Question

Pioneer Chicken advertises "lite" chicken with 30% fewer calories than standard chicken. When the process for "lite" chicken breast production is in control, the average chicken breast contains 420 calories, and the standard deviation in caloric content of the chicken breast population is 25 calories.
Pioneer wants to design a mean chart to monitor the caloric content of chicken breasts, where 25 chicken breasts would be chosen at random to form each sample. What are the limits with three standard deviations (z=3) from the target?

Answer 1

Answer:

$LL = 420 -3\frac{25}{\sqrt{25}}= 405$ represent the lower limit

$UL = 420 +3\frac{25}{\sqrt{25}}= 435$ represent the Upper limit

So then the limits for this case are (405, 435)

Explanation:

Let's define X as our random variable that represent the "number of calories for a chicken breast", and we have the following data:

$\bar X = 420 , \sigma= 25$

The select a sample of 25 chickens , n = 25. And we want to find the limits for a confidence interval within 3 deviations from the mean with z =3.

We assume that the distribution for X is normal. So then the distribution for the sample mean is also normal.

And for this case the confidence interval would be given by:

$(\bar X -z\frac{\sigma}{\sqrt{n}} < \mu < \bar X -z\frac{\sigma}{\sqrt{n}})$

So the limits are defined as:

$LL = \bar X -z\frac{\sigma}{\sqrt{n}}$ represent the lower limit

$UL = \bar X +z\frac{\sigma}{\sqrt{n}}$ represent the Upper limit

Since we have all the values given we cn replace like this:

$LL = 420 -3\frac{25}{\sqrt{25}}= 405$ represent the lower limit

$UL = 420 +3\frac{25}{\sqrt{25}}= 435$ represent the Upper limit

So then the limits for this case are (405, 435)