Statistics Paper 4 June 2016

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.

Writing the condition as $P\left(Z < \dfrac{2 - \mu}{0.1}\right) = 0.97$, state the value of z for which $\Phi(z) = 0.97$.

The diameters of washers produced by a machine follow a Normal distribution with standard deviation 0.1 mm. The mean is to be set so that there is a probability of only 3% that a diameter exceeds 2.0 mm.

Given that $\Phi(1.881) = 0.97$, find the mean diameter in mm.

A continuous random variable X has probability density function f(x) with $f(x) = 0$ for $x < 1$ and for $x > 5$. Between 1 and 5 the density is made of three straight pieces: it is constant at k from $x = 1$ to $x = 3$, rises in a straight line from k at $x = 3$ to 2k at $x = 4$, then falls in a straight line from 2k at $x = 4$ to 0 at $x = 5$.

Which equation in k does the total area under the density give?

A continuous random variable X has a probability density function that is constant at $\dfrac{2}{9}$ for $1 \le x \le 3$, rises in a straight line from $\dfrac{2}{9}$ at $x = 3$ to $\dfrac{4}{9}$ at $x = 4$, then falls in a straight line from $\dfrac{4}{9}$ at $x = 4$ to 0 at $x = 5$, and is 0 elsewhere.

Which expression gives f(x) on the rising piece $3 \le x \le 4$?

Bag A contains 3 red balls and 2 white balls. Bag B contains 2 red balls and 3 white balls. A bag is selected at random and two balls are drawn from it, one after the other without replacement.

Find the probability that the two balls drawn are red.

Bag A contains 3 red balls and 2 white balls. Bag B contains 2 red balls and 3 white balls. A bag is selected at random and two balls are drawn from it, one after the other without replacement.

Given that the two balls drawn are red, find the probability that they came from bag A.

An unbiased tetrahedral die has faces marked 1, 2, 3 and 4. Two such dice are tossed and X is the sum of the two scores.

Find $P(X = 5)$.

Two unbiased tetrahedral dice with faces marked 1, 2, 3 and 4 are tossed and X is the sum of the two scores, so that $P(X = x)$ is $\dfrac{1}{16},\ \dfrac{2}{16},\ \dfrac{3}{16},\ \dfrac{4}{16},\ \dfrac{3}{16},\ \dfrac{2}{16},\ \dfrac{1}{16}$ for $x = 2, 3, 4, 5, 6, 7, 8$ in turn.

A player pays $1 to play. If the sum is 2, 3 or 4 the player wins nothing; if the sum is 5, 6 or 7 the player wins $2; for a sum greater than 7 the player wins $4.

Find the expected gain, in dollars, from each game.

A stem and leaf diagram records the number of hectares owned by farmers around a small town, with the key 1/3 meaning 13 hectares:

Find the median number of hectares.

The number of hectares owned by 25 farmers, arranged in order, is

1, 4, 7, 11, 13, 18, 19, 20, 21, 22, 24, 27, 28, 30, 30, 32, 33, 34, 35, 37, 37, 42, 43, 45, 47.

Using the position $\dfrac{1}{4}(n + 1)$, find the lower quartile.

The number of hectares owned by 25 farmers, arranged in order, is

1, 4, 7, 11, 13, 18, 19, 20, 21, 22, 24, 27, 28, 30, 30, 32, 33, 34, 35, 37, 37, 42, 43, 45, 47.

Using the positions $\dfrac{1}{4}(n + 1)$ and $\dfrac{3}{4}(n + 1)$ for the quartiles, find the interquartile range.

A dozen loaves of bread taken at random from a large batch had masses, in grammes, of

741; 701; 834; 829; 808; 660; 659; 739; 472; 865; 851 and 801.

Assuming the masses come from a Normal distribution, find the unbiased estimate of the population mean, in grammes, correct to 1 decimal place.

For a random sample of 12 loaves of bread the masses, in grammes, gave $\sum x = 8960$ and $\sum x^{2} = 6\ 828\ 956$.

Assuming the masses come from a Normal distribution, find the unbiased estimate of the population variance, correct to 2 decimal places.

A batch of 50 loaves gave a mean mass of 750 grammes, and the population variance of the loaves is known to be 10 816.

Using the table value $z = 1.64$ for 90% confidence, find the upper limit of the 90% confidence interval for the mean mass, in grammes, correct to 2 decimal places.

The number of passengers arriving at a taxi rank per hour has a Poisson distribution with mean 2.

Calculate the probability that in a particular hour no passenger arrives, correct to 3 decimal places.

The number of passengers arriving at a taxi rank per hour has a Poisson distribution with mean 2. At the beginning of an hour there are 4 taxis available for hire, and each taxi takes only one passenger.

Calculate the probability that 4 taxis is an insufficient number for that hour, correct to 4 decimal places.

The number of passengers arriving at a taxi rank per hour has a Poisson distribution with mean 2.

Find the probability that exactly 2 passengers arrive in 2 consecutive hours, correct to 4 decimal places.

A fair coin is tossed 100 times and X is the number of heads obtained, so $X \sim B\left(100,\ \dfrac{1}{2}\right)$.

The distribution of X is to be approximated by a Normal distribution with the same mean and variance. State the variance of that Normal distribution.

A fair coin is tossed 100 times and X is the number of heads obtained, so $X \sim B\left(100,\ \dfrac{1}{2}\right)$, which is approximated by $N(50,\ 25)$.

Using a continuity correction, calculate the probability that the number of heads obtained is less than 37, correct to 4 decimal places.

For a fair coin tossed 100 times, the probability of obtaining fewer than 37 heads is 0.0035. Now 2 000 such coins are each tossed 100 times, and Y is the number of those 2 000 sets that give fewer than 37 heads, so $Y \sim B(2\ 000,\ 0.0035)$.

Which approximation to the distribution of Y is appropriate, and why?

The number Y of sets, out of 2 000 sets of 100 coin tosses, that give fewer than 37 heads has $Y \sim B(2\ 000,\ 0.0035)$, which is approximated by a Poisson distribution with mean 7.

Find the probability that fewer than 37 heads are obtained more than 3 times, correct to 3 decimal places.

The mass of the contents of a packet of brand M powdered milk is Normally distributed with mean 500 g and standard deviation 8 g. The mass of its packaging is Normally distributed with mean 20 g and standard deviation 2 g, independently of the contents.

State the variance of the total mass of a randomly chosen packet.

The mass of the contents of a packet of brand M powdered milk is Normally distributed with mean 500 g and standard deviation 8 g, and the mass of its packaging is Normally distributed with mean 20 g and standard deviation 2 g, independently of the contents.

Find the probability that a randomly chosen packet has a total mass exceeding 525 grammes, correct to 4 decimal places.

The mass of the contents of a packet of brand M powdered milk is Normally distributed with mean 500 g and standard deviation 8 g, and packets are independent of one another.

Find the probability that the total mass of the contents of three randomly chosen packets exceeds 1 515 grammes, correct to 4 decimal places.

The contents of a packet of brand N powdered milk are Normally distributed with mean 405 g and standard deviation 6 g. The contents of a packet of brand M are Normally distributed with mean 500 g and standard deviation 8 g. All packets are independent.

Let $V = (U_{1} + \ldots + U_{5}) - (X_{1} + \ldots + X_{4})$, where the U are five brand N packets and the X are four brand M packets. State the variance of V.

The contents of a packet of brand N powdered milk are Normally distributed with mean 405 g and standard deviation 6 g, and the contents of a packet of brand M are Normally distributed with mean 500 g and standard deviation 8 g, all packets independent.

Find the probability that the contents of five randomly chosen brand N packets weigh more than the contents of four randomly chosen brand M packets, correct to 4 decimal places.

For 7 cars of the same make and model, m is the total distance travelled since the car was new, in thousands of km, and d is the distance the car travels on one litre of petrol, in km:

Which statement best describes the relationship between m and d?

For 7 cars, m is the total distance travelled since new, in thousands of km, and d is the distance travelled on one litre of petrol, in km:

The totals are $\sum m = 1700$, $\sum d = 130$, $\sum m^{2} = 575\ 000$ and $\sum md = 27\ 850$.

Find the gradient of the regression line of d on m, correct to 4 significant figures.

For 7 cars, m is the total distance travelled since new, in thousands of km, and d is the distance travelled on one litre of petrol, in km, with $\sum m = 1700$ and $\sum d = 130$.

The regression line of d on m has gradient $-0.02295$. Find its intercept on the d axis, correct to 4 significant figures.

For 7 cars of one model, the regression line of d on m is

$$d = 24.15 - 0.02295m,$$

where m is the total distance travelled since the car was new, in thousands of km, and d is the distance travelled on one litre of petrol, in km.

Use the line to estimate the distance travelled per litre by a car which has travelled a total distance of 450 000 km. Give your answer in km correct to 3 significant figures.

For 7 cars of one model, the regression line of d on m is $d = 24.15 - 0.02295m$, where m is the total distance travelled since the car was new, in thousands of km, and d is the distance travelled on one litre of petrol, in km.

What does the gradient $-0.02295$ mean in this context?

The maximum heights cleared by 100 male athletes in a high jump competition were grouped as follows, with the class centres used being 163, 168, 173, 178, 183 and 188 cm in turn:

Find the mean height, in cm.

For the maximum heights cleared by 100 male athletes, grouped with class centres 163, 168, 173, 178, 183 and 188 cm and frequencies 4, 18, 37, 26, 10 and 5, the mean is 174.75 cm and $\sum fx^{2} = 3\ 057\ 075$.

Find the unbiased estimate of the variance, correct to 2 decimal places.

The maximum heights cleared by 100 male athletes are to be modelled by a Normal distribution with mean 174.75 cm and standard deviation 5.7899 cm. The expected frequencies for the first three classes are 3.73 for $161 < h \le 165$, 16.00 for $165 < h \le 170$ and 31.11 for $170 < h \le 175$.

Calculate the expected frequency for the class $175 < h \le 180$, correct to 2 decimal places.

(You may use $\Phi(0.043) = 0.5172$ and $\Phi(0.907) = 0.8178$.)

The maximum heights cleared by 100 male athletes are modelled by a Normal distribution, and the expected frequencies for the six classes are 3.73, 16.00, 31.11, 30.05, 14.39 and z, the last class being $185 < h \le 190$ which carries the whole upper tail.

Find z, correct to 2 decimal places.

A chi-squared goodness of fit test is to be carried out on 100 athletes' heights in six classes, with observed frequencies 4, 18, 37, 26, 10 and 5 and expected frequencies 3.73, 16.00, 31.11, 30.05, 14.39 and 4.73 respectively.

Which classes must be combined before the test, and why?

After pooling, a chi-squared goodness of fit test has four classes with observed frequencies 22, 37, 26 and 15 and expected frequencies 19.73, 31.11, 30.05 and 19.12 respectively.

Calculate the value of $\chi^{2} = \sum \dfrac{(O - E)^{2}}{E}$, correct to 2 decimal places.

A chi-squared goodness of fit test compares 100 athletes' heights with a Normal model. After pooling there are four classes, and both the mean and the variance of the Normal model were estimated from the data itself. The test statistic is 2.81 and the test is at the 5% level.

What are the degrees of freedom and the conclusion?