Respuesta :
Complete Question
The complete question is shown on the first uploaded image
Answer:
Using 2SD Method
The confidence interval is
The interval notation : [tex](0.108 , 0.212 )[/tex]
The interval notation and the sample proportion ± the margin of error
notation:[tex]0.16 \pm 0.052[/tex]
The interpretation
There is 95% confidence that the true proportion of those that dislike cilantro lie within the upper(0.212) and the lower(0.108) limit of the calculate confidence interval
Using theory-based method
The 95% confidence interval is
The interval notation : [tex](0.109 , 0.211 )[/tex]
The interval notation and the sample proportion ± the margin of error
notation:[tex]0.16 \pm 0.051[/tex]
The interpretation
There is 95% confidence that the true proportion of those that dislike cilantro lie within the upper(0.211) and the lower(0.109) limit of the calculate confidence interval
Using theory-based method
The 88% confidence interval is
The interval notation : [tex](0.120 , 0.200 )[/tex]
The interval notation and the sample proportion ± the margin of
error notation: [tex]0.16 \pm 0.040[/tex]
The interpretation is
There is 88% confidence that the true proportion of those that dislike cilantro lie within the upper(0.200 ) and the lower(0.120) limit of the calculate confidence interval.
Step-by-step explanation:
From the question we are told that
The number sample size is n = 200
The number of people that dislike cilantro is k = 32
Generally the sample proportion is mathematically represented as
[tex]\r p = \frac{k}{N}[/tex]
=> [tex]\r p = \frac{32}{200}[/tex]
=> [tex]\r p = 0.16[/tex]
Generally 2SD confidence interval is mathematically represented as
[tex]\r p \pm 2\sqrt{\frac{\r p(1 - \r p)}{n} }[/tex]
substituting value
[tex]0.16 \pm 2\sqrt{\frac{0.16(1 - 0.16)}{200} }[/tex]
=> [tex]0.16 \pm 0.052[/tex]
This can also be represented as
[tex](0.16 - 0.052 , 0.06 + 0.052)[/tex]
=> [tex](0.108 , 0.212 )[/tex]
Generally this confidence interval can be interpreted as
There is 95% confidence that the true proportion of those that dislike cilantro lie within the upper and the lower limit of the calculate confidence interval
Using theory-based method estimate 95% confidence interval
Generally from the question the confidence level is 95% hence the level of significance is calculated as
[tex]\alpha = (100 - 95)\%[/tex]
=> [tex]\alpha = 0.05[/tex]
The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the confidence level using theory base method is
[tex]0.16 \pm 1.96 \sqrt{\frac{0.16 (1- 0.16)}{200} }[/tex]
=> [tex]0.16 \pm 0.051[/tex]
This can also be represented as
[tex](0.16 - 0.051 , 0.06 + 0.051)[/tex]
=> [tex](0.109 , 0.211 )[/tex]
Generally this confidence interval can be interpreted as
There is 95% confidence that the true proportion of those that dislike cilantro lie within the upper(0.211) and the lower(0.109) limit of the calculate confidence interval
Using theory-based method estimate 88% confidence interval
Generally from the question the confidence level is 95% hence the level of significance is calculated as
[tex]\alpha = (100 - 88)\%[/tex]
=> [tex]\alpha = 0.12[/tex]
The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table is
[tex]Z_{\frac{\alpha }{2} } =Z_{\frac{0.12 }{2} } = 1.55[/tex]
Generally the confidence level using theory base method is
[tex]0.16 \pm 1.55 \sqrt{\frac{0.16 (1- 0.16)}{200} }[/tex]
=> [tex]0.16 \pm 0.040[/tex]
This can also be represented as
[tex](0.16 - 0.040 , 0.06 + 0.040)[/tex]
=> [tex](0.120 , 0.200 )[/tex]
Generally this confidence interval can be interpreted as
There is 88% confidence that the true proportion of those that dislike cilantro lie within the upper(0.200 ) and the lower(0.120) limit of the calculate confidence interval
