so that $X \sim B(n,\ 0.8)$. Requiring a probability of at least 0.9 that 60 or more seeds germinate leads to the inequality $n^{2} - 149.2n + 5531.6 \ge 0$.

Solve $n^{2} - 149.2n + 5531.6 = 0$ and give the larger of the two roots, correct to 1 decimal place.

The probability that a seed grown under specified conditions will germinate and produce a plant is 0.8. A number n of seeds are planted under these conditions and X is the number of them that germinate, so that $X \sim B(n,\ 0.8)$. Requiring a probability of at least 0.9 that 60 or more seeds germinate leads to $n^{2} - 149.2n + 5531.6 \ge 0$, whose roots are $n = 68.8$ and $n = 80.4$.

Find the minimum value of n, the number of seeds to be planted.

A random sample of eight observations of a normal variable gave $\bar{x} = 4.65$ and $\sum (x - \bar{x})^{2} = 0.74$.

Calculate the unbiased estimate of the population variance.

A random sample of eight observations of a normal variable gave $\bar{x} = 4.65$ and $\sum (x - \bar{x})^{2} = 0.74$. The unbiased estimate of the population variance is $s^{2} = \dfrac{0.74}{7} = 0.10571$.

Testing $H_{0}: \mu = 4.32$ against $H_{1}: \mu \ne 4.32$, calculate the value of the test statistic t.

A random sample of eight observations of a normal variable gave $\bar{x} = 4.65$ and $\sum (x - \bar{x})^{2} = 0.74$. Testing $H_{0}: \mu = 4.32$ against $H_{1}: \mu \ne 4.32$ at the 5% level of significance gives a test statistic of $t = 2.87$.

What is the conclusion of the test?

The switchboard of a small company handles both incoming and outgoing calls. During lunch hour on any day, the numbers of incoming and outgoing calls are independent and have Poisson distributions with parameters 5 and 3 respectively.

Find the probability that during lunch hour of a randomly chosen day there will be exactly 3 outgoing calls.

The switchboard of a small company handles both incoming and outgoing calls. During lunch hour on any day, the numbers of incoming and outgoing calls are independent and have Poisson distributions with parameters 5 and 3 respectively.

Find the probability that during lunch hour of a randomly chosen day there will be at least 6 incoming calls.

The switchboard of a small company handles both incoming and outgoing calls. During lunch hour on any day, the numbers of incoming and outgoing calls are independent and have Poisson distributions with parameters 5 and 3 respectively.

Find the probability that during lunch hour of a randomly chosen day there will be a combined number of 3 calls through the switchboard.

A random variable X is normally distributed with mean 15 and standard deviation 6. A random sample of 40 is chosen and found to have mean $\bar{X}$.

What is the distribution of $\bar{X}$?

A random variable X is normally distributed with mean 15 and standard deviation 6. A random sample of 40 is chosen and found to have mean $\bar{X}$.

Find $P(\bar{X} > 16)$.

In a one-tailed calculation the value z is needed for which the area to the right of z under the standard normal curve is 0.05, that is $P(Z > z) = 0.05$.

Write down the value of z, correct to 3 decimal places.

A random variable X is normally distributed with mean 15 and standard deviation 6. A random sample of size n is chosen and has mean $\bar{X}$.

Find the sample size n such that $P(\bar{X} > 15.5) = 0.05$.

The data below summarises the altitude x (in metres) above sea level and the mean air temperature y (in $^{\circ}$C) for 10 weather stations.

$$\sum x = 1076 \qquad \sum x^{2} = 211642$$

$$\sum y = 94.2 \qquad \sum xy = 9383.1$$

$$\sum y^{2} = 894.58$$

Calculate the product moment correlation coefficient between x and y.

For 10 weather stations, x is the altitude in metres above sea level and y is the mean air temperature in $^{\circ}$C. The product moment correlation coefficient between x and y works out as $r = -0.91$.

Which comment on the relationship between x and y is correct?

The data below summarises the altitude x (in metres) above sea level and the mean air temperature y (in $^{\circ}$C) for 10 weather stations.

$$\sum x = 1076 \qquad \sum x^{2} = 211642$$

$$\sum y = 94.2 \qquad \sum xy = 9383.1$$

$$\sum y^{2} = 894.58$$

Calculate the gradient of the regression line of y on x.

For 10 weather stations, x is the altitude in metres above sea level and y is the mean air temperature in $^{\circ}$C, and the regression line of y on x is

$$y = 10.265 - 0.00785x.$$

Use this line to estimate the mean air temperature, in $^{\circ}$C, at a place 250 m above sea level.

A manufacturer sells heavy and light vehicles. The cost of a light vehicle, in thousands of dollars, is normally distributed with mean 252 and standard deviation 2. The cost of a heavy vehicle, in thousands of dollars, is normally distributed with mean 1 012 and standard deviation 5. Costs are independent of one another.

One vehicle of each type is selected at random. Write X for the cost of the light vehicle and Y for the cost of the heavy vehicle. Find the mean of $Y - 4X$, in thousands of dollars.

A manufacturer sells heavy and light vehicles. The cost of a light vehicle, in thousands of dollars, is normally distributed with mean 252 and standard deviation 2. The cost of a heavy vehicle, in thousands of dollars, is normally distributed with mean 1 012 and standard deviation 5. Costs are independent of one another.

One vehicle of each type is selected at random. Write X for the cost of the light vehicle and Y for the cost of the heavy vehicle. Find the variance of $Y - 4X$.

A manufacturer sells heavy and light vehicles. The cost of a light vehicle, in thousands of dollars, is normally distributed with mean 252 and standard deviation 2. The cost of a heavy vehicle, in thousands of dollars, is normally distributed with mean 1 012 and standard deviation 5. Costs are independent of one another.

A vehicle of each type is selected at random. Find the probability that the heavy vehicle costs less than 4 times the light vehicle.

A manufacturer sells heavy and light vehicles. The cost of a light vehicle, in thousands of dollars, is normally distributed with mean 252 and standard deviation 2. The cost of a heavy vehicle, in thousands of dollars, is normally distributed with mean 1 012 and standard deviation 5. Costs are independent of one another.

One heavy and four light vehicles are selected at random. Find the probability that the cost of the heavy vehicle is less than the total cost of the four light vehicles.

A policeman claims that the type of accident he attends depends on the colour of the car involved. The table shows the results of 200 accidents he attended.

Assuming that the type of accident and the colour of the car are independent, calculate the expected frequency of fatal accidents involving black cars.

A policeman claims that the type of accident he attends depends on the colour of the car involved. The table shows the results of 200 accidents he attended.

Assuming that the type of accident and the colour of the car are independent, calculate the expected frequency of minor accidents involving white cars.

A policeman claims that the type of accident he attends depends on the colour of the car involved. The table shows the results of 200 accidents he attended.

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

A policeman claims that the type of accident he attends depends on the colour of the car involved. The table shows the results of 200 accidents he attended.

Under the hypothesis that the type of accident and the colour of the car are independent, the expected frequencies are 21, 21 and 18 for black cars, 24.5, 24.5 and 21 for white cars, and 24.5, 24.5 and 21 for red cars, taking minor, serious and fatal in that order.

Calculate the value of the chi-squared test statistic.

A policeman claims that the type of accident he attends depends on the colour of the car involved. The table shows the results of 200 accidents he attended.

A chi-squared test of independence on this table gives $\chi^{2} = 14.7$ on 4 degrees of freedom. The test is carried out at the 10% level of significance.

What is the conclusion?

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

Calculate the mean number of defects per component.

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

The mean of this distribution is 2.25. Calculate the expected frequency of components with 0 defects for the Poisson distribution having the same mean, correct to 1 decimal place.

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

The mean of this distribution is 2.25. Calculate the expected frequency of components with 3 defects for the Poisson distribution having the same mean, correct to 1 decimal place.

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

The expected Poisson frequencies are 10.5, 23.7, 26.7, 20.0, 11.3, 5.1, 1.9 and 0.8 for 0, 1, 2, 3, 4, 5, 6 and $\ge 7$ defects. The classes 5, 6 and $\ge 7$ are pooled into a single class $\ge 5$, and the mean is estimated from the data.

State the number of degrees of freedom for the goodness of fit test.

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

The classes 5, 6 and $\ge 7$ are pooled into one class $\ge 5$, giving observed frequencies 11, 22, 26, 24, 9 and 8 against expected Poisson frequencies 10.5, 23.7, 26.7, 20.0, 11.3 and 7.8.

Calculate the value of the chi-squared test statistic.

One hundred electrical components are tested to see how many defects each has. The results are shown in the table.

A chi-squared goodness of fit test at the 5% level of significance, against the Poisson distribution with the same mean, gives $\chi^{2} = 1.437$ on 4 degrees of freedom.

What is the conclusion?