Question

The mean weight of a breed of yearling cattle is 11921192 pounds. Suppose that weights of all such animals can be described by the Normal model ​N(11921192​,7878​). ​a) How many standard deviations from the mean would a steer weighing 10001000 pounds​ be? ​b) Which would be more​ unusual, a steer weighing 10001000 ​pounds, or one weighing 12501250 ​pounds? ​
a) A steer weighing 10001000 pounds is 2.462.46 standard deviations below the mean. ​(Round to two decimal places as​ needed.) ​
b) The steer weighing 1000 pounds is more unusual.

Answer 1

Answer:

a) 2.46 standard deviations below the mean

b) The steer weighing 1000 pounds is more unusual.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 1192, \sigma = 78$

a) How many standard deviations from the mean would a steer weighing 1000 pounds​ be?

This is Z when X = 1000.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1000 - 1192}{78}$

$Z = -2.46$

$Z = -2.46$ is 2.46 standard deviations below the mean.

​b) Which would be more​ unusual, a steer weighing 1000 ​pounds, or one weighing 1250 ​pounds?

The higher the absolute value of the z-score, the more unusual the measure is.

X = 1000 pounds

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1000 - 1192}{78}$

$Z = -2.46$

The absolute value of -2.46 is 2.46

X = 1250 pounds

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1250 - 1192}{78}$

$Z = 0.74$

The absolute value of 0.74 is 0.74.

So the steer weighing 1000 pounds is more unusual.