Standard +0.3 This is a straightforward application of the sampling distribution of the mean with a normal population. Students must recognize that X̄ ~ N(40, 9/n), standardize to find P(X̄ > 42) < 0.05, use z = 1.645, and solve algebraically for n. While it requires understanding of the CLT and inverse normal use, it's a standard textbook exercise with no conceptual surprises, making it slightly easier than average.
2. A random sample of size \(n\) is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).
1st B1 for stating the correct distribution for \(\bar{X}\). May be implied if correctly used in line 2 and no incorrect version seen elsewhere.
1st M1 for an attempt to standardise with 42, 40 and their \(\sqrt{\frac{9}{n}}\), must have \(n\). Allow \(\pm\)
2nd B1 for using \(z = \pm 1.6449\) (or better)
Answer
Marks
Guidance
2nd dM1 Dep on 1st M1 for forming an equation in \(n\) or \(\sqrt{n}\). Allow "=" or "<". i.e. setting their standardised expression = their \(z\) value (\(
z
> 1.5\))
A1 for \(n = 7\) only. The A1 must follow from correct working so e.g. \(n < 6.087\) leading to \(n = 7\) is A0
Total: 5 marks
$X \sim N(40, 3^2)$, $\bar{X} \sim N(40, \frac{9}{n})$ (Condone $Y \sim N(40, \frac{9}{n})$) | B1 |
$P(\bar{X} > 42) = P(Z > \frac{42-40}{\sqrt{\frac{9}{n}}})$ | M1 |
$\frac{42-40}{\sqrt{\frac{9}{n}}} \geq 1.6449$ | B1 dM1 |
$n \geq 6.087$ | A1 |
$n = 7$ | |
| | | 1st B1 for stating the correct distribution for $\bar{X}$. May be implied if correctly used in line 2 and no incorrect version seen elsewhere.
1st M1 for an attempt to standardise with 42, 40 and their $\sqrt{\frac{9}{n}}$, must have $n$. Allow $\pm$
2nd B1 for using $z = \pm 1.6449$ (or better)
2nd dM1 Dep on 1st M1 for forming an equation in $n$ or $\sqrt{n}$. Allow "=" or "<". i.e. setting their standardised expression = their $z$ value ($|z| > 1.5$)
A1 for $n = 7$ only. The A1 must follow from correct working so e.g. $n < 6.087$ leading to $n = 7$ is A0
**Total: 5 marks**
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2. A random sample of size $n$ is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than $5 \%$.\\
\hfill \mbox{\textit{Edexcel S3 2013 Q2 [5]}}