Answer:
a
The upper bound of the 99% prediction level is [tex] 98.2 [/tex]
b
The 95% confidence interval is [tex]9.7383 < \mu < 10.2617 [/tex]
Step-by-step explanation:
Considering first question
From the question we are told that
The sample size is n = 30
The sample mean is [tex]\= x = 96.2\%[/tex]
The standard deviation is [tex]s = 0.8\%[/tex]
Generally the degree of freedom is mathematically represented as
[tex]df = n - 1[/tex]
=> [tex]df = 30 - 1[/tex]
=> [tex]df = 29[/tex]
From the question we are told the confidence level is 99% , hence the level of significance is
[tex]\alpha = (100 - 99 ) \%[/tex]
=> [tex]\alpha = 0.01[/tex]
Generally from the t distribution table the critical value of at a degree of freedom of is
[tex]t_{\alpha , 29} = 2.462[/tex]
Generally the 99% prediction level is mathematically represented as
[tex]\= x \pm [(t_{\alpha , df }) * s * (\sqrt{1 + \frac{1}{ n} } )}] [/tex]
Generally the upper bound of the 99% prediction level is mathematically represented as
[tex]\= x + [(t_{\alpha , df }) * s * (\sqrt{1 + \frac{1}{ n} } )}] [/tex]
=> [tex] 96.2 + (2.462 ) * 0.8 * (\sqrt{1 + \frac{1}{ 30} } )}] [/tex]
=> [tex] 98.2 [/tex]
Considering second question
Generally the sample is mathematically represented as
[tex]\= x = \frac{\sum x_i}{n}[/tex]
=> [tex]\= x = \frac{ 9.8 + 10.2 + \cdots +9.6 }{7}[/tex]
=> [tex]\= x = 10[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ \frac{ \sum ( x_ i - \= x)}{n-1} }[/tex]
=> [tex]\sigma = \sqrt{ \frac{ ( 9.8 -10)^2 + ( 10.2 -10)^2 + \cdots + ( 9.6 -10)^2 }{7-1} }[/tex]
=> [tex]\sigma = 0.283[/tex]
Generally the degree of freedom is mathematically represented as
[tex] df = n- 1 [/tex]
=> [tex] df = 7- 1 [/tex]
=> [tex] df = 6 [/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the t distribution table the critical value of at a degree of freedom of is
[tex]t_{\frac{\alpha }{2} , 6 } = 2.447[/tex]
Generally the margin of error is mathematically represented as
[tex]E = t_{\frac{\alpha }{2} , 6 } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E =2.447* \frac{0.283 }{\sqrt{7} }[/tex]
=> [tex]E =0.2617[/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \=x +E[/tex]
=> [tex]10 -0.2617 < \mu < 10 + 0.2617[/tex]
=> [tex]9.7383 < \mu < 10.2617 [/tex]
