Answer :
Answer:
The answers to the questions are
(a) P = 0.0176
(b) P = 0.5
(c) P = 0.0571
Step-by-step explanation:
The P value is the calculated probability of obtaining extreme reults which are as extreme as observed res ult when the study null hypothesis H₀ is true
The p value is given by
P ([tex]\overline{\rm x}[/tex] = 185 when μ = 175)
z = [tex]\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} }[/tex]
z([tex]\overline { \rm x}[/tex] = 185) = [tex]\frac{ 185 -175}{20/\sqrt{10} }[/tex] = 1.5811
Therefore the P ([tex]\overline{\rm x}[/tex] = 185 when μ = 175) = P(z = 1.5811)
0.94295
The probability density function is
[tex]f(x) = \frac{1}{\sigma\sqrt{2\pi } } e^{-\frac{1}{2}(\frac{x-\mu}{\sigma} )^2 }[/tex] Therefore the probability is
[tex]f(x) = \frac{1}{20\sqrt{2\pi } } e^{-\frac{1}{2}(\frac{185-175}{20} )^2 }[/tex] = 0.0176
b) The type II error is given by
β = P(type II error) = P(Fail to reject the null hypothesis when it is false)
Therefore
β = P ([tex]\overline { \rm x}[/tex] < 185 when μ = 185)
P[tex](\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} } <\frac{185-185}{20/\sqrt{10} } )[/tex], which gives
P(z<0) and β = 0.5
(c) For true mean foam height of 195 mm we have
β = P ([tex]\overline { \rm x}[/tex] < 185 when μ = 195)
and P[tex](\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} } <\frac{185-195}{20/\sqrt{10} } )[/tex]
Which gives P(z < -1.58)
From z table the probability is β = 0.0571