Complete Question
The complete question is shown on the first uploaded image
Answer:
a
The 95% confidence interval is [tex] 228.5< \mu < 231.5 [/tex]
b
The width will reduce by one
c
The width will remain the same
Step-by-step explanation:
Considering question a
From the question we are told that
The sample size is n = 36
The standard deviation is [tex]\sigma =4.44 \ inches[/tex]
The mean is [tex]\mu = 230 \ inches[/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 normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E_1 = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E_1 = 1.96* \frac{4.44}{\sqrt{36} }[/tex]
=> [tex]E_1 = 1.4504 [/tex]
Generally the width of the confidence interval is
[tex]W_1 = 2 * E_1[/tex]
=> [tex]W_1 = 2 * 1.4504[/tex]
=> => [tex]W_1 = 3[/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \=x +E[/tex]
=> [tex] 230 -1.4504 < \mu < 230 + 1.4504 [/tex]
=> [tex] 228.5< \mu < 231.5 [/tex]
Considering question b
when [tex]\sigma_1 = 3.33 \ inches[/tex]
Generally the margin of error is mathematically represented as
[tex]E_2 = Z_{\frac{\alpha }{2} } * \frac{\sigma_1 }{\sqrt{n} }[/tex]
=> [tex]E_2 = 1.96* \frac{3.33}{\sqrt{36} }[/tex]
=> [tex]E_2 = 1.0878 [/tex]
Generally the width of the confidence interval is
[tex]W_2 = 2 * E_2[/tex]
=> [tex]W_2 = 2 * 1.0878[/tex]
=> [tex]W_2 = 2[/tex]
So comparing [tex]W_1 \ and \ W_2[/tex] we see that the width will decrease by 1
Considering question b
When [tex]\= x = 71[/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \=x +E[/tex]
=> [tex] 69.55< \mu < 72.45[/tex]
Generally the width is mathematically represented as
[tex]W_3 = 72.45 - 69.55[/tex]
=> [tex]W_3 = 3[/tex]
Comparing [tex]W_2 \ and \ W_1[/tex] we see that the width of the confidence interval remain the same
