CAIE A-Level Further Mathematics 4.2.4 Mean Confidence Intervals

Practise calculating a small-sample confidence interval for a normal-population mean using unbiased variance, the correct t value and an exact standard error.

Exam points

calculate the sample mean and unbiased variance from raw data or supplied summary statistics
select the two-sided t critical value using the confidence level and n minus one degrees of freedom
form the interval from sample mean plus or minus critical value times the standard error

[Maximum number: 4]

Nassa is researching the lengths of a particular type of snake in two countries, A and B.

He takes a random sample of 10 snakes of this type from country A and measures the length, $x \mathrm{~m}$ , of each snake. He then calculates a 90% confidence interval for the population mean length, $\mu \mathrm{m}$ , for snakes of this type, assuming that snake lengths have a normal distribution. This confidence interval is $3.36 \leqslant \mu \leqslant 4.22$ .

Find the sample mean and an unbiased estimate for the population variance.

$\bar{x}=\frac{1}{2}(4.22+3.36)=3.79$

B1

$4.22-3.36=\frac{2 t s}{\sqrt{10}}$

М1

Using z value implies M0

With t=1.833

A1

( s=0.7418 ) variance =0.55(0)

A1

4

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4.2.4—Mean confidence intervals question 1