Question

gallup regularly conducts a poll using a \cantril scale,\ which asks respondents to imagine a ladder with 10 rungs. rung 0 represents the worst possible life and rung 10 represents the best possible life. respondents are asked what rung they would say they are on. responses are classified as \thriving\ (standing on rung 7 or higher and expecting to be on rung 8 or higher five years from now), \suffering\ (standing on rung 4 or lower and expecting to be on rung 4 or lower 5 years from now), or \struggling\ (not thriving or suffering). in may of 2020, gallup found that the index had reached the lowest level since the great recession of 2008, at 48.8% thriving, with a sample size of 20,006. complete parts a through c below.
b) calculate the standard error of \\(\hat{p}\\).
\\(se(\hat{p}) = \\) \\(0.0035340\\) (round to seven decimal places as needed.)
c) explain what this standard error means in the context of this situation. choose the correct answer below.
a. the standard error is the best estimate of the margin of error of the poll. the researchers can be confident that the difference between \\(p\\) and \\(\hat{p}\\) is not greater than the standard error.
b. the standard error is the best estimate of the standard deviation of the sampling distribution of the proportions, which measures the amount of variation in the sample and population proportions.
c. the standard error is the best estimate of the margin of error of the poll, which measures the amount of variation in the sample size expected to be seen from sample to sample when 20,006 people are asked the polling question.
d. the standard error is the best estimate of the standard deviation of the sampling distribution of the proportions, which measures the amount of variation in the sample proportion expected to be seen from sample to sample of 20,006 people.

Explanation:

Step1: Identify given values

Sample proportion $\hat{p}=0.488$, sample size $n=20006$

Step2: Apply standard error formula

The formula for the standard error of a sample proportion is $SE(\hat{p})=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Substitute values:

$$ SE(\hat{p})=\sqrt{\frac{0.488\times(1-0.488)}{20006}}=\sqrt{\frac{0.488\times0.512}{20006}} $$

Calculate numerator: $0.488\times0.512=0.249856$
Then:

$$ SE(\hat{p})=\sqrt{\frac{0.249856}{20006}}=\sqrt{0.000012489}≈0.0035340 $$

Step3: Interpret standard error

Match the correct description of the standard error for a sample proportion.

Answer:

b) $0.0035340$
c) D. The standard error is the best estimate of the standard deviation of the sampling distribution of the proportions, which measures the amount of variation in the sample proportion expected to be seen from sample to sample of 20,006 people.