Edexcel A-Level Maths: Statistics

A cycling-campaign group claims that more than $35\%$35% of commuters in a large city would cycle to work if dedicated cycle lanes were built. A researcher takes a random sample of $30$30 commuters and finds that $16$16 of them say they would cycle to work.

The researcher will model the number of commuters in the sample who say they would cycle as a binomial distribution and test the campaign group's claim 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)

In a survey of the adult residents of a town, each resident was asked whether they hold a full driving licence (event $L$L) and whether they use a bus at least once a week (event $B$B). The survey found that

$P(L) = 0.55, \qquad P(B) = 0.40, \qquad P(L \cap B) = 0.22.$P(L)=0.55,P(B)=0.40,P(L∩B)=0.22.

(a) Draw a Venn diagram to represent these events, and use it to find $P(L \cup B)$P(L∪B) and the probability that a resident chosen at random holds neither a licence nor uses a bus weekly. (4 marks)

(b) Given that a resident uses a bus at least once a week, find the probability that they hold a full driving licence. (3 marks)

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

A farm shop weighs a random sample of $50$50 eggs from one day's production. The masses, $m$m grams, are summarised in the grouped frequency table below.

Mass, $m$m (g)	Frequency, $f$f
$40 \leq m < 44$40≤m<44	$5$5
$44 \leq m < 48$44≤m<48	$12$12
$48 \leq m < 52$48≤m<52	$20$20
$52 \leq m < 56$52≤m<56	$9$9
$56 \leq m < 64$56≤m<64	$4$4

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

(b) The shop labels an egg "extra large" if its mass is more than $2$2 standard deviations above the mean. Using your answers to part (a), estimate the mass above which an egg is labelled "extra large". (2 marks)

(c) State one reason why your answers to parts (a) and (b) are only estimates, and comment briefly on what the table suggests about the distribution of the masses. (2 marks)

A machine produces glass tumblers, and past records show that $25\%$25% of the tumblers it produces have a small flaw. A quality inspector takes a random sample of $18$18 tumblers from the machine's output. Let $X$X be the number of tumblers in the sample that have a flaw, and assume $X$X may be modelled as a binomial distribution.

(a) Write down the distribution of $X$X, and find $P(X = 4)$P(X=4), giving your answer to $4$4 decimal places. (2 marks)

(b) Find the probability that at least $5$5 of the tumblers in the sample have a flaw. (2 marks)

(c) Find the mean and variance of $X$X. (2 marks)

The masses of apples sold by a grocer are modelled by a normal distribution with mean $180$180 g and standard deviation $15$15 g.

(a) Find the probability that a randomly chosen apple has a mass greater than $200$200 g. (2 marks)

(b) Apples in the heaviest $10\%$10% by mass are sold as "premium". Find, to the nearest gram, the smallest mass an apple can have to be sold as premium. (3 marks)

The lengths, in minutes, of $80$80 telephone calls made to a help-line one morning are summarised in the grouped frequency table below.

Length, $t$t (min)	Frequency, $f$f
$0 \leq t < 5$0≤t<5	$11$11
$5 \leq t < 10$5≤t<10	$18$18
$10 \leq t < 15$10≤t<15	$25$25
$15 \leq t < 20$15≤t<20	$14$14
$20 \leq t < 30$20≤t<30	$12$12

Use linear interpolation to estimate the median length of a call. Give your answer to $3$3 significant figures. (4 marks)