Answer:
The 95% confidence interval is
[tex] 5.92 < \mu < 7.284 [/tex]
Step-by-step explanation:
From the question we are told that
The data is 6.5 6.7 7.2 6.4 7.3
Generally the sample mean is mathematically represented as
[tex]\= x = \frac{\sum x }{n}[/tex]
=> [tex]\= x = \frac{ 5.5 + 6.5 + \cdot 7.3 }{ 6}[/tex]
=> [tex]\= x = 6.6[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ \frac{ \sum (x_i - \= x )^2 }{ n-1} }[/tex]
=> [tex]\sigma = \sqrt{ \frac{ (5.5 - 6.6 )^2 +(6.5 - 6.6 )^2 + \cdot + (7.3 - 6.6 )^2 }{ 6-1} }[/tex]
=> [tex]\sigma = 0.6512[/tex]
Gnerally the degree of freedom is mathematically represented as
[tex]df = n- 1[/tex]
=> [tex]df = 6 -1[/tex]
=> [tex]df = 5[/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 [tex]\frac{\alpha }{2}[/tex] is
[tex]t_{\frac{\alpha }{2} , 5 } = 2.571 [/tex]
Generally the margin of error is mathematically represented as
[tex]E = t_{\frac{\alpha }{2}, 5 } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E = 2.571 * \frac{0.6512}{\sqrt{6} }[/tex]
=> [tex]E = 0.6835 [/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \= x +E[/tex]
=> [tex]6.6 - 0.6835 < \mu < 6.6 + 0.6835[/tex]
=> [tex] 5.92 < \mu < 7.284 [/tex]
