Question

Thirty-five more than 0.8 times a number is the same as a 43 less than the product of -7 and the number

Answer 1

The information given is that p=0.38 , hence q=1−p=0.62 . First we use the formulas to compute the mean and standard deviation of p^ :

μp^=p=0.38andσP^=pqn−−−√=(0.38)(0.62)900−−−−−−−−−−√=0.01618

Then 3σP^=3(0.01618)=0.04854≈0.05 so

[p^−3p^(1−p^)n−−−−−−−−√,p^+3p^(1−p^)n−−−−−−−−√]=[0.38−0.05,0.38+0.05]=[0.33,0.43]

which lies wholly within the interval [0,1] , so it is safe to assume that p^ is approximately normally distributed.

To be within 5 percentage points of the true population proportion 0.38 means to be between 0.38−0.05=0.33 and 0.38+0.05=0.43 . Thus

P(0.33<P^<0.43)=P(0.33−μP^σP^<Z<0.43−μP^σP^)=P(0.33−0.380.01618<Z<0.43−0.380.01618)=P(−3.09<Z<3.09)=P(3.09)−P(−3.09)=0.9990−0.0010=0.9980