Question 2 - A-Level Maths

OCR MEI S2 2005 June — Question 2 18 marks

Exam Board	OCR MEI
Module	S2 (Statistics 2)
Year	2005
Session	June
Marks	18
Paper	Download PDF ↗
Topic	Normal Distribution
Type	Finding unknown boundaries
Difficulty	Standard +0.3 This is a straightforward normal distribution question requiring standard techniques: z-score calculations, inverse normal for percentiles, and finding unknown parameters. Part (v) involves basic probability rules with independence. All parts are routine applications of S2 content with no novel problem-solving required, making it slightly easier than average.
Spec	2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation

2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .

Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
The manufacturer wishes to quote a fuel economy figure which will be exceeded on \(90 \%\) of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation \(\sigma \mathrm { mpg }\).
Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that \(\sigma = 5.63\).
Draw a sketch to illustrate both distributions on a single diagram.
Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.

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2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg).

The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .\\
(i) Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .\\
(ii) The manufacturer wishes to quote a fuel economy figure which will be exceeded on $90 \%$ of days. What figure should be quoted?

The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation $\sigma \mathrm { mpg }$.\\
(iii) Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that $\sigma = 5.63$.\\
(iv) Draw a sketch to illustrate both distributions on a single diagram.\\
(v) Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.

\hfill \mbox{\textit{OCR MEI S2 2005 Q2 [18]}}

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