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

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. 

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 = 0.4187.

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

The snack shop owner calculates the regression line of on  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. 

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
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.

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.

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).

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: 

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

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

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.

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,, 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  against  

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  at the 5% level of significance, calculate the minimum possible value for the missing weight, . Give your answer correct to 1 decimal place.

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 .

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

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

Given that the correct probability mass function is of the form

where  is a constant,

 Find the exact value of .

Find 

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