A machine used to regulate the amount of a certain chemical dispensed in the production of a particular type of cough syrup can be set so that it discharges an average of milliliters (ml) of the chemical in each bottle of cough syrup. The amount of chemical placed into each bottle of cough syrup is known to have a normal distribution with a standard deviation of 0.250 ml. If this machine discharges more than 2 ml of the chemical when preparing a given bottle of this cough syrup, the bottle is considered to be unacceptable by industry standards. Determine the setting for so that no more than 1% of the bottles of cough syrup prepared by this machine will be rejected.

The mean value of the chemical to be added to every bottle of syrup so that the bottle wont be rejected would be = 1.41750 ml

Step-by-step explanation:

Standard deviation ( σ )= 0.250 ml

assume the required amount of chemical to be deposited in each syrup bottle to be X

Hence the distribution of X in each bottle can be represented as :

X Г N ( u , 0.250 )

for a bottle not to be rejected  :  P ( X ≤ 2 )

To determine the setting so that no more than 1% of the bottles of cough syrup prepared by this machine will be rejected we have to standardize the variable X

at X = 2 ml

z = = z ----------- ( 1 )

probability for a bottle to be rejected : P( x >2 ) = 1 - P ( x ≤ 2 )

                                                                             = 0.01

attached below is a normal distribution curve showing probability values

from the curve

p( 0 < Z < z ) = 0.5 - 0.01 = 0.49  

Area = 0.49 hence from the Normal table where area = 0.49 the value of z = 2.33

Referring back to equation 1 above:  z = = 2.33 --------- (2)

u = mean value that should be added so that the bottle will not be rejected

make u subject of the equation : u = 2 - ( 2.33 * 0.250 )

                                                       u = 1.41750

The mean value of the chemical to be added to every bottle of syrup so that the bottle wont be rejected would be = 1.41750 ml

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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