Respuesta :
Answer:
[tex]Mean=p=0.75[/tex]
[tex]sd=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.75(1-0.75)}{450}}=0.0204[/tex]
Step-by-step explanation:
1) Data given
[tex]p=75\%=0.75[/tex] represent the population proportion of complaints settled for new car dealers
[tex]n=450[/tex] represent the sample of complaints involving new car dealers.
2) Find the distribution of [tex]\hat p[/tex]
First we can begin with the expected value
[tex]E(\hat p)=p[/tex] and that represent the mean
Now we can find the variance for [tex]\hat p[/tex]
When we use a proportion p, when we draw n items each from a Bernoulli distribution. The variance of each Xi distribution is p(1−p) and hence the standard error is p(1−p)/n. for this reason the variance for [tex]\hat p[/tex] is given by:
[tex]Var(\hat p)= \frac{p(1-p)}{n}[/tex]
So then the deviation would be given by:
[tex]Sd(\hat p)=\sqrt{\frac{p(1-p)}{n}}[/tex]
The sample distribution of the sample proportion [tex]\hat p[/tex] is normal, so then we have this:
[tex]\hat p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]
3) Calculating the mean and standard deviation
We can replace the values given in order to find the mean and deviation:
[tex]Mean=p=0.75[/tex]
[tex]sd=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.75(1-0.75)}{450}}=0.0204[/tex]
