2. Duckworth Drug Company is a large manufacturer of various kinds of liquid vit

Question

2. Duckworth Drug Company is a large manufacturer of various kinds of liquid vitamins. The quality control department has noted that the bottles of vitamins marked 6 ounces vary in content with a standard deviation of 0.3 ounces. a. What percentage of all bottles produced contains more than 6.51 ounces of vitamins? b. What percentage of all bottles produced contains less than 5.415 ounces? C. What percentage of bottles produced contains between 5.46 and 6.495 ounces? d. Ninety-five percent of the bottles will contain at least how many ounces? e. What percentage of the bottles contains between 6.3 and 6.6 ounces?

Explanation / Answer

2.

Mean ( u ) =6

Standard Deviation ( sd )=0.3

Normal Distribution = Z= X- u / sd ~ N(0,1)

a)

P(X > 6.51) = (6.51-6)/0.3

= 0.51/0.3 = 1.7

= P ( Z >1.7) From Standard Normal Table

= 0.0446

4.46% of the bottles produced contains more than 6.51 ounces of vitamins

b)

P(X < 5.415) = (5.415-6)/0.3

= -0.585/0.3= -1.95

= P ( Z <-1.95) From Standard Normal Table

= 0.0256

2.56% of all bottles produced contains less than 5.415 ounces of vitamins

c)

To find P(a < = Z < = b) = F(b) - F(a)

P(X < 5.46) = (5.46-6)/0.3

= -0.54/0.3 = -1.8

= P ( Z <-1.8) From Standard Normal Table

= 0.03593

P(X < 6.495) = (6.495-6)/0.3

= 0.495/0.3 = 1.65

= P ( Z <1.65) From Standard Normal Table

= 0.95053

P(5.46 < X < 6.495) = 0.95053-0.03593 = 0.9146

91.46% of bottles produced contains between 5.46 and 6.495 ounces of vitamins

d)

P ( Z > x ) = 0.95

Value of z to the cumulative probability of 0.95 from normal table is -1.6449

P( x-u/ (s.d) > x - 6/0.3) = 0.95

That is, ( x - 6/0.3) = -1.6449

--> x = -1.6449 * 0.3+6 = 5.5065

e)

To find P(a < = Z < = b) = F(b) - F(a)

P(X < 6.3) = (6.3-6)/0.3

= 0.3/0.3 = 1

= P ( Z <1) From Standard Normal Table

= 0.84134

P(X < 6.6) = (6.6-6)/0.3

= 0.6/0.3 = 2

= P ( Z <2) From Standard Normal Table

= 0.97725

P(6.3 < X < 6.6) = 0.97725-0.84134 = 0.1359

13.59% of bottle produced contains between 6.3 and 6.6 ounces of vitamins

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