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For many years businesses have struggled with the rising cost of health care. But recently, the increases have slowed due to less inflation in health care prices and employees paying for a larger portion of health care benefits. A recent Mercer survey showed that 52% of U.S. employers were likely to require higher employee contributions for health care coverage. Suppose the survey was based on a sample of 1,000 companies. Compute the margin of error and a 95% confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage. If required, round your answer to four decimal places. Round intermediate calculations to four decimal places.

Respuesta :

Answer:

The confidence interval would be given by this formula

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.96[/tex]

The margin of error for this case is given by:

[tex] ME= z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

And replacing we got:

[tex]ME = 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.0259[/tex]

And replacing into the confidence interval formula we got:

[tex]0.52 - 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.4941[/tex]

[tex]0.52 + 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.5459[/tex]

And the 95% confidence interval would be given (0.4941;0.5459).

Step-by-step explanation:

Data given and notation  

n=1000 represent the random sample taken    

[tex]\hat p=0.52[/tex] estimated proportion of of U.S. employers were likely to require higher employee contributions for health care coverage

[tex]\alpha=0.05[/tex] represent the significance level (no given, but is assumed)    

Solution to the problem

The confidence interval would be given by this formula

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.96[/tex]

The margin of error for this case is given by:

[tex] ME= z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

And replacing we got:

[tex]ME = 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.0259[/tex]

And replacing into the confidence interval formula we got:

[tex]0.52 - 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.4941[/tex]

[tex]0.52 + 1.64*\sqrt{\frac{0.52(1-0.52)}{1000}}=0.5459[/tex]

And the 95% confidence interval would be given (0.4941;0.5459).

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