Question

A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to test millimeters versus millimeters, using the results of samples. Calculate the P-value if the observed statistic is Round your answer to four decimal places (e.g. 98.7654).

Answer 1

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 ($\overline{\rm x}$ = 185 when μ = 175)

z =  $\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} }$

z($\overline { \rm x}$ = 185)  = $\frac{ 185 -175}{20/\sqrt{10} }$ = 1.5811

Therefore the P ($\overline{\rm x}$ = 185 when μ = 175) =  P(z = 1.5811)  

    0.94295

The probability density function is

$f(x) = \frac{1}{\sigma\sqrt{2\pi } } e^{-\frac{1}{2}(\frac{x-\mu}{\sigma} )^2 }$ Therefore the probability is

$f(x) = \frac{1}{20\sqrt{2\pi } } e^{-\frac{1}{2}(\frac{185-175}{20} )^2 }$ = 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 ($\overline { \rm x}$ < 185 when μ = 185)

P$(\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} }$, which gives

P(z<0) and β = 0.5

(c) For true mean foam height of 195 mm we have

β = P ($\overline { \rm x}$ < 185 when μ = 195)

and P$(\frac{ \overline { \rm x} -\mu}{\sigma /\sqrt{n} }$

Which gives P(z < -1.58)

From z table the probability is       β = 0.0571