Normal Distribution: Weight of Bags

[Monkeyseat]

Question:

Bags of sugar are sold as 1kg. To ensure bags are not sold underwieght, the machine is set to put a mean weight of 1004g in each bag. The manufacturer claims that the process works to a standard deviation of 2.4g. What proportion of bags are underweight?

Working:

Sorry, don't really have any - I don't know where to start... By underweight, does it mean 1003g, 999g or what? Apologies for the lack of working, I'd like to show some but I don't know how to approach this. We have generally been using the formula; z value = (X - mean)/standard deviation, and then looking the z value up in a normal distribution table. I thought this might be to do with the distribution of the sample mean:

z value = (X - mean)/(standard deviation/sqr. n)

Where n is the sample size and sqr. the square root sign.

But I still didn't know how to start. I've done a few other different ones but don't really understand this topic, so if someone could give me some clear pointers or steer me in the right direction that would be great.

Cheers.

 

[Deleted member 4993]

Question:

Bags of sugar are sold as 1kg. To ensure bags are not sold underwieght, the machine is set to put a mean weight of 1004g in each bag. The manufacturer claims that the process works to a standard deviation of 2.4g. What proportion of bags are underweight?

Working:

Sorry, don't really have any - I don't know where to start... By underweight, does it mean 1003g, 999g or what? Apologies for the lack of working, I'd like to show some but I don't know how to approach this. We have generally been using the formula; z value = (X - mean)/standard deviation, and then looking the z value up in a normal distribution table. I thought this might be to do with the distribution of the sample mean but I still didn't know how to start. I've done a few other different ones but don't really understand this topic, so if someone could give me some clear pointers or steer me in the right direction that would be great.

Cheers.

What is the z value for 1000 g bag?

What % of bags are under that weight when the mean is 1004 with a standard deviation of 2.3?

 

[Monkeyseat]

What is the z value for 1000 g bag?

What % of bags are under that weight when the mean is 1004 with a standard deviation of 2.3?

Thank you for the reply. I'm not sure how to find the z value of the 1000g bag, sorry.

I did (1000-1004)/2.4 but that gave me the wrong z value I think. Is it 0.95254 for the z value? I don't fully understand this, so it may take a while.

 

[Monkeyseat]

I've worked on it for a bit and I think I might have the right answer now:

z value = (X - mean)/standard deviation

z value = (1000 - 1004)/2.4

z value = -1.67 (rounded)

When looked up in the table, this is: 0.95254

We are therefore trying to find the probability of less than the negative z value, so the probability that it is less than 1000g that is 1 - 0.95254 which is 0.04746.

Is that correct? Why do we use 1000 as the X value? I though 1000 wasn't under weight.

Thanks. I made a typo in my last post (2.3 instead of 2.4 for the standard deviation - now corrected) which slowed me down a bit in answering, also forgot to do 1 take the z value.