Further Math ASs 4.2.4 Confidence intervals

[Maximum number: 6]

Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, $x \mathrm{~m}$ , for each kangaroo. He then calculates a 90% confidence interval for the population mean tail length, $\mu \mathrm{m}$ , of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that $1.11 \leqslant \mu \leqslant 1.14$ .

Find the values of $\sum x$ and $\sum x^{2}$ for this sample.

$\bar{x}+1.771 \sqrt{\frac{s^{2}}{14}}=1.14$ or $\bar{x}-1.771 \sqrt{\frac{s^{2}}{14}}=1.11$
SOI
Allow incorrect t-value, not z-value.

[Add: $\left.\bar{x}=\frac{1}{2}(1.14+1.11)=1.125\right] \quad \sum x=15.75$
Does not depend on use of a t-value.

Subtract or substitute: $\sqrt{\frac{s^{2}}{14}}=\frac{1}{2}\left(\frac{1.14-1.11}{1.771}\right)$
Allow incorrect t-value, but not a z-value.

$s^{2}=0.00100[4]$ or s=0.0316[9] $\frac{450}{448063}$ , implied by correct final answer.

$\sum x^{2}=13 s^{2}+\frac{\left(\sum x\right)^{2}}{14}$

М1

OE

$\sum x^{2}=17.7$ (3)
CWO

6

CAIE A-Level Further Math AS 4.2.4 Mean Confidence Intervals Questions

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CAIE A-Level Further Math AS 4.2.4 Mean Confidence Intervals Questions question 1