Question

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a)Suppose n = 43 and p = 0.20. (For each answer, enter a number. Use 2 decimal places.)
n·p =
n·q =

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank:

Yes or No

second blank:

can or cannot    

third blank:

n·p and n·q do not exceed, n·q does not exceed, n·p exceeds, n·q exceeds, both n·p and n·q exceed, or n·p does not exceed

fourth blank (Enter an exact number.) ____

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ =  

σ_p̂ =  

(b) Suppose n = 25 and​​​​​​​ p = 0.15. Can we safely approximate p̂ by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank:

Yes or No    

second blank:

can or cannot    

third blank:

n·p and n·q do not exceed, n·q does not exceed, n·p exceeds, n·q exceeds, both n·p and n·q exceed, or n·p does not exceed

fourth blank (Enter an exact number.) ______

(c) Suppose n = 62 and​​​​​​​ p = 0.24.(For each answer, enter a number. Use 2 decimal places.)
n·p =
n·q =

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank:

Yes or No    

second blank:

can or cannot    

third blank:

n·p and n·q do not exceed, n·q does not exceed, n·p exceedsn·q exceeds, both n·p and n·q exceed, or n·p does not exceed

fourth blank (Enter an exact number.) _____


What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ =  

σ_p̂ =

Answer 1

$\mathbf{Solution}$

a) n = 43, p=0.20,9=1-p=0.80

$np=43*0.20=\mathbf{{\color{Red} 8.60}}$

ng 43 * 0.80 = 34.40

${\color{Red} \mathbf{Yes}}$, $\hat p$${\color{Red} \mathbf{can }}$ be approximated by a normal random variable because ${\color{Red} \mathbf{both\; n·p \;and\; n·q \;exceed}}\;\;\;\;\mathbf{{\color{Red} 5}}$

$\mu_{\hat p}=p=\mathbf{{\color{Red} 0.20}}$

$\sigma_{\hat p}=\sqrt{\frac{p*q}{n}}=\sqrt{\frac{0.20*0.80}{43}}=\mathbf{{\color{Red} 0.061}}$

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${\color{Blue} \mathbf{(b)}}\;n=25,\;p=0.15,q=1-p=0.85$

$np=25*0.15=\mathbf{3.75{\color{Red} }}$

$nq=25*0.85=\mathbf{21.25{\color{Red} }}$

${\color{Red} \mathbf{No}}$, $\hat p$${\color{Red} \mathbf{cannot}}$ be approximated by a normal random variable because ${\color{Red} \mathbf{np\;do\;not \;exceed}\;\;\;\;\mathbf{5}}$

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${\color{Blue} \mathbf{(c)}}\;n=62,\;p=0.24,\;q=1-p=0.76$

$np=62*0.24=\mathbf{{\color{Red} 14.88}}$

$nq=62*0.76=\mathbf{{\color{Red} 47.12}}$

${\color{Red} \mathbf{Yes}}$, $\hat p$${\color{Red} \mathbf{can }}$ be approximated by a normal random variable because ${\color{Red} \mathbf{both\; n·p \;and\; n·q \;exceed}}\;\;\;\;\mathbf{{\color{Red} 5}}$

$\mu_{\hat p}=p=\mathbf{{\color{Red} 0.24}}$

$\sigma_{\hat p}=\sqrt{\frac{p*q}{n}}=\sqrt{\frac{0.24*0.76}{62}}=\mathbf{{\color{Red} 0.054}}$