Paper 4 · confidence intervals

The diameters of 25 steel rods have a sample mean of 0.980 cm and a standard deviation of 0.015 cm. Assuming the diameters are normally distributed with this standard deviation, find the 99% confidence interval for the population mean diameter.

(0.9741,\ 0.9859)

(0.9723,\ 0.9877)

(0.9414,\ 1.0186)

(0.9730,\ 0.9870)

Explanation: The 99% CI is

\bar{x} \pm z_{0.005}\cdot\frac{s}{\sqrt n}

with

z_{0.005}=2.576

s=0.015

n=25

. The margin is

2.576\times\frac{0.015}{\sqrt{25}}=2.576\times0.003=0.00773

, giving

(0.980-0.00773,\,0.980+0.00773)=(0.9723,\,0.9877)

cm. Using the 95% multiplier

z=1.96

instead of the 99% value gives the first option. Forgetting to divide the standard deviation by

\sqrt n

before multiplying by

z

gives the much wider third option. Using the one-tailed 99% value

z=2.326

instead of the two-tailed

z_{0.005}=2.576

gives the fourth option.

Derived from ZIMSEC Maths Paper 1, June 2008, Q1 (Statistics)

More questions from this paper