Complete Question
The complete question is shown on the first uploaded image
Answer:
The sales level that is the cut-off between quarters that are considers as "failure" and those that are not is evaluated as
[tex]y = 6.3 \ million \ dollars[/tex]
Step-by-step explanation:
From the question we are told that the
Mean is [tex]\mu = 8 \ million \ dollars[/tex]
Standard deviation is [tex]\sigma = 1.3 \ million \ dollars[/tex]
Let Y be the random variable that denotes the sales made quarterly among health care information system
The normal distribution for this data is mathematically represented as
[tex]Y[/tex] ~ N [tex](\mu = 8 , \sigma^2 = 1.69)[/tex]
From the question we are told that a quarter is consider a failure by the company if the sales level that quarter in the bottom 10% of the of all quarterly sales
So
The the Probability of obtaining quarterly sales level that is equal to 10% of all quarterly sale (It is not below 10%) is mathematically represented as
[tex]P[Y < y ] = 0.10[/tex]
This can be represented as a normal distribution in this manner
[tex]P [ \frac{Y- \mu}{\sigma } < \frac{y -\mu}{\sigma} ] = 0.10[/tex]
Where [tex]\frac{Y - \mu }{\sigma } = Z[/tex] ~ [tex]N(0 , 1)[/tex]
{This mean that this equal to the normal distribution between 0, 1 which is the generally range of every probability }
Therefor we have
[tex]P[Z < \frac{y- 8}{1.3} ][/tex]
The cumulative distribution function for a normal distribution of y is mathematically represented as
[tex]\phi [\frac{y- 8}{1.3} ] = 0.10[/tex]
This is because a cumulative distribution function of a random value Y or a distribution of Y evaluated at y is the probability that Y will take will take a value that is less or equal to y
[tex]\frac{y- 8}{1.3} = \phi ^{-1}(0.10)[/tex]
Calculating the inverse of the cumulative distribution function value of 0.10 which is negative the the critical value(Critical values determine what probability a particular variable will have when a sampling distribution is normal or close to normal.) of 0.01
[tex]\phi ^{-1}(0.10) =- [1 - \frac{0.10}{2}][/tex]
[tex]= - 1.28155[/tex]
[tex]\frac{y - 8}{1.3} = - 1.2816[/tex]
[tex]=> y = 8 - (1.2816)(1.3)[/tex]
[tex]y = 6.3 \ million \ dollars[/tex]
