Q.A sample of 100 measurements at breaking strength of cotton thread gave a mean $7.4$ g and a standard deviation of $1.2$ g. Find 95% confidence limits for the mean breaking strength of cotton thread.

$\bar{x}\pm1.96\,\dfrac{s}{\sqrt{n}}=7.4\pm1.96(0.12)=7.4\pm0.2352\Rightarrow(7.16,\ 7.64)$.

In the TN HSC Class-12 Business Maths estimation topic, the $95\%$ confidence limits for the population mean (large sample) are $\bar{x}\pm1.96\,\dfrac{\sigma}{\sqrt{n}}$, with the sample SD used for $\sigma$ when the population value is unknown.

Given: $n=100,\ \bar{x}=7.4,\ s=1.2$.

Step 1 — standard error.

$\text{S.E.}=\frac{s}{\sqrt{n}}=\frac{1.2}{\sqrt{100}}=\frac{1.2}{10}=0.12.$

Step 2 — $95\%$ confidence limits. …