Respuesta :
Answer:
[tex]\mu_{\hat p}= 0.06[/tex]
[tex] \sigma_{\hat p}= \sqrt{\frac{0.06*(1-0.06)}{280}}= 0.0142[/tex]
And we want to find this probability:
[tex] p(\hat p <0.09)[/tex]
[tex] z = \frac{p \mu_{\hat p}}{\sigma_{\hat p}}[/tex]
[tex] z = \frac{0.09-0.06}{0.0142}= 2.114[/tex]
And we can find this probability:
[tex] P(z<2.114)[/tex]
And using the normal table or excel we got:
[tex] P(z<2.114)= 0.9827[/tex]
Step-by-step explanation:
For this case we have the following info given:
[tex] n = 280[/tex] represent the sample size
[tex] p =0.06[/tex] represent the true proportion
The sample proportion can be approximated with this distribution:
[tex] \hat p \sim N (p ,\sqrt{\frac{p(1-p)}{n}})[/tex]
The mean is given by:
[tex]\mu_{\hat p}= 0.06[/tex]
And the deviation is given by:
[tex] \sigma_{\hat p}= \sqrt{\frac{0.06*(1-0.06)}{280}}= 0.0142[/tex]
And we want to find this probability:
[tex] p(\hat p <0.09)[/tex]
And we can use the z score formula given by:
[tex] z = \frac{p \mu_{\hat p}}{\sigma_{\hat p}}[/tex]
And replacing we got:
[tex] z = \frac{0.09-0.06}{0.0142}= 2.114[/tex]
And we can find this probability:
[tex] P(z<2.114)[/tex]
And using the normal table or excel we got:
[tex] P(z<2.114)= 0.9827[/tex]
