Practice Paper Statistics 2 (Edexcel International A Level Maths: Pure 2)

Practice Paper Questions

1a
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2 marks

In the town of Wooster, Ohio, it is known that 90% of the residents prefer the locally produced Woostershire brand sauce when preparing a Caesar salad.  The other 10% of residents prefer another well-known brand.

30 residents are chosen at random by a pollster.  Let the random variable X represent the number of those 30 residents that prefer Woostershire brand sauce.

Suggest a suitable distribution for X and comment on any necessary assumptions.

1b
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3 marks

Find the probability that

(i)
90% or more of the residents chosen prefer Woostershire brand sauce

(ii)
none of the residents chosen prefer the other well-known brand.
1c
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2 marks

The pollster knows that there is a greater than 97% chance of at least k of the 30 residents preferring Woostershire brand sauce, where k is the largest possible value that makes that statement true.

Find the value of k.

 

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2a
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1 mark

Chloe’s sister likes to play a game in which she hides a coin in one of her hands and asks Chloe to guess which hand it is in.  Chloe claims that she can predict which hand the coin is hidden in more often than not.  Her sister thinks Chloe is just guessing so she decides to conduct a hypothesis test to test Chloe’s claim.

State suitable null and alternative hypotheses for Chloe’s sister’s test.

2b
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3 marks

They play the game 100 times, and Chloe guesses correctly 60 times.

Using a suitable approximation, test at the 5% significance level whether or not Chloe’s claim is justified.

2c
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2 marks

Justify the use of your approximation in part (b).

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3a
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1 mark

The diagram below shows the probability density function, f open parentheses x close parentheses, of a random variable X. f left parenthesis x right parenthesis equals k when 0 less or equal than x less or equal than a, otherwise f left parenthesis x right parenthesis equals 0.

hq4

Write down an expression for k in terms of a.

3b
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1 mark

Write down an expression for E open parentheses X close parentheses in terms of a.

3c
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3 marks

Find a simplified expression for V a r open parentheses X close parentheses in terms of a.

3d
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2 marks

Given that P left parenthesis X greater than 5 right parenthesis equals 0.6 find the value of a.

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4a
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4 marks

Harry is trying to draw specific lengths without measuring equipment. He draws a straight line and stops when he thinks it is 10 cm long. The actual length of Harry’s line, L cm, can be modelled by the probability density function

f left parenthesis l right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 20 over 421 open parentheses l minus 10.5 close parentheses to the power of 4 space space space space space space space space space space space space space space space space 8 less or equal than l less or equal than 12 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Estimate the lower quartile of the lengths of Harry’s lines.

4b
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1 mark

Harry tries to draw a 10 cm line 80 times.

Write down how many of Harry’s line you would expect to be less than the lower quartile.

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5a
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4 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

 straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 2 end cell row cell a minus b over x end cell cell 2 less or equal than x less than 4 end cell row cell fraction numerator x squared plus 17 over denominator c end fraction end cell cell 4 less or equal than x less than 7 end cell row 1 cell x greater or equal than 7 end cell end table close

(i)
Explain why b equals 2 a.
(ii)
Show that c equals 66.
(iii)
Show that 4 a minus b equals 2 and hence find the values of a and b.
5b
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1 mark

Write down the median of X.

5c
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3 marks

Find the lower quartile and the upper quartile of X.

5d
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2 marks

Describe the skewness of X. Justify your answer.

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6a
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2 marks

Grace, a grumpy toddler, attends nursery five days a week. The number of tantrums that Grace has in a day follows a Poisson distribution with variance 3.14.

Find the probability that Grace has exactly 17 tantrums during a week at nursery.

6b
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3 marks

Find the probability that Grace has fewer than four tantrums in a two-day period at nursery.

6c
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4 marks

Given that Grace has fewer than four tantrums at nursery one day, find the probability that she had no tantrums at nursery that day.

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7a
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2 marks

A walk-in vaccination centre can model the number of people arriving every 5 minutes by the random variable X with distribution Po(0.6).

State two assumptions required for the Poisson model to be valid.

7b
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4 marks

The centre launches a new advertising campaign and conducts a hypothesis test to see if it has increased the number of people arriving for a vaccine. They choose a half hour period at random and find that 7 people arrive for a vaccine during this time.

Clearly stating your hypotheses, test at the 10% significance level whether the number of arrivals has increased.

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8a
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4 marks

A teacher keeps bronze, silver and gold star stickers as prizes for his students.  The teacher takes a sample of five stickers at random from a full pack.

(i)
Identify the population and the sampling units.
(ii)
Write down the number of possible samples and find how many distinct samples the teacher could take.
8b
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6 marks

The school office contains a large number of the bronze, silver and gold star stickers in the ratio 5:3:2. At the end of each term the students are awarded 50 points for a gold star, 20 points for a silver star and 10 points for a bronze star. Their final score is the product of the points their stars are worth. The teacher takes a sample of 3 stickers at random to show his students how to find the product of the points their stickers are worth.

Given that the teacher chose his sample only from bronze and silver stars, write down the sampling distribution of the product of the values the stickers are worth. Clearly define your random variable and statistic.

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9a
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8 marks

An amusement park has a challenge where guests attempt to remain on a mechanical bull for as long as possible. The times, in seconds, that the guests stay on the mechanical bull can be modelled using a continuous random variable, T, with probability density function straight f open parentheses t close parentheses.

straight f open parentheses t close parentheses equals open curly brackets table row 0 cell t less than 0 end cell row cell 3 over 50 end cell cell 0 less or equal than t less than 5 end cell row cell fraction numerator 6 t minus 15 over denominator 250 end fraction end cell cell 5 less or equal than t less than 10 end cell row cell 1 over t squared end cell cell t greater or equal than 10 end cell end table close

Specify fully the cumulative distribution function straight F open parentheses t close parentheses .

9b
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4 marks

Find the median length of time that a guest is able to stay on the mechanical bull.

9c
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3 marks

Given that a guest has remained on the mechanical bull for 5 seconds, find the probability that the guest will still be on the mechanical bull after another 5 seconds. 

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