Respuesta :
Answer:
The 99% two-sided confidence interval for p, the proportion of bearings with surface finish rougher than allowed specification is (0.1430, 0.3788). The upper bound of this 2-sided confidence interval is 0.3788.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
In a random sample of 92 automobile engine crankshaft bearings, 24 have a surface finish that is rougher than the specifications allow.
This means that [tex]n = 92, \pi = \frac{24}{92} = 0.2609[/tex]
99% confidence level
So [tex]\alpha = 0.01[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2609 - 2.575\sqrt{\frac{0.2609*0.7391}{92}} = 0.1430[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2609 + 2.575\sqrt{\frac{0.2609*0.7391}{92}} = 0.3788[/tex]
The 99% two-sided confidence interval for p, the proportion of bearings with surface finish rougher than allowed specification is (0.1430, 0.3788). The upper bound of this 2-sided confidence interval is 0.3788.
