5.7 A laundry detergent manufacturing industry produces a...

5.7 A laundry detergent manufacturing industry produces a machine that fills boxes of laundry flakes. The machine is set for 16 oz net weight. The net weights of boxes filled from the machine have a mean of 16 oz and a standard deviation of 0.3 oz, and they are normally distributed. a. What proportion of the boxes will have net weights of more than 15 oz? b. What proportion of the boxes will have net weights of more than 16.5 oz? c. What proportion of the boxes weights differing from the mean weight for all boxes by more than 0.4 oz? d. If a sample of 32 boxes is drawn, what is the probability that its mean net weight will be greater than 16.3 oz? e. If a sample of 32 boxes is drawn, what is the probability that its mean net weight will be less than 15.8 oz?

Transcript

00:01 Hello student, here in the given problem, here given nu which is equal to 16 o's and here sigma is 0 .3 o's.

00:14 So here first we need to find out the proportion of the boxes with net weight of more than 15 o's that is the probability of x is greater than 15.

00:25 So by using the z square formula z is equal to x minus nu divided by sigma.

00:31 So here put the values x which is 15 minus 16 divided by 0 .3.

00:37 So the answer is minus 3 .33.

00:40 So the probability of z is greater than minus 3 .33.

00:44 Therefore the probability of z is less than 33.

00:49 Therefore 1 minus probability of z is greater than 3 .33.

00:54 So the answer is 0 .9996.

00:58 Therefore, first answer x is greater than 15 o's 0 .9996.

01:07 Now to find the b part to find the proportion of the boxes with net less than more net weight more than 16 .5 that is probability of x is greater than 16 .5 o's.

01:26 So by using the z square formula z is equal to 16 .5 minus 16 divided by 0 .3.

01:36 Therefore the answer is 1 .6667.

01:41 Therefore the probability of z is greater than 1 .667.

01:45 Therefore the answer is 0 .0475.

01:51 Therefore the final answer the probability of z is greater than 1 .667 which is equal to 0 .0475.

02:05 Now to find the c part proportion of the boxes with weight differing from the mean weight by more than 0 .0's.

02:14 Therefore we can write it as probability of x mod of x minus nu is greater than 0 .4.

02:22 Therefore we can write it as the probability of x minus nu which is greater than minus 0 .4 and x minus nu which is greater than 0 .4.

02:39 Therefore probability of x is less than 15 .6 and or the probability of x is greater than 16 .4 to find this probability.

02:53 So by using the z square formula z1 that is 15 .6 minus 16 divided by 0 .3.

03:01 Therefore the answer is minus 1 .33 and for z2 for 16 .4 minus 16 divided by 0 .3.

03:12 Therefore answer is 1 .33...

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