Paper 3: Statistics and Mechanics

The times, in minutes, that 40 customers spent in a shop are summarised below.

Time, t (min)	0 ≤ t < 10	10 ≤ t < 20	20 ≤ t < 30	30 ≤ t < 60
Frequency	8	14	10	8

(a) Use linear interpolation to estimate the median. (b) A histogram is drawn. Find the frequency density for the class 30 ≤ t < 60. (c) Estimate the mean time.

For 10 days, a café recorded the maximum temperature, x °C, and the number of cold drinks sold, y. The temperatures ranged from 10 °C to 30 °C. The regression line of y on x is y = 12.5 + 3.2x and the product moment correlation coefficient is 0.87. (a) Interpret the value 3.2 in context. (b) Estimate the number of cold drinks sold on a day with a maximum temperature of 25 °C, and comment on the reliability of your estimate. (c) Explain why the line should not be used to estimate sales when x = 40. (d) Test, at the 5% level, whether there is evidence of positive correlation. (One-tailed critical value for n = 10: 0.5494.)

In a sixth form, 30% of students study Spanish. Of the students who study Spanish, 40% also study French. Of the students who do not study Spanish, 20% study French. (a) Find the probability that a randomly chosen student studies French. (b) Given that a student studies French, find the probability that they also study Spanish. (c) Determine whether studying Spanish and studying French are independent. Give a reason.

Mo suspects that a six-sided dice is biased. He rolls it 60 times and gets 17 sixes. (a) Find the critical region for a two-tailed test, at the 5% level, of whether the probability of a six differs from 1/6. Use 2.5% in each tail. (b) State the actual significance level of the test. (c) Carry out the test, stating your conclusion in context.

The masses, W grams, of loaves from a bakery are normally distributed. 10% of loaves weigh more than 520 g and 5% weigh less than 470 g. (a) Find the mean and standard deviation of W. (b) Find P(W > 510). (c) Three loaves are chosen at random. Find the probability that all three weigh more than 510 g.

The lengths of bolts made by a machine are normally distributed with standard deviation 3 mm. The mean length should be 25 mm. A random sample of 16 bolts has mean length 26.4 mm. (a) Test, at the 1% level of significance, whether the mean length differs from 25 mm. (b) Find the critical region for the sample mean in this test. (c) Explain why it is important that the sample is random.

A car starts from rest and accelerates uniformly to 20 m s^-1 in 10 s. It travels at this speed for T seconds, then decelerates uniformly to rest in 8 s. Figure 1 shows the velocity–time graph. The total distance travelled is 1180 m. (a) Find the value of T. (b) Find the deceleration of the car.

Particles A and B, of masses 4 kg and 6 kg, are connected by a light inextensible string over a smooth fixed pulley. The particles hang with the string taut and are released from rest. (a) Find the acceleration of the particles and the tension in the string. (b) After 2 s the string breaks. A has not reached the pulley. Find the further height that A rises after the string breaks.

A box of mass 10 kg rests on rough horizontal ground. The coefficient of friction is 0.3. The box is pulled by a force of 60 N acting at 30° above the horizontal. (a) Find the acceleration of the box. (b) When the box reaches a speed of 6 m s^-1, the force is removed. Find the further distance travelled before the box stops.

A uniform ladder of length 6 m and mass 20 kg rests with its top against a smooth vertical wall and its foot on rough horizontal ground. The ladder makes an angle of 60° with the ground. (a) The ladder is on the point of slipping. Find the coefficient of friction between the ladder and the ground. (b) The ladder is moved so that the coefficient of friction is 0.4, at the same angle. A man of mass 70 kg climbs the ladder. Modelling the man as a particle, find how far up the ladder he can climb before it slips.

A particle P is projected from a point O on horizontal ground with velocity (12 i + 16 j) m s^-1, where i and j are horizontal and vertically upward unit vectors. (a) Find the speed of projection and the angle of projection above the horizontal. (b) Find the time of flight and the horizontal range. (c) Find the length of time for which P is at least 10 m above the ground. (d) Find the direction of motion of P when t = 2. (e) State one limitation of the model.