The lengths of female snakes of a particular species are normally distributed with mean $$\(54 \mathrm{~cm}\)$$ and standard deviation $$\(6.1 \mathrm{~cm}\)$$. The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than $$\(45 \mathrm{~cm}\)$$ and 17 have lengths more than $$\(56 \mathrm{~cm}\)$$. Find estimates for the mean and standard deviation of the lengths of male snakes of this species. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s20_qp_51 Year：2020 Question No：6(b)
Answer:
$
\begin{array}{l}
\frac{45-\mu}{\sigma}=-0.994
\frac{56-\mu}{\sigma}=1.372
\end{array}
$
One appropriate standardisation equation with \(\mu, \sigma, \mathrm{z}\)-value (not probability) and 45 or 56.
$
11=2.366 \sigma
$
(M1 for correct algebraic elimination of \(\mu\) or \(\sigma\) from their two simultaneous equations to form an equation in one variable) \(
\sigma=4.65, \mu=49.6\)
\begin{array}{l}
\frac{45-\mu}{\sigma}=-0.994
\frac{56-\mu}{\sigma}=1.372
\end{array}
$
One appropriate standardisation equation with \(\mu, \sigma, \mathrm{z}\)-value (not probability) and 45 or 56.
$
11=2.366 \sigma
$
(M1 for correct algebraic elimination of \(\mu\) or \(\sigma\) from their two simultaneous equations to form an equation in one variable) \(
\sigma=4.65, \mu=49.6\)
Knowledge points:
5.5.1 understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables (Sketches of normal curves to illustrate distributions or probabilities may be required.)
5.5.2.2 finding a relationship between $x_{1}, \mu$ and $\sigma $ given the value of $P\left(X>x_{1}\right)$ or a related probability (For calculations involving standardisation, full details of the working should be shown.) (e.g. $Z=\frac{(X-\mu)}{\sigma}$)
Solution:
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