Question

According to a Yale program on climate change communication survey, 71% of Americans think global warming is happening.†

(a) For a sample of 16 Americans, what is the probability that at least 13 believe global warming is occurring? Use the binomial distribution probability function discussed in Section 5.5 to answer this question. (Round your answer to four decimal places.)

(b)For a sample of 170 Americans, what is the probability that at least 120 believe global warming is occurring? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(c)As the number of trials in a binomial distribution application becomes large, what is the advantage of using the normal approximation of the binomial distribution to compute probabilities?

As the number of trials becomes large, the normal approximation gives a more accurate answer than the binomial probability function.As the number of trials becomes large, the normal approximation simplifies the calculations required to obtain the desired probability.    

(d)When the number of trials for a binomial distribution application becomes large, would developers of statistical software packages prefer to use the binomial distribution probability function shown in Section 5.5 or the normal approximation of the binomial distribution discussed in Section 6.3? Explain.

Answer 1

a)

Sample size , n =    16
Probability of an event of interest, p =   0.71
and probability is given by

P(X=x) = C(n,x)*px*(1-p)(n-x)

P(X≥13)=P(X=13) +P(X=14) + P(X=15) +P(X=16) = 0.2740(answer)

b)

Sample size , n =    170                      
Probability of an event of interest, p =   0.71                      
right tailed                          
X ≥   120                      
                          
Mean = np =    120.7                      
std dev ,σ=√np(1-p)=   5.9163                      
P(X ≥   120   ) = P(Xnormal ≥   119.5   )          
                          
Z=(Xnormal - µ ) / σ = (   119.5   -   120.7   ) /   5.9163   =   -0.203
                          
=P(Z ≥   -0.203   ) =    0.5804              

c)

As the number of trials becomes large, the normal approximation simplifies the calculations required to obtain the desired probability

d)

When the number of trials for a binomial distribution application becomes large, would developers of statistical software packages prefer to use the normal approximation of the binomial distribution .

because As the number of trials becomes large, the normal approximation simplifies the calculations required to obtain the desired probability