Respuesta :
Answer:
(a) The point estimate of the proportion of the population who would answer yes is 0.295.
(b) The margin of error for a 95% confidence interval is 0.0259.
(c) 95% confidence interval for population proportion is [0.2691 , 0.3209].
Step-by-step explanation:
We are given that a survey asked subjects whether they would be willing to accept cuts in their standard of living to protect the environment, 352 of 1195 subjects said yes.
Let [tex]\hat p[/tex] = sample proportion of subjects who said yes.
(a) The point estimate of the proportion of the population who would answer yes = [tex]\hat p[/tex] = [tex]\frac{X}{n}[/tex]
So, [tex]\hat p = \frac{352}{1195}[/tex] = 0.295
(b) Margin of error is given by = [tex]Z_(_\frac{\alpha}{2}_) \times \text{Standard of Error}[/tex]
where, [tex]\alpha[/tex] = level of significance = 1 - 0.95 = 5%
At 5% level of significance, z table gives critical value of 1.96 for two-sided interval.
Standard of error = [tex]\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] = [tex]\sqrt{\frac{0.295(1-0.295)}{1195} }[/tex] = 0.0132
So, Margin of error for 95% confidence interval = [tex]1.96 \times 0.0132[/tex]
= 0.0259
(c) 95% confidence interval for population proportion is given by =
Point estimate [tex]\pm[/tex] Margin of error
⇒ 0.295 [tex]\pm[/tex] 0.0259
⇒ [0.295 - 0.0259 , 0.295 + 0.0259]
⇒ [0.2691 , 0.3209]
So, 95% confidence interval = [0.2691 , 0.3209]
The numbers in this interval represent that we are 95% confident that the population proportion will lie between 0.2691 and 0.3209.
(d) Assumptions needed for constructing a confidence interval are;
- The data must be sampled randomly.
- Sample values must be independent of each other.
- Data must follow normal distribution.
