Question

the weights of a certain dog breed are approximately normally distributed with a mean of \\(\mu = 53\\) pounds, and a standard deviation of \\(\sigma = 6\\) pounds.
a dog of this breed weighs 49 pounds. what is the dog’s z - score? round your answer to the nearest hundredth as needed.
\\(z = \square\\)
a dog has a z - score of - 0.04. what is the dog’s weight? round your answer to the nearest tenth as needed.
\\(\square\\) pounds
a dog has a z - score of 0.04. what is the dog’s weight? round your answer to the nearest tenth as needed.
\\(\square\\) pounds

Explanation:

First Sub - Question: Find the z - score for a 49 - pound dog

Step 1: Recall the z - score formula

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

Step 2: Identify the values

We know that $x = 49$, $\mu=53$, and $\sigma = 6$.

Step 3: Substitute the values into the formula

Substitute the values into the formula: $z=\frac{49 - 53}{6}=\frac{- 4}{6}\approx - 0.67$ (rounded to the nearest hundredth).

Second Sub - Question: Find the weight of a dog with a z - score of - 0.04

Step 1: Rearrange the z - score formula to solve for $x$

Starting from $z=\frac{x - \mu}{\sigma}$, we can multiply both sides by $\sigma$ and then add $\mu$ to both sides to get $x=\mu+z\sigma$.

Step 2: Identify the values

We know that $\mu = 53$, $z=-0.04$, and $\sigma = 6$.

Step 3: Substitute the values into the formula

Substitute the values: $x=53+(-0.04)\times6=53 - 0.24 = 52.76\approx52.8$ (rounded to the nearest tenth).

Third Sub - Question: Find the weight of a dog with a z - score of 0.04

Step 1: Use the formula for $x$ from the z - score formula

We use the formula $x=\mu+z\sigma$.

Step 2: Identify the values

We know that $\mu = 53$, $z = 0.04$, and $\sigma=6$.

Step 3: Substitute the values into the formula

Substitute the values: $x=53+(0.04)\times6=53 + 0.24=53.24\approx53.2$ (rounded to the nearest tenth).

Answer:

• For the first sub - question: $z\approx - 0.67$
• For the second sub - question: The weight is approximately $52.8$ pounds
• For the third sub - question: The weight is approximately $53.2$ pounds