Question

ment: 7.5 Practice ns Exercise 07.19 (Self-Test) Algorithmic Question 1 of Check My Wor A population has a mean of 300 and a standard deviation of 60. Suppose a sample of size 125 is selected and Z is used to estimate μ. Use z-table. a. What is the probability that the sample mean will be withn +i- s of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.) b. What is the probebility that the sample mean will be within +/-At of the population mean (to 4 decimals)? (Round z value in ntermediate c Type here to search up

Answer 1

$\mathbf{Solution}$

$\mathbf{given:}\;\mu=300,\;\sigma=60,\;n=125$

$\mathbf{formula:}\;z=\frac{\bar x -\mu}{\frac{\sigma }{\sqrt{n}}}$

${\color{Blue} \mathbf{a.}}\;within\;\pm5\;of\;\mu\Rightarrow P(295<\bar x<305)$

$P(295<\bar x<305)\Rightarrow P\left (\frac{295-300}{\frac{60}{\sqrt{125}}}< \frac{\bar x -\mu}{\frac{\sigma }{\sqrt{n}}}<\frac{305-300}{\frac{60}{\sqrt{125}}}\right )$

$\Rightarrow P(-0.93<z<0.93)$

$Required\;area\;under\;normal\;curve$

$\Rightarrow P(z<0.93)-P(z<-0.93)$

$\Rightarrow 0.8238-0.1762\;(from\;z-table)$

$\Rightarrow \mathbf{{\color{Red} 0.6476}}$

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${\color{Blue} \mathbf{b.}}\;within\;\pm11\;of\;\mu\Rightarrow P(289<\bar x<311)$

$P(289<\bar x<311)\Rightarrow P\left (\frac{289-300}{\frac{60}{\sqrt{125}}}< \frac{\bar x -\mu}{\frac{\sigma }{\sqrt{n}}}<\frac{311-300}{\frac{60}{\sqrt{125}}}\right )$

$\Rightarrow P(-2.05<z<2.05)$

2.05

$Required\;area\;under\;normal\;curve$

$\Rightarrow P(z<2.05)-P(z<-2.05)$

$\Rightarrow 0.9798-0.0202\;(from\;z-table)$

$\Rightarrow \mathbf{{\color{Red} 0.9596}}$