During a national debate on changes to health care, a cable news service performs an opinion poll of 570 small-business owners. It shows that 72 percent of small-business owners do not approve of the changes.
(a) Develop a 90 percent confidence interval for the proportion opposing health care changes
Find the confidence intervals for the proportion
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To develop a confidence interval for the proportion of small-business owners opposing health care changes, we can use the following steps:
1. Determine the sample proportion: The sample proportion is calculated by dividing the number of small-business owners who do not approve of the changes (72% of 570) by the total number of small-business owners surveyed.
Sample proportion (p̂) = (Sample size of small-business owners opposing health care changes) / (Total sample size of small-business owners surveyed)
p̂ = (0.72) * (570)
2. Determine the standard error: The standard error (SE) is the measure of the uncertainty or variability in the sample proportion.
Standard error (SE) = √((p̂*(1-p̂))/n)
Where:
p̂ is the sample proportion,
1-p̂ is the proportion of small-business owners approving of the changes,
n is the sample size (570).
3. Determine the critical value: The critical value depends on the desired level of confidence. Here we want a 90 percent confidence interval, meaning there is a 90 percent probability that the true proportion lies within the interval.
For a 90 percent confidence interval, the critical value is 1.645 (based on the z-distribution).
4. Calculate the confidence interval: The confidence interval formula is given by:
Confidence interval = p̂ ± (critical value * standard error)
Using the calculated values from steps 1-3, we can calculate the confidence interval for the proportion opposing health care changes.
Confidence interval = p̂ ± (1.645 * SE)
Substituting the values:
Confidence interval = 0.72 ± (1.645 * √((0.72*(1-0.72))/570))
Now we can evaluate and calculate the confidence interval.