Question

the weights of a certain dog breed are approximately normally distributed with a mean of \\( \mu = 49 \\) pounds, and a standard deviation of \\( \sigma = 7 \\) pounds.

fill in the indicated boxes.

image of a normal distribution curve with labels \\( \mu - 3\sigma \\), \\( \mu - 2\sigma \\), \\( \mu - \sigma \\), \\( \mu \\), \\( \mu + \sigma \\), \\( \mu + 2\sigma \\), \\( \mu + 3\sigma \\) and empty boxes below each label

a dog of this breed weighs 53 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 0.13. what is the dog’s weight? round your answer to the nearest tenth as needed.
\\( \square \\) pounds

a dog has a z - score of - 0.13. what is the dog’s weight? round your answer to the nearest tenth as needed.
\\( \square \\) pounds

Explanation:

Step1: Calculate the z - score for a dog weighing 53 pounds

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 53\), \(\mu=49\), and \(\sigma = 7\).

$$z=\frac{53 - 49}{7}=\frac{4}{7}\approx0.57$$

Step2: Calculate the weight for a dog with \(z = 0.13\)

The formula for \(x\) (raw score) is \(x=\mu+z\sigma\). Substitute \(\mu = 49\), \(z = 0.13\), and \(\sigma=7\) into the formula.

$$x=49+(0.13\times7)=49 + 0.91=49.91\approx49.9$$

Step3: Calculate the weight for a dog with \(z=- 0.13\)

Using the formula \(x=\mu+z\sigma\), substitute \(\mu = 49\), \(z=-0.13\), and \(\sigma = 7\)

$$x=49+(-0.13\times7)=49-0.91 = 48.09\approx48.1$$

Answer:

For the z - score of a 53 - pound dog: \(z = 0.57\)
For the dog with \(z = 0.13\): \(49.9\) pounds
For the dog with \(z=-0.13\): \(48.1\) pounds