Question

The 2004 presidential election exit polls from the critical state of Ohio provided the following results. There were 2020 respondents in the exit polls and 768 were college graduates. Of the college graduates, 412 voted for George Bush.
(a) Calculate a 99% confidence interval for the proportion of college graduates in Ohio that voted for George Bush. Round the answers to 3 decimal places.

_______________\leq p\leq_______________

(b) Calculate a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for George Bush. Round the answer to 3 decimal places.

_________________\leq p

Answer 1

Answer: a)  $0.490\leq p\leq0.582$

b)  $0.501\leq p$

Step-by-step explanation:

Given : Sample size of respondents in the exit polls  : n= 2020

Number of respondents voted for George Bush = 412

Sample proportion: $\hat{p}=\dfrac{412}{768}\approx0.536$

a) Critical value for 99% confidence level : $z_{\alpha/2}=2.576$

Confidence interval for proportion:-

$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

$=0.536\pm (2.576)\sqrt{\dfrac{0.536(1-0.536)}{768}}\\\\=0.536\pm0.046\\\\=(0.490,\ 0.582)$

Hence, the 99% confidence interval for the proportion of college graduates in Ohio that voted for George Bush: $0.490\leq p\leq0.582$

b) Critical value for 95% confidence level : $z_{\alpha/2}=1.96$

Lower confidence bound for the proportion :

$\hat{p}- z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.536-(1.96)\sqrt{\dfrac{0.536(1-0.536)}{768}}\\\\=0.536-0.035=0.501$

Hence,  a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for George Bush : $0.501\leq p$

Answer 2

a. 99% confidence interval for the proportion of college graduates in Ohio that voted for George Bush is $0.490 \leq p \leq 0.582$

b. a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for George Bush is $P \geq 0.507$

Explanation:

The 2004 presidential election exit polls from the critical state of Ohio provided the following results. There were 2020 respondents in the exit polls and 768 were college graduates. Of the college graduates, 412 voted for George Bush.

(a) Calculate a 99% confidence interval for the proportion of college graduates in Ohio that voted for George Bush. Round the answers to 3 decimal places.

Sample size n = 2020

Number of college graduates who voted for George Bush x = 412

 The point estimate of proportion g college graduates in Ohio who voted for George Bush is $p = \frac{x}{h} =\frac{412}{768} =0.536$

For using normal distribution, check ($np$) and $n(1-p) \geq 5$

$np$ -> $768(0.5365)$ -> $412>5$

$n(1-p)$ -> $768(0.4635)$ -> $356>5$

For 99% CI ->  $Z_{\frac{\alpha}{2}} = z_{0.025} = 2.576$

Resulting CI ->

$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\=0.536\pm (2.576)\sqrt{\dfrac{0.536(1-0.536)}{768}}\\\\=0.536\pm0.046\\\\=(0.490,\ 0.582)$

Therefore the result is $0.490 \leq p \leq 0.582$

(b) Calculate a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for George Bush. Round the answer to 3 decimal places.

For 95% CI then $Z_{0.05} = Z_{\alpha} = 1.645$ (lower bound)

Resulting CI -> $P \geq 0.5365 - 1.645 \sqrt{\frac{0.2487}{768} }$

Simplifies to -> $P \geq 0.507$

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