Chapter EP, Problem 12.6.4EP

To determine

To find: The hypotheses.

Answer to Problem 12.6.4EP

The hypothesis $H_0=45$ and accept the alternative hypothesis $H_1<45$ .

Explanation of Solution

Given information:

The given values is $H_0=45$ , $H_1<45$ , $\overline{x}=39.5$ , $s=1.2$ and $n=75$ .

Calculation:

The confidence interval formula is,

  $CI=\overline{x}\pm 1.96\frac{s}{\sqrt{n}}$ ....(1)

Substitute the values in equation −(1)

  $\begin{array}{c}CI=\overline{x}\pm 1.96\frac{s}{\sqrt{n}}\\ =39.5\pm 1.96\frac{1.2}{\sqrt{75}}\\ =39.5\pm 1.96\times 0.1386\\ \approx 39.5\pm 0.2716\end{array}$

The lower bound of the interval is,

  $\begin{array}{c}l=39.5-0.2716\\ =39.2284\end{array}$

The upper bound of the interval is,

  $\begin{array}{c}u=39.5+\mathrm{0.27.16}\\ =39.7716\end{array}$

Thus, the confidence interval mean is $39.2284\le \mu \le 39.7716$

Since the confidence interval for mean is. $39.2284\le \mu \le 39.7716$ , therefore $95\%$ of the time the mean lies between the values $39.2284$ and $39.7716$ .

Therefore, the null hypothesis $H_0=45$ and accept the alternative hypothesis $H_1<45$ .