Question

2. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard deviation of 1.5 oz.
a. The probability that a random bottle weights between 35.94 and 36.06 oz. is __________. (in decimal format, round to 4 decimal digits, i.e. 5%=.0500

b. Suppose we take a random sample of 36 bottles filled by this machine. The probability that the mean of the sample is between 35.94 and 36.06 oz. is __________. (in decimal format, round to 4 decimal digits, i.e. 5%=.0500)

c. Suppose we take a random sample of 36 bottles filled by this machine. 90% of sample means will exceed _______oz. (round to the nearest 2 decimal digits)

d. Suppose we take a random sample of 36 bottles filled by this machine. So, the middle 95% of the sample means based on samples of size 36 will be between __________oz and __________oz (symmetrically distributed around the mean).

Multiple choice below

		33.53 grams and 38.47 grams
		33.06 grams and 38.94 grams
		35.51 grams and 36.49 grams

Answer 1

a)

Here, μ = 36, σ = 1.5, x1 = 35.94 and x2 = 36.06. We need to compute P(35.94<= X <= 36.06). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (35.94 - 36)/1.5 = -0.04
z2 = (36.06 - 36)/1.5 = 0.04

Therefore, we get
P(35.94 <= X <= 36.06) = P((36.06 - 36)/1.5) <= z <= (36.06 - 36)/1.5)
= P(-0.04 <= z <= 0.04) = P(z <= 0.04) - P(z <= -0.04)
= 0.516 - 0.484
= 0.0320

b)

Here, μ = 36, σ = 0.25, x1 = 35.94 and x2 = 36.06. We need to compute P(35.94<= X <= 36.06). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (35.94 - 36)/0.25 = -0.24
z2 = (36.06 - 36)/0.25 = 0.24

Therefore, we get
P(35.94 <= X <= 36.06) = P((36.06 - 36)/0.25) <= z <= (36.06 - 36)/0.25)
= P(-0.24 <= z <= 0.24) = P(z <= 0.24) - P(z <= -0.24)
= 0.5948 - 0.4052
= 0.1896

c)

z value at 90% = 1.28

z = (x - mean)/s
1.28 = (x - 36)/0.25
x = 1.28 * 0.25 + 36
x = 36.32

d)

z value at 95% = +/- 1.64

z = (x - mean)/s
-1.64 = (x - 36)/0.25
x = -1.64 * 0.25 + 36
x = 35.51

z = (x - mean)/s
1.64 = (x - 36)/0.25
x = 1.64 * 0.25 + 36
x = 36.49