An article about the relative priorities of protecting the environment and the economy in America
summarized a survey of 1,600 adult U.S. citizens. When asked if they expect to feel the effects of climate
change in their lifetime, 912 of the survey participants responded yes. The article states that participants
were selected to be representative of all adult U.S. citizens.
(a) Estimate the proportion of adult U.S. citizens who think that they will feel the effects of climate
change in their lifetime
(b) Use the sample data to estimate the standard error of p̂. (Round your answer to three decimal
places.)
(c) Calculate the margin of error associated with the estimate in Part (a). (Hint: See Example 9.3. Round
your answer to three decimal places.)
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(a) The proportion of adult U.S. citizens who think that they will feel the effects of climate change in their lifetime is estimated by the number of participants who responded yes divided by the total number of participants in the survey.
Proportion = Number of participants who responded yes / Total number of participants
Proportion = 912 / 1600
Proportion ≈ 0.570
So, approximately 57% of adult U.S. citizens in the survey think that they will feel the effects of climate change in their lifetime.
(b) The standard error of p̂, the sample proportion, can be estimated using the formula:
Standard Error = √((p̂ * (1 - p̂)) / n)
where p̂ is the sample proportion and n is the sample size.
In this case, p̂ ≈ 0.570 and n = 1600
Standard Error = √((0.570 * (1 - 0.570)) / 1600)
Standard Error ≈ 0.014
So, the estimated standard error of p̂ is approximately 0.014.
(c) The margin of error associated with the estimate in part (a) can be calculated using the formula:
Margin of Error = Critical value * Standard Error
To approximate the critical value, we can use a 95% confidence level, which corresponds to a z-score of approximately 1.96.
Margin of Error ≈ 1.96 * 0.014
Margin of Error ≈ 0.027
So, the margin of error associated with the estimate in part (a) is approximately 0.027 or 2.7%.
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