Question

A random sample of 175 measurements possessed a mean $\overline{\mathbf{x}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{2}$ and a standard deviation s = .79.

a. Test ${\mathit{H}}_{\mathbf{0}}\mathbf{:}\mathit{\mu }\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{3}$ against ${\mathit{H}}_{\mathbf{a}}\mathbf{:}\mathit{\mu }\mathbf{\ne }\mathbf{8}\mathbf{.}\mathbf{3}$Use $\mathit{a}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$

Short answer

⦁ The Z-statistic does not fall into the rejection region. So, we fail to reject the

null hypothesis.

Step-by-step solution

Given Information

The sample size, n=175

The mean,$\overline{x}=8.2$

The standard deviation, s=0.79

State the concept used in the test.

The statistical hypothesis test is used here. A statistical hypothesis is a statement about the numerical value of a population parameter.

Compute the hypothesis test H0:μ=8.3 against Ha:μ≠8.3 at the significance level a=0.05 

The null and alternative hypothesis are:

$$\begin{gathered} H_0:\mu =8.3 \\ {H}_a:\mu \ne 8.3 \end{gathered}$$

The significance level,$a=0.05$

The test statistic is computed as:

$$\begin{gathered} Z=\frac{\overline{x}-\mu }{s/\sqrt{n}} \\ =\frac{8.2-8.3}{0.79/\sqrt{175}} \\ =\frac{-0.1}{0.0597} \\ =-1.675 \end{gathered}$$

This is a two-tailed test. So, the $Z_{\frac{a}{2}}$ obtained from the standard normal table is 1.96.

Here, $\left|z\right|<z_{\frac{a}{2}}\Rightarrow \left|-1.675\right|<1.96$. So, we fail to reject the null hypothesis.

Hence, the Z-statistic does not fall into the rejection region. So, we fail to reject the null hypothesis.