Answer: 0.8461
Step-by-step explanation:
Let p be the proportion of residents are against the increase of taxes to fund alternatives to drug addiction treatment.
As per given , we have p=0.40
A random sample is taken with size : n= 400
Expecting sample proportion : [tex]\hat{p}=\dfrac{150}{400}=0.375[/tex]
Now , the probability that more than 150 of the residents surveyed will be against increasing taxes if a random sample of 400 residents are surveyed will be :
[tex]P(p>0.375)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.375-0.40}{\sqrt{\dfrac{0.40(0.60)}{400}}})[/tex]
[tex]=P(z>\dfrac{0.375-0.40}{\sqrt{\dfrac{0.40(0.60)}{400}}})\ \ [\because\ z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}][/tex]
[tex]=P(z>-1.02)=P(z<1.02)\ \ [\because \ P(Z>-z)=P(Z<z)][/tex]
[tex]\approx0.8461[/tex] [By z-table]
Hence, the approximate probability that more than 150 of the residents surveyed will be against increasing taxes if a random sample of 400 residents are surveyed is 0.8461 .
