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The weights of a certain dog breed are approximately normally distributed with a mean of [tex]\mu = 46[/tex] pounds and a standard deviation of [tex]\sigma = 6[/tex] pounds.

1. A dog of this breed weighs 52 pounds. What is the dog's [tex]z[/tex]-score? Round your answer to the nearest hundredth as needed.

[tex]z = \square[/tex]

2. A dog has a [tex]z[/tex]-score of 1.37. What is the dog's weight? Round your answer to the nearest tenth as needed.

[tex]\square[/tex] pounds

3. A dog has a [tex]z[/tex]-score of -1.37. What is the dog's weight? Round your answer to the nearest tenth as needed.

[tex]\square[/tex] pounds

Answer :

Sure, I'd be happy to help with this!

### Step-by-step Solution

1. Finding the [tex]$z$[/tex]-score for a dog weighing 52 pounds:

The formula for finding the [tex]$z$[/tex]-score is:

[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]

where [tex]\(X\)[/tex] is the dog's weight, [tex]\(\mu\)[/tex] is the mean weight, and [tex]\(\sigma\)[/tex] is the standard deviation.

- Given [tex]\(X = 52\)[/tex] pounds, [tex]\(\mu = 46\)[/tex] pounds, and [tex]\(\sigma = 6\)[/tex] pounds, we substitute these values into the formula:

[tex]\[
z = \frac{(52 - 46)}{6} = \frac{6}{6} = 1.0
\][/tex]

The [tex]$z$[/tex]-score for a dog weighing 52 pounds is 1.0.

2. Finding the weight of a dog with a [tex]$z$[/tex]-score of 1.37:

To find the weight from a [tex]$z$[/tex]-score, we can rearrange the [tex]$z$[/tex]-score formula to solve for [tex]\(X\)[/tex]:

[tex]\[
X = \mu + z\sigma
\][/tex]

- Given [tex]\(z = 1.37\)[/tex], [tex]\(\mu = 46\)[/tex], and [tex]\(\sigma = 6\)[/tex]:

[tex]\[
X = 46 + 1.37 \times 6 = 46 + 8.22 = 54.2
\][/tex]

The weight of the dog with a [tex]$z$[/tex]-score of 1.37 is 54.2 pounds.

3. Finding the weight of a dog with a [tex]$z$[/tex]-score of -1.37:

Again, using the formula to solve for [tex]\(X\)[/tex]:

[tex]\[
X = \mu + z\sigma
\][/tex]

- Given [tex]\(z = -1.37\)[/tex], [tex]\(\mu = 46\)[/tex], and [tex]\(\sigma = 6\)[/tex]:

[tex]\[
X = 46 + (-1.37 \times 6) = 46 - 8.22 = 37.8
\][/tex]

The weight of the dog with a [tex]$z$[/tex]-score of -1.37 is 37.8 pounds.

I hope this helps! Let me know if you have any other questions.

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