The weights of a certain dog breed are approximately normally distributed with a mean of 52 pounds, and a standard deviation of 6.6 pounds. Use your graphing calculator to answer the following questions. Write your answers in *percent* form. Round your answers to the nearest tenth of a percent.

a) Find the percentage of dogs of this breed that weigh less than 52 pounds.  %
b) Find the percentage of dogs of this breed that weigh less than 50 pounds.  %
c) Find the percentage of dogs of this breed that weigh more than 50 pounds.  %

Given that the weights of this dog breed are approximately normally distributed with a **mean (μ) = 52 pounds** and a **standard deviation (σ) = 6.6 pounds**, we can answer these questions by using the **standard normal distribution** (also called the **Z-distribution**).

We'll calculate the **Z-scores** for the values, which tells us how many standard deviations away a given value is from the mean. The Z-score is given by the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Once we have the Z-scores, we can use standard normal distribution tables (or a graphing calculator) to find the corresponding percentages.

### a) Find the percentage of dogs that weigh less than 52 pounds:

Since 52 pounds is the **mean**, the Z-score for 52 pounds is:

\[
Z = \frac{52 - 52}{6.6} = 0
\]

From the Z-table (or calculator), a Z-score of 0 corresponds to **50%**. So, **50% of the dogs weigh less than 52 pounds**.

### b) Find the percentage of dogs that weigh less than 50 pounds:

Now, for **X = 50 pounds**, the Z-score is:

\[
Z = \frac{50 - 52}{6.6} = \frac{-2}{6.6} \approx -0.303
\]

Using a Z-table or calculator, a Z-score of **-0.303** gives a cumulative probability of approximately **38.1%**. So, **38.1% of the dogs weigh less than 50 pounds**.

### c) Find the percentage of dogs that weigh more than 50 pounds:

To find the percentage of dogs weighing **more than 50 pounds**, we subtract the cumulative probability for 50 pounds from 100%:

\[
100\% - 38.1\% = 61.9\%
\]

Thus, **61.9% of the dogs weigh more than 50 pounds**.

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Let me know if you need further clarifications! 

Here are some related questions:
1. What percentage of dogs weigh more than 52 pounds?
2. How would the results change if the standard deviation were 5 pounds?
3. What is the weight corresponding to the 90th percentile for this dog breed?
4. How can you calculate the percentage of dogs weighing between 50 and 55 pounds?
5. If the weight of a dog is 60 pounds, what is its Z-score?

**Tip**: Always check the Z-score table for exact values when approximating percentages from Z-scores!

The weights of a certain dog breed are approximately normally distributed with a mean of 52 pounds, and a standard deviation of 6.6 pounds. Use your graphing calculator to answer the following questions. Write your answers in percent form. Round your answers to the nearest tenth of a percent.

a) Find the percentage of dogs of this breed that weigh less than 52 pounds.
b) Find the percentage of dogs of this breed that weigh less than 50 pounds.
c) Find the percentage of dogs of this breed that weigh more than 50 pounds.

Explore how to calculate percentages of weights for a dog breed using normal distribution with a mean of 52 pounds and a standard deviation of 6.6 pounds. Learn to find probabilities using Z-scores for weights less than 52 pounds, less than 50 pounds, and more than 50 pounds. Perfect for high school statistics students.

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