Question 1 - A-Level Maths

OCR MEI S2 2007 June — Question 1 19 marks

Exam Board	OCR MEI
Module	S2 (Statistics 2)
Year	2007
Session	June
Marks	19
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Z-tests (known variance)
Type	One-tail z-test (upper tail)
Difficulty	Standard +0.3 This is a straightforward hypothesis testing question with standard normal distribution calculations. Parts (i)-(iii) involve routine z-score lookups and binomial/normal approximation, while parts (iv)-(v) require a basic one-tail z-test with known variance—all standard S2 procedures requiring minimal problem-solving beyond applying learned techniques.
Spec	2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation

1 The random variable \(X\) represents the time taken in minutes for a haircut at a barber's shop. \(X\) is Normally distributed with mean 11 and standard deviation 3 .

Find \(\mathrm { P } ( X < 10 )\).
Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
Write down suitable null and alternative hypotheses for the test.
Carry out the test at the \(5 \%\) level.

Show LaTeX source

1 The random variable $X$ represents the time taken in minutes for a haircut at a barber's shop. $X$ is Normally distributed with mean 11 and standard deviation 3 .\\
(i) Find $\mathrm { P } ( X < 10 )$.\\
(ii) Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.\\
(iii) Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes.

A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.\\
(iv) Write down suitable null and alternative hypotheses for the test.\\
(v) Carry out the test at the $5 \%$ level.

\hfill \mbox{\textit{OCR MEI S2 2007 Q1 [19]}}

This paper (4 questions)

View full paper

Q1 19 Q2 19 Q3 16 Q4 18