Answer:
a
The upper bound of the 99% prediction level is
b
The 95% confidence interval is
Step-by-step explanation:
Considering first question
From the question we are told that
The sample size is n = 30
The sample mean is ![\= x = 96.2\%](https://tex.z-dn.net/?f=%5C%3D%20x%20%20%3D%20%2096.2%5C%25)
The standard deviation is ![s = 0.8\%](https://tex.z-dn.net/?f=s%20%20%3D%200.8%5C%25)
Generally the degree of freedom is mathematically represented as
![df = n - 1](https://tex.z-dn.net/?f=df%20%20%3D%20%20n%20-%201)
=> ![df = 30 - 1](https://tex.z-dn.net/?f=df%20%20%3D%20%2030%20-%201)
=> ![df = 29](https://tex.z-dn.net/?f=df%20%20%3D%20%2029)
From the question we are told the confidence level is 99% , hence the level of significance is
![\alpha = (100 - 99 ) \%](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%28100%20-%2099%20%29%20%5C%25)
=> ![\alpha = 0.01](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.01)
Generally from the t distribution table the critical value of at a degree of freedom of is
![t_{\alpha , 29} = 2.462](https://tex.z-dn.net/?f=t_%7B%5Calpha%20%2C%2029%7D%20%3D%202.462)
Generally the 99% prediction level is mathematically represented as
![\= x \pm [(t_{\alpha , df }) * s * (\sqrt{1 + \frac{1}{ n} } )}]](https://tex.z-dn.net/?f=%5C%3D%20x%20%5Cpm%20%5B%28t_%7B%5Calpha%20%20%2C%20df%20%7D%29%20%2A%20s%20%2A%20%28%5Csqrt%7B1%20%2B%20%5Cfrac%7B1%7D%7B%20n%7D%20%7D%20%29%7D%5D%20)
Generally the upper bound of the 99% prediction level is mathematically represented as
=>
=>
Considering second question
Generally the sample is mathematically represented as
![\= x = \frac{\sum x_i}{n}](https://tex.z-dn.net/?f=%5C%3D%20x%20%20%3D%20%5Cfrac%7B%5Csum%20x_i%7D%7Bn%7D)
=>
=>
Generally the standard deviation is mathematically represented as
![\sigma = \sqrt{ \frac{ \sum ( x_ i - \= x)}{n-1} }](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%20%5Csqrt%7B%20%5Cfrac%7B%20%5Csum%20%28%20x_%20i%20-%20%5C%3D%20x%29%7D%7Bn-1%7D%20%7D)
=> ![\sigma = \sqrt{ \frac{ ( 9.8 -10)^2 + ( 10.2 -10)^2 + \cdots + ( 9.6 -10)^2 }{7-1} }](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%20%5Csqrt%7B%20%5Cfrac%7B%20%28%209.8%20%20-10%29%5E2%20%2B%20%20%28%2010.2%20%20-10%29%5E2%20%2B%20%5Ccdots%20%2B%20%28%209.6%20%20-10%29%5E2%20%20%7D%7B7-1%7D%20%7D)
=> ![\sigma = 0.283](https://tex.z-dn.net/?f=%5Csigma%20%3D%200.283)
Generally the degree of freedom is mathematically represented as
![df = n- 1](https://tex.z-dn.net/?f=%20df%20%3D%20%20n-%201%20)
=> ![df = 7- 1](https://tex.z-dn.net/?f=%20df%20%3D%20%207-%201%20)
=> ![df = 6](https://tex.z-dn.net/?f=%20df%20%3D%20%206%20)
From the question we are told the confidence level is 95% , hence the level of significance is
![\alpha = (100 - 95 ) \%](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%28100%20-%2095%20%29%20%5C%25)
=> ![\alpha = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.05)
Generally from the t distribution table the critical value of at a degree of freedom of is
![t_{\frac{\alpha }{2} , 6 } = 2.447](https://tex.z-dn.net/?f=t_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%2C%206%20%7D%20%3D%20%202.447)
Generally the margin of error is mathematically represented as
![E = t_{\frac{\alpha }{2} , 6 } * \frac{\sigma }{\sqrt{n} }](https://tex.z-dn.net/?f=E%20%3D%20t_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%2C%206%20%7D%20%2A%20%20%5Cfrac%7B%5Csigma%20%7D%7B%5Csqrt%7Bn%7D%20%7D)
=> ![E =2.447* \frac{0.283 }{\sqrt{7} }](https://tex.z-dn.net/?f=E%20%3D2.447%2A%20%20%20%20%5Cfrac%7B0.283%20%7D%7B%5Csqrt%7B7%7D%20%7D)
=> ![E =0.2617](https://tex.z-dn.net/?f=E%20%3D0.2617)
Generally 95% confidence interval is mathematically represented as
=>
=>