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kari74 [83]
3 years ago
13

A machine fills containers with a particular product. Assume the filling weights are normally distributed with a variance of 0.1

6 ounces2. If 85% of the containers hold greater than 12 ounces, find the machine's mean filling weight (in ounces).
Mathematics
1 answer:
Ghella [55]3 years ago
8 0

Answer:

z=-0.674

And if we solve for \mu we got

\mu=12 +0.674*0.4=12.270

So then the mean is 'mu = 12.270 for this case.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

X \sim N(\mu,\sqrt{0.16}= 0.4)  

For this part we want to find a value a, such that we satisfy this condition:

P(X>12)=0.85   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.25 of the area on the left and 0.85 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.85

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=-0.674

And if we solve for \mu we got

\mu=12 +0.674*0.4=12.270

So then the mean is 'mu = 12.270 for this case.

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3 years ago
I have X, I just need Y.
GarryVolchara [31]

I'm getting x = 9.3223 which rounds to 9.32. Nice work.

Once you know the value of x, you can use the law of sines to find the value of y

sin(A)/a = sin(B)/b

sin(y)/7 = sin(70)/x

sin(y)/7 = sin(70)/9.3223

sin(y) = 7*(sin(70)/9.3223)

sin(y) = 0.70560358983312

y = arcsin(0.70560358983312)

y = 44.8783278056846

<h3>y = 44.88 approximately</h3>

------------------------------

If you must use the law of cosines, then here's how you could do it

c^2 = a^2 + b^2 - 2*a*b*cos(C)

7^2 = x^2 + 9^2 - 2*x*9*cos(y)

7^2 = (9.3223)^2 + 9^2 - 2*9.3223*9*cos(y)

49 = 86.90527729+81-167.8014cos(y)

49 = 167.90527729-167.8014cos(y)

-167.8014cos(y) = 49-167.90527729

-167.8014cos(y) = -118.90527729

cos(y) = -118.90527729/(-167.8014)

cos(y) = 0.708607182598

y = arccos(0.708607182598)

y = 44.8782954209662 ... this is slightly different from before due to rounding error

<h3>y = 44.88 approximately</h3>
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