Respuesta :
Answer: a) (2.57, 3.01)
b) B) We can be 99% confident that the mean gross earnings of all Rolling Stones concerts ies in the interval
Step-by-step explanation:
a) Confidence interval formula :
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
, where n = sample size .
[tex]\sigma[/tex] = Population standard deviation.
[tex]\overline{x}[/tex] = Sample mean
[tex]z_{\alpha/2}[/tex] = Two-tailed z-value for significance level ([tex]\alpha[/tex]).
As per given :
n= 30
[tex]\overline{x}=2.79[/tex]
[tex]\sigma=0.47[/tex]
[tex]\alpha=1-0.99=0.01[/tex]
Using z-value table ,
Two-tailed z-value : [tex]z_{\alpha/2}=z_{0.005}=2.576[/tex]
Confidence interval :
[tex]2.79\pm (2.576)\dfrac{0.47}{\sqrt{30}}[/tex]
[tex]2.79\pm (2.576)(0.0858)[/tex]
[tex]\approx2.79\pm 0.22\\\\=(2.79-0.22,\ 2.79+0.22)=(2.57,\ 3.01)[/tex]
Thus , a 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions) : (2.57, 3.01)
Interpretation of a 99% confidence interval : We are 99% confident that the true population mean lies in it.
Thus , the correct interpretation of the given situation :
We can be 99% confident that the mean gross earnings of all Rolling Stones concerts lies in the interval.
Using the z-distribution, it is found that the 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions) is (2.57, 3.01).
The interpretation is:
(B) We can be 99% confident that the mean gross earnings of all Rolling Stones concerts lies in the interval.
We are given the standard deviation for the population, which is why the z-distribution is used to solve this question. We also have that:
- The sample mean is of [tex]\overline{x} = 2.79[/tex].
- The population standard deviation is of [tex]\sigma = 0.47[/tex].
- The sample size is of [tex]n = 30[/tex].
The confidence interval is:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
The first step is finding the critical value, which is z with a p-value of [tex]\frac{1 + \alpha}{2}[/tex], in which [tex]\alpha[/tex] is the confidence level.
In this problem, [tex]\alpha = 0.99[/tex], thus, z with a p-value of [tex]\frac{1 + 0.95}{2} = 0.995[/tex], which means that it is z = 2.575.
Thus, the interval is given by:
[tex]\overline{x} - z\frac{\sigma}{\sqrt{n}} = 2.79 - 2.575\frac{0.47}{\sqrt{30}} = 2.57[/tex]
[tex]\overline{x} + z\frac{\sigma}{\sqrt{n}} = 2.79 + 2.575\frac{0.47}{\sqrt{30}} = 3.01[/tex]
The 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions) is (2.57, 3.01).
The interpretation is that we are 99% sure that the mean for all shows is in this interval, thus option B.
A similar problem is given at https://brainly.com/question/22596713
