Question

show all your work. indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. a certain type of bird lives in two regions of a state. the distribution of weight for birds of this type in the northern region is approximately normal with mean 16 ounces and standard deviation 3 ounces. the distribution of weight for birds of this type in the southern region is approximately normal with mean 10 ounces and standard deviation 2.5 ounces. (a) calculate the z - scores for a weight of 13 ounces for a bird living in the southern region and for a weight of 13 ounces for a bird living in the northern region. (b) is it more likely that a bird of this type with a weight greater than 13 ounces lives in the northern region or the southern region? justify your answer.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Calculate z - score for northern region

For the northern region, $\mu = 16$ ounces, $\sigma=3$ ounces and $x = 13$ ounces.
$z_{north}=\frac{13 - 16}{3}=\frac{- 3}{3}=-1$

Step3: Calculate z - score for southern region

For the southern region, $\mu = 10$ ounces, $\sigma = 2.5$ ounces and $x = 13$ ounces.
$z_{south}=\frac{13 - 10}{2.5}=\frac{3}{2.5}=1.2$

Step4: Analyze probabilities using z - scores

The probability of a value being greater than $x$ in a normal distribution is $P(Z>z)=1 - P(Z < z)$. Looking up in the standard - normal table, $P(Z < - 1)=0.1587$, so $P(Z>-1)=1 - 0.1587 = 0.8413$ for the northern region. And $P(Z < 1.2)=0.8849$, so $P(Z>1.2)=1 - 0.8849 = 0.1151$ for the southern region. Since $0.8413>0.1151$, it is more likely that a bird with a weight greater than 13 ounces lives in the northern region.

Answer:

(a) $z_{north}=-1$, $z_{south}=1.2$
(b) It is more likely that a bird of this type with a weight greater than 13 ounces lives in the northern region. Justification: The z - score for the northern region is $z_{north}=-1$ and the probability $P(Z > - 1)=0.8413$, while the z - score for the southern region is $z_{south}=1.2$ and the probability $P(Z>1.2)=0.1151$. Since $0.8413>0.1151$, the northern region has a higher probability.