Question

When a survey asked subjects whether they would be willing to accept cuts in their standard of living to protect the​ environment, 352352 of 11951195 subjects said yes. a. Find the point estimate of the proportion of the population who would answer yes. b. Find the margin of error for a​ 95% confidence interval. c. Construct the​ 95% confidence interval for the population proportion. What do the numbers in this interval​ represent? d. State and check the assumptions needed for the interval in ​(c) to be valid.

Answer 1

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 $\hat p$ = sample proportion of subjects who said yes.

(a) The point estimate of the proportion of the population who would answer yes = $\hat p$ =  $\frac{X}{n}$

So,  $\hat p = \frac{352}{1195}$  = 0.295

(b) Margin of error is given by =  $Z_(_\frac{\alpha}{2}_) \times \text{Standard of Error}$

where, $\alpha$ = 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 =  $\sqrt{\frac{\hat p(1-\hat p)}{n} }$ =  $\sqrt{\frac{0.295(1-0.295)}{1195} }$  = 0.0132

So, Margin of error for 95% confidence interval = $1.96 \times 0.0132$

                                                                               = 0.0259

(c) 95% confidence interval for population proportion is given by =

                   Point estimate $\pm$ Margin of error

              ⇒ 0.295 $\pm$ 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.