help me wit my statistics question

Question

The weights of a set of boxes are normally distributed with mean 16g. and standard deviation 0.6g. If a box is selected at random, find the probability that its weight is more than 15.8g.

Answer 1

Since 15.8g is less than 1 standard deviation (s.d.) from the mean, we need to find out how many s.d. from the mean 15.8g is. 16-15.8=0.2g. Since 0.2 is 1/3 of 0.6, we are 0.33(repeating)s.d. from our mean. If we look up in a Normal Table, they don't necessarily have z=0.33, but they do have z=0.30 and 0.35. Since 0.33 is in between 0.30 and 0.35, we need can use those Areas to assist us z=0.30 has an Area of 23.58 z=0.35 has an Area of 27.37 so z=0.33 will have an Area in between 23.58 and 27.37 To get a z of 0.33, we could multiplying z=0.30 by 1.11111 since 0.3*1.11111=0.33. Then we could multiply the corresponding area by 1.1111 as well: 23.58*1.1111=26.1999 (which is in between 23.58 and 27.37 as we expected it to be) This tells us that we should expect 26.1999% of weights of boxes to be between 15.8g and 16.2g. Dividing this by 2 gives us 13.1%, so 13.1% of the weights are expected to be between 15.8g and 16g. But to find out the probability that it's weight is MORE than 15.8g, we need to take into account the probability that the box is greater than 16g as well. Since that value is our mean, we know that 50% of boxes are greater than 16g. Therefore, adding 13.1%+50%=63.1% So 63.1% of boxes should be greater than 15.8 g

Answer 2

We are given μ = 16 g, σ = 0.6 g, and x = 15.8 g.

 

 

Find the z-score to use a standard normal distribution table to find the probability of X > x.

 

z = (x - μ)/σ = (15.8 - 16)/0.6 = -0.33333

 

 

Consult a table or use an online statistics calculator to find the probability using this z-score:

 

P(z < -0.33333) = 0.36944

 

P(z > -0.33333) = 1 - P(z < -0.33333) = 1 - 0.36944 = 0.63056

Answer 3

Hello Carlos -

 

We are told the weights are distributed normally. So you use a normal curve. You need a representation of one, which you probably have in your book. On such a curve, 50% of the population lies below the mean, and 50% above. The width of the curve is 2 standard deviations. In our case here that means the curve is 1.2 g wide, with  the center at 16 g.

 

So from a common sense way of thinking, you have to find the probability that a chosen box is more that 15.8 g. Is this going to be more or less than 50%? I know you picked more. :) To find out how much more, we find normalcdf, using your calculator, for 15.8. This is the probability of being less than 15.8, or from -infinity to 15.8. For example on TI calculators, normalcdf(-1E99, 15.8, 16.0, 0.6).

 

 

Then subtract that from 1, and you will have what you are looking for.

 

Hope this helps.

 

Ken