A bottle of liquid cough syrup sold over the counter may contain no more than 24

Question

A bottle of liquid cough syrup sold over the counter may contain no more than 240 mg of dextromethorphan. The mixing and bottling process is subject to random variation: the quantity of dextromethorphan dispensed in a bottle is normally distributed with a standard deviation of 6 mg and with a mean controlled by the setting of the bottling process. The bottling process can be set to any desired quantity of dextromethorphan dispensed in each bottle.

1. What mean should be set for the bottling process so that the probability that a randomly selected bottle will contain no more than 240 mg of dextromethorphan equals 0.01?

2. Suppose that a random sample of 25 bottles is to be obtained. What is the standard deviation of the sample average quantity of dextromethorphan in the sampled bottles?

3. From each batch of cough syrup that is produced, a random sample of 25 bottles is selected and the quantity of dextromethorphan in each bottle is measured. If the sample average is less than 230 mg, then the batch of cough syrup is accepted for shipping to customers. Suppose that the bottling process is set so that the mean quantity of dextromethorphan dispensed in a bottle is 227 mg. What is the probability that a sample of 25 bottles yields average contents less than 230 mg?

Explanation / Answer

HEre standard deviation of bottling process 6mg

(1) Here lets say the mean set is mg.

so,

Pr(X> 240) < 0.01

Pr( X > 240 ; ; 6) = 0.01

so Here Z- value for p - value = 0.01

Z = +2.326

(240 - )/ 6 = 2.326

240 - = 6 * 2.326

= 240 - 6 * 2.326 = 226.04 mg

(b) standard deviation of the sample average quantity of dextromethorphan in the sampled bottles

se0 = /sqrt(n) = 6/ sqrt (25) = 6/ 5 = 1.2 mg

(c) Here mean set = 227 mg

standard deviation of the sample average quantity of dextromethorphan in the sampled bottles = 1.2 mg

so Pr(x < 230 mg ) = NORMAL (x < 230 mg ; 227 ; 1.2)

Z = ( 230 - 227)/1.2 = 2.5

P- value = Pr(Z < 2.5) = 0.9934

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