91Ó°ÊÓ

A random sample of n=64 observations is drawn from a population with$\mathrm{μ}=20$ and$\mathrm{σ}=16$ .

According to properties of the Sampling distribution of $\stackrel{\mathbf{¯}}{\mathbf{x}}\mathbf{}{\mathit{μ}}_{\stackrel{\mathbf{¯}}{\mathbf{x}}}\mathbf{=}\mathit{μ}$

and${\mathit{σ}}_{\stackrel{\mathbf{¯}}{\mathbf{x}}}\mathbf{=}\frac{\mathbf{σ}}{\sqrt{\mathbf{n}}}$

Therefore,

${\mathrm{μ}}_{\stackrel{¯}{x}}=20$and ${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{16}{\sqrt{64}}$ i.e.${\mathrm{σ}}_{\stackrel{¯}{x}}=2$

Now,

$$\begin{gathered} P\left(\stackrel{¯}{x}<16\right)=P\left(\frac{\stackrel{¯}{x}\mathrm{μ}}{\mathrm{σ}l\sqrt{n}}<\frac{16-\mathrm{μ}}{\mathrm{σ}\sqrt{n}}\right) \\ =P\frac{\stackrel{¯}{x}-20}{2}<\frac{16-20}{2} \\ =P\left(z<-2\right) \end{gathered}$$

Therefore, from z-score table,

$P\left(\stackrel{¯}{x}<16\right)=0.0228$

Thus, probability that $\stackrel{¯}{x}$is less than 16 is 0.0228.

According to properties of the Sampling distribution of localid="1660821731616" $\stackrel{\mathbf{¯}}{\mathbf{x}}\mathbf{}{\mathit{μ}}_{\stackrel{\mathbf{¯}}{\mathbf{x}}}\mathbf{=}\mathit{μ}$

andlocalid="1660821735110" ${\mathit{σ}}_{\stackrel{\mathbf{¯}}{\mathbf{x}}}\mathbf{=}\frac{\mathbf{σ}}{\sqrt{\mathbf{n}}}$

Therefore,${\mathrm{μ}}_{\stackrel{¯}{x}}=20$

${\mathrm{μ}}_{\stackrel{¯}{x}}=20$and ${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{16}{\sqrt{64}}$i.e. ${\mathrm{σ}}_{\stackrel{¯}{x}}=2$

Now,

localid="1658202322115" $$\begin{gathered} P(\stackrel{¯}{x}>23)=P\left(\frac{\stackrel{¯}{x}-\mathrm{μ}}{\mathrm{σ}l\sqrt{n}}>\frac{23-\mathrm{μ}}{\mathrm{σ}l\sqrt{n}}\right) \\ =P\left(\frac{\stackrel{¯}{x}-20}{2}>\frac{23-20}{2}\right) \\ =P(z>1.5) \end{gathered}$$

Therefore, from z-score table,

$$\begin{gathered} P\left(\overline{x}>23\right)=1-P\left(z<1.5\right) \\ =-0.9332 \\ =0.0668 \end{gathered}$$

Thus, probability that $\overline{x}$is greater than 23 is 0.0668.

According to properties of the Sampling distribution of $\overline{\mathbf{x}}$

localid="1660821743033" ${\mathit{μ}}_{\overline{\mathbf{x}}}\mathbf{=}\mathit{μ}$andlocalid="1660821747012" ${\mathit{σ}}_{\overline{\mathbf{x}}}\mathbf{=}\frac{\mathbf{σ}}{\sqrt{\mathbf{n}}}$

Therefore,

${\mathrm{μ}}_{\overline{x}}=20$and ${\mathrm{σ}}_{\overline{x}}=\frac{16}{\sqrt{64}}$ i.e. ${\mathrm{σ}}_{\overline{x}}=2$

Now,

$$\begin{gathered} P\overline{x}>25=P\left(\frac{\overline{x}-\mathrm{μ}}{\mathrm{σ}\sqrt{n}}>\frac{25-\mathrm{μ}}{\mathrm{σ}l\sqrt{n}}\right) \\ =P\left(\frac{\overline{x}-20}{2}>\frac{25-20}{2}\right) \\ =P\left(z>2.5\right) \end{gathered}$$

Therefore, from z-score table,

$$\begin{gathered} P\left(\overline{x}>25\right)=1-P\left(z<2.5\right) \\ =1-0.9937 \\ =0.0062 \end{gathered}$$

Thus, probability that $\overline{x}$is greater than 15 is 0.0062.

According to properties of the Sampling distribution of $\overline{\mathbf{x}}\mathbf{}\mathbf{}\mathbf{}{\mathit{μ}}_{\overline{\mathbf{x}}}\mathbf{=}\mathit{μ}$

and${\mathit{σ}}_{\overline{\mathbf{x}}}\mathbf{=}\frac{\mathbf{σ}}{\sqrt{\mathbf{n}}}$

Therefore,

${\mathrm{μ}}_{\overline{x}}=20$and ${\mathrm{σ}}_{\overline{x}}=\frac{16}{\sqrt{64}}$i.e.${\mathrm{σ}}_{\overline{x}}=2$

Now,

Therefore, from z-score table,

Thus, probability that $\overline{x}$falls between 16 and 22 is 0.8185.

According to properties of the Sampling distribution of $\overline{\mathbf{x}}\mathbf{}\mathbf{}\mathbf{}{\mathit{μ}}_{\overline{\mathbf{x}}}\mathbf{=}\mathit{μ}$

and${\mathit{σ}}_{\overline{\mathbf{x}}}\mathbf{=}\frac{\mathbf{σ}}{\sqrt{\mathbf{n}}}$

Therefore,${\mathrm{μ}}_{\overline{x}}=20$

and ${\mathrm{σ}}_{\overline{x}}=\frac{16}{\sqrt{64}}$ i.e. ${\mathrm{σ}}_{\overline{x}}=2$

Now,

$$\begin{gathered} P\left(\overline{x}<14\right)=P\left(\frac{\overline{x}-\mathrm{μ}}{\mathrm{σ}\sqrt{n}}<\frac{14-\mathrm{μ}}{\mathrm{σ}l\sqrt{n}}\right) \\ =P\left(\frac{\overline{x}-20}{2}<\frac{14-20}{2}\right) \\ =P\left(z<-3\right) \end{gathered}$$

Therefore, from z-score table,

$$\begin{gathered} P\left(\overline{x}<14\right)=\left(Pz<-3\right) \\ =0.00135 \end{gathered}$$

Thus, probability that $\overline{x}$is less than 14 is 0.00135.