Practice Paper 3 (Statistics) (Edexcel A Level Maths: Mechanics)

Practice Paper Questions

1a
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4 marks

Abner, an American baseball fanatic, has just moved to a town in which it is known that 25% of the residents are familiar with the rules of the game.

 Abner takes a random sample of 40 residents of the town. Find the probability that

(i)
fewer than 13

(ii)
no more than 13

(iii)
more than 13

(iv)
at most 13 but at least 5

of the residents in Abner’s sample are familiar with the rules of baseball.

1b
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3 marks

Abner would like to field a town team for an upcoming regional baseball tournament.  Abner is intending to play himself, but he needs to find enough other players to fill up the team.  As he does not yet know anyone in the town, he decides to take another random sample of residents in hopes of finding enough other players.  Only people familiar with the rules of baseball are able to be included in the team, but it may be assumed that anyone in the town familiar with the rules would also be willing to join Abner’s team.

Given that an American baseball team must have a minimum of nine players, find the smallest number of people that Abner should include in his sample in order to have at least a 90% chance of filling up his team.

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2a
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2 marks

A snack shop owner has noticed that the sale of energy drinks seems to increase later in the school term.  He conducts a hypothesis test at the 1% level of significance to see if the sale of the drinks, d, increases as the number of days until the school holidays, h, decreases. 

(i)
What type of correlation is the snack shop owner testing for?

(ii)
State which of the two variables is the explanatory variable.
2b
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4 marks

Over the final thirty days of term the owner keeps a record of the number of sales of energy drinks and, using this data, calculates the product moment correlation coefficient to be  r = minus0.4187.

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient,  r, for a sample size of 30. 

Level 10% 5% 2.5% 1% 0.5%
n = 30 0.2407 0.3061 0.3610 0.4226 0.4629

(i)

Write down the critical region for the hypothesis test.

(ii)
Stating your hypotheses clearly, test the snack shop owner’s suspicion that more energy drinks are sold closer to the school holidays.
2c
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2 marks

The snack shop owner calculates the regression line of d on h and uses it to predict the number of energy drinks he will sell on the first day of the new term, when there are still 90 days until the holidays.  State two reasons why this is unlikely to give a reliable prediction. 

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3a
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2 marks

Summary statistics from the large data set for the daily mean windspeed (knots) measured in Heathrow throughout October 1987 and October 2015 are given in the table below.

  Min Max Median straight capital sigma x straight capital sigma x squared
1987 2 16 5 185 1401
2015 3 10 6 197 1357


Calculate the mean of the daily mean windspeeds for each of the two years.

3b
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3 marks

The standard deviation for 2015 was 1.84.

Calculate the standard deviation for 1987 and compare the daily mean windspeeds for each of the two years.

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4
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8 marks

A lifestyle magazine for ancient historians recently conducted a poll of its readership.  The following Venn diagram shows some of the results for the numbers of the historians polled who said that they were fans of any, all or none of the following three ancient Near Eastern rulers – the Hittite king Suppiluliuma I (S),  the Assyrian king Ashurbanipal (A), and the Akkadian king Manishtushu (M).q4-very-hard-3-2-further-probability-edexcel-a-level-maths-statistics

180 historians were polled in total.  For each historian who was only a fan of Manishtushu, there were 7 who were only fans of Suppiluliuma I.  For each two historians who were fans of both Ashurbanipal and Manishtushu but not of Suppiluliuma I, there were nine historians who were fans of all three rulers.  Most of the historians were, as expected, fans of at least one of the three rulers, but it was also discovered that being a fan of Ashurbanipal and being a fan of Manishtushu were independent events.

One of the historians who was polled is selected at random.  Given that the historian chosen is a fan of at least one of the three rulers, find the exact probability that the historian is a fan of: 

(i)
Suppiluliuma I

(ii)
at least two the three rulers

(iii)
exactly two of the three rulers.

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5a
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3 marks

A machine is used to fill bags of potatoes for a supermarket chain.  The weight, W kg, of potatoes in the bags is normally distributed with mean 3 kg and standard deviation sigma kg. 

Given that 7% of the bags contain a weight of potatoes that is at least 50 g more than  the mean, find:

straight P left parenthesis 2.9 less or equal than W less or equal than 3.1 right parenthesis.

5b
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4 marks

Twelve of the bags of potatoes are chosen at random.

Find the probability that not more than one of the bags will contain less than 2.96 kg of potatoes.

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6a
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4 marks

Pizza Prince is a fast-food restaurant which is known for their Crown pizza.  The weights of Crown pizza are normally distributed with standard deviation 42 g.  It is thought that the mean weight,mu, is 350 g.

A restaurant inspector believes that the mean weight of the Crown pizza is less than
350 g.  She visits the restaurant over the period of a week, and samples and weighs five randomly selected Crown pizzas.  She uses the data to carry out a hypothesis test at the 5% level of significance.

She tests straight H subscript 0 ∶ mu equals 350  against  straight H subscript 1 ∶ mu less than 350.

When the inspector writes up her report, she can only find the values for four of the weights, these are shown below:

325.2              356.1              319.7              300.5

Given that the result of the hypothesis test is that there is insufficient evidence to reject straight H subscript 0 at the 5% level of significance, calculate the minimum possible value for the missing weight, w. Give your answer correct to 1 decimal place.

6b
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4 marks

The inspector remembers her assistant claiming that if she had used a 10% level of significance then the outcome to the hypothesis test would have been different.

Using this information, write down an inequality for w.

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7a
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2 marks

A student claims that a random variable  has a probability distribution defined by the following probability mass function:

P open parentheses X equals x close parentheses equals open curly brackets table row cell fraction numerator 1 over denominator 3 x squared end fraction space space space space space space space space space space space space space space space space x equals negative 3 comma negative 1 end cell row cell fraction numerator 1 over denominator 3 x cubed end fraction space space space space space space space space space space space space space space space space space x equals 1 comma 3 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 otherwise end cell end table close

Explain how you know that the student’s function does not describe a probability distribution.

7b
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2 marks

Given that the correct probability mass function is of the form

 P open parentheses X equals x close parentheses equals open curly brackets table row cell k over x squared space space space space space space space space space space space space space space space space space x equals negative 3 comma negative 1 end cell row cell k over x cubed space space space space space space space space space space space space space space space space space x equals 1 comma 3 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 otherwise end cell end table close

where k is a constant,

 Find the exact value of k.

7c
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2 marks

Find straight P left parenthesis X less than 2 right parenthesis.

7d
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1 mark

State, with a reason, whether or not X is a discrete random variable.

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