Answer :
The 99% confidence interval for the population proportion of printers that are used in small businesses, using the z-distribution, is of:
(0.2743, 0.5493)
What is a confidence interval of proportions?
A confidence interval of proportions has the bounds given by the equation presented next:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which the variables used to calculated these bounds are listed as follows:
- [tex]\pi[/tex] is the sample proportion, which is also the point estimate of the parameter.
- z is the critical value of the z-distribution.
- n is the sample size.
The confidence level is of 99%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.99}{2} = 0.995[/tex], so the critical value is z = 2.575.
(the critical value can be found either with the z-table or with a z-distribution calculator).
The sample size and the estimate are listed as follows:
[tex]n = 85, \pi = \frac{35}{85} = 0.4118[/tex]
The lower bound of the interval is obtained as follows:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.4118 - 2.575\sqrt{\frac{0.4118(0.5882)}{85}} = 0.2743[/tex]
The upper bound of the interval is obtained as follows:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.4118 + 2.575\sqrt{\frac{0.4118(0.5882)}{85}} = 0.5493[/tex]
More can be learned about the z-distribution at https://brainly.com/question/25890103
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