AQA A-Level Maths: Statistics

A components factory states that the proportion of micro-switches it produces that are faulty is $20\%$20%. A purchasing manager believes the factory has improved its process and that the proportion of faulty switches is now less than $20\%$20%. To investigate, the manager takes a random sample of $40$40 switches from the factory's output and finds that $3$3 of them are faulty.

The number of faulty switches in the sample is to be modelled by a binomial distribution, and the manager's belief is tested at the $5\%$5% significance level.

(a) State suitable null and alternative hypotheses for this test, defining the parameter you use. (2 marks)

(b) Find the critical region for the test, and state the actual significance level of this region to $3$3 significant figures. (6 marks)

(c) Using your critical region, carry out the test and state your conclusion in context. (4 marks)

A bottling machine, machine $A$A, dispenses sparkling water into bottles. The volume dispensed, $X$X ml, is modelled by a normal distribution with mean $500$500 ml and standard deviation $20$20 ml.

(a) Find the probability that a randomly chosen bottle filled by machine $A$A contains between $470$470 ml and $530$530 ml. (3 marks)

(b) A bottle is rejected as "overfilled" if its volume is among the heaviest-filled $2.5\%$2.5%. Find, to the nearest millilitre, the smallest volume for which a bottle is rejected as overfilled. (3 marks)

(c) A second machine, machine $B$B, fills small cartons. The volume it dispenses, $Y$Y ml, is modelled by a normal distribution with unknown mean $\mu$μ and unknown standard deviation $\sigma$σ. It is known that

$P(Y < 46) = 0.0228 \qquad \text{and} \qquad P(Y < 70) = 0.8413.$P(Y<46)=0.0228andP(Y<70)=0.8413.

Find the values of $\mu$μ and $\sigma$σ. (4 marks)

A college surveys all $200$200 of its first-year students. Each student is recorded according to whether they own a bicycle (event $A$A) and whether they live within two miles of the college (event $B$B). The results are summarised in the two-way table below.

	Lives within 2 miles ($B$B)	Does not live within 2 miles ($B'$B′)	Total
Owns a bicycle ($A$A)	$33$33	$47$47	$80$80
Does not own a bicycle ($A'$A′)	$27$27	$93$93	$120$120
Total	$60$60	$140$140	$200$200

A student is chosen at random from those surveyed.

(a) Find $P(A \cap B)$P(A∩B) and $P(A \cup B)$P(A∪B). (3 marks)

(b) Given that the chosen student lives within two miles of the college, find the probability that they own a bicycle. (2 marks)

(c) Determine whether the events $A$A and $B$B are statistically independent, justifying your answer. (3 marks)

A study group records the time, $t$t minutes, that each of $50$50 randomly selected students spends using a particular revision app on one day. The results are summarised in the grouped frequency table below.

Time, $t$t (min)	Frequency, $f$f
$0 \leq t < 10$0≤t<10	$2$2
$10 \leq t < 20$10≤t<20	$6$6
$20 \leq t < 30$20≤t<30	$15$15
$30 \leq t < 40$30≤t<40	$14$14
$40 \leq t < 50$40≤t<50	$8$8
$50 \leq t < 70$50≤t<70	$5$5

(a) Use the midpoints of the classes to estimate the mean and the standard deviation of these times. Give your answers to $3$3 significant figures. (4 marks)

(b) An outlier is defined as any value more than $2$2 standard deviations from the mean. Using your answers to part (a), explain whether the table could contain any outliers. (2 marks)

A researcher records, for each of $12$12 seaside towns, the mean daily hours of sunshine $x$x during one week and the takings $y$y (in hundreds of pounds) of an ice-cream kiosk that week. The summary statistics are

$S_{xx} = 200, \qquad S_{yy} = 180, \qquad S_{xy} = 168, \qquad \bar{x} = 6, \qquad \bar{y} = 8.$Sxx​=200,Syy​=180,Sxy​=168,xˉ=6,yˉ​=8.

(a) Calculate the product moment correlation coefficient $r$r for these data, giving your answer to $3$3 significant figures. (2 marks)

(b) The researcher believes there is positive correlation between sunshine and takings. Test this belief at the $5\%$5% significance level, given that the critical value for a one-tailed test with $n = 12$n=12 is $0.4973$0.4973. (2 marks)

(c) The equation of the regression line of $y$y on $x$x is $y = 2.96 + 0.84x$y=2.96+0.84x. The hours of sunshine in the towns ranged from $2$2 to $10$10. Comment on the reliability of using this line to predict the takings for a town with a mean of $15$15 hours of sunshine per day. (1 mark)

The discrete random variable $X$X has the probability distribution shown in the table, where $k$k is a constant.

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$P(X = x)$P(X=x)	$0.15$0.15	$k$k	$0.25$0.25	$2k$2k	$0.15$0.15

(a) Find the value of $k$k. (2 marks)

(b) Find $E(X)$E(X). (2 marks)