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A recent poll of 500 registered Democrats found that 315 of them worry a great deal about the...

Question:

A recent poll of 500 registered Democrats found that 315 of them worry a great deal about the environment.

(a) Make a 95% confidence interval for the proportion of all Democrats who think that climate change is a serious problem.

(b) In the same poll (at Gallop.org), 20% of Republicans surveyed worry a great deal about the environment. Is that value significantly different from the statistic reported for Democrats? Explain mathematically.

Confidence interval

The confidence interval provides a lower upper limit which contains an unknown population parameter. The sample mean and sample proportion are the best estimate of the confidence interval and the value of the sample proportion expressed between 0 to 1.

Answer and Explanation:

Given information

Sample size: 500

Number of success: 315


The value of sample proportion is calculated as follow.

{eq}\begin{align*} \hat P& = \dfrac{X}{n}\\ & = \dfrac{{315}}{{500}}\\ & = 0.63 \end{align*}{/eq}

a)

The 95% confidence interval for the proportion of all Democrats who think that climate change is a serious problem is calculated as follow.

{eq}\begin{align*} P\left( {\hat P - {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} < p < \hat P + {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} } \right) &= 0.95\\ P\left( {0.63 - 1.96\sqrt {\dfrac{{0.63\left( {1 - 0.63} \right)}}{{500}}} < p < 0.63 + 1.96\sqrt {\dfrac{{0.63\left( {1 - 0.63} \right)}}{{500}}} } \right) &= 0.95\\ P\left( {0.58 < \mu < 0.67} \right) &= 0.95 \end{align*}{/eq}

Therefore, required confidence interval is 0.58 to 0.67

b)

Therefore, the 20% of Republicans surveyed worry a great deal about the environment is does not lies between confidence limits which means the value significantly different from the statistic reported for Democrats.


Learn more about this topic:

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Finding Confidence Intervals for Proportions: Formula & Example

from Statistics 101: Principles of Statistics

Chapter 9 / Lesson 8
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