Complete Question
The complete question is shown on the first uploaded image
Answer:
The null hypothesis is rejected this means that [tex]\mu \ne 0.5025[/tex]
The 95% confidence interval is [tex]0.5045608 < \mu < 0.5046392 [/tex]
Step-by-step explanation:
From the question we are told that
The standard deviation is s= 0.0001
The sample size is n = 25
The sample mean is [tex]\= x = 0.5046 \ in[/tex]
The population mean is [tex]\mu = 0.5025 \ in[/tex]
The null hypothesis is [tex]H_o : \mu = 0.5025[/tex]
The alternative hypothesis is [tex]H_a : \mu \ne 0.5025[/tex]
The test statistics is mathematically represented as
[tex]t = \frac{0.5046 - 0.5025}{ \frac{0.0001}{\sqrt{25} } }[/tex]
[tex]t = 105[/tex]
So the p-value from the z-table is mathematically represented as
[tex]p-value = 2 * P( z > 105)[/tex]
[tex]p-value = 0.000[/tex]
seeing that
[tex]p-value < \alpha[/tex] we reject the null hypothesis
The critical value of
[tex]\frac{\alpha }{2}[/tex] obtained from the normal distribution table is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
The margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} }*\frac{s}{\sqrt{n} }[/tex]
=> [tex]E = 1.96 *\frac{0.0001}{\sqrt{25} }[/tex]
=> [tex]E =3.92 *10^{-5} [/tex]
The 95% confidence level is mathematically represented as
[tex]\= x - E < \mu < \= x + E[/tex]
=> [tex]0.5046 - 3.92 *10^{-5} < \mu < 0.5046 + 3.92 *10^{-5} [/tex]
=> [tex]0.5045608 < \mu < 0.5046392 [/tex]
