Question

The mean weight of a breed of yearling cattle is 1065 pounds. Suppose that weights of all such animals can be described by the Normal model ​N(1065​,56​). ​a) How many standard deviations from the mean would a steer weighing 1000 pounds​ be? ​b) Which would be more​ unusual, a steer weighing 1000 ​pounds, or one weighing 1250 ​pounds?

Answer 1

Answer:

a)$z=\frac{1000-1065}{56}=-1.16$

And this value means that the weight X=1000 pounds it's 1.6 deviations below the true mean of 1065 pounds

b) $z=\frac{1250-1065}{56}=3.304$

And this value means that the weight X=1250 pounds it's 3.304 deviations above the true mean of 1065 pounds.

So we can cocnlude on this case that the weight of 1250 would me more unusual since represent a value with a more distance respect to the mean than the weight of 1000 pounds.

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Part a

Let X the random variable that represent the weights of a breed of yearling cattle of a population, and for this case we know the distribution for X is given by:

$X \sim N(1065,56)$  

Where $\mu=1065$ and $\sigma=56$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we find the Z score for the value of X=1000 we got:

$z=\frac{1000-1065}{56}=-1.16$

And this value means that the weight X=1000 pounds it's 1.6 deviations below the true mean of 1065 pounds

3) Part b

For this case if we find the z score for the value of 1250 we got:

$z=\frac{1250-1065}{56}=3.304$

And this value means that the weight X=1250 pounds it's 3.304 deviations above the true mean of 1065 pounds.

So we can cocnlude on this case that the weight of 1250 would me more unusual since represent a value with a more distance respect to the mean than the weight of 1000 pounds.