When a survey asked subjects whether they would be willing to accept cuts in their standard of living to protect the environment, 387 of 1160 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.

Step-by-step explanation:

The sample proportion is the point estimate for the population proportion.

Confidence interval is written as

Sample proportion ± margin of error

Margin of error = z × √pq/n

Where

z represents the z score corresponding to the confidence level

p = sample proportion. It also means probability of success

q = probability of failure

q = 1 - p

p = x/n

Where

n represents the number of samples

x represents the number of success

a) From the information given,

n = 1160

x = 387

p = 387/1160 = 0.33

q = 1 - 0.33 = 0.67

the point estimate of the proportion of the population who would answer yes = 0.33

b) To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.5 = 0.05

α/2 = 0.05/2 = 0.025

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.025 = 0.975

The z score corresponding to the area on the z table is 1.96. Thus, confidence level of 95% is 1.96

Therefore,

Margin of error = 1.96√(0.33)(0.67)/1160 = 0.027

c) the 95% confidence interval for the population proportion is

0.33 ± 0.027

The numbers represent the sample proportion and the margin of error

d) The assumptions are

1) the sampling must be random

2) the sample size shouldn't be more than 10% of the population

10/100 × 1160 = 116

387 > 116

3) the sample size should be sufficiently large. That is, greater than 30

387 > 30

25

step-by-step explanation:

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