show work please Suppose that the proportion of voters who w

show work please

Suppose that the proportion of voters who will vote for Mr. Smith is p = 0.47. Let p-hat be the sample proportion of voters for Mr. Smith in a random sample of size 106. Find the following probabilities. Do not use the normal approximation. (a)P(p-hat

Solution

(a) P(phat<=0.48) = P((phat-p)/sqr(p*(1-p)/n) <(0.48-0.47)/sqrt(0.47*(1-0.47)/106))

=P(Z<0.21) =0.5832 (from standard normal table)

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(b) P(0.33<=phat<=0.71)

= P((0.33-0.47)/sqrt(0.47*(1-0.47)/106) <Z< (0.48-0.47)/sqrt(0.71*(1-0.47)/106))

=P(-2.89<Z<0.17) =0.5656 (from standard normal table)

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(c)mean=n*p= 106*0.47 = 49.82

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(d) standard deviation =sqrt(n*p*(1-p))

=sqrt(106*0.47*(1-0.47))

=5.138541