The weights of a certain dog breed are approximately normally distributed with a mean of 49 pounds, and a standard deviation of 5.8 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 49 pounds. %
b) Find the percentage of dogs of this breed that weigh less than 45 pounds. %
c) Find the percentage of dogs of this breed that weigh more than 45 pounds. %

To solve these problems involving a normal distribution, we can use the properties of the normal distribution and a standard normal distribution table (or a calculator).

### Given:
- Mean (\( \mu \)) = 49 pounds
- Standard Deviation (\( \sigma \)) = 5.8 pounds

### a) Percentage of dogs that weigh less than 49 pounds

Since 49 pounds is the mean, we can find the cumulative probability:

\[
P(X < 49) = 50\% \text{ (since it's the mean)}
\]

### Answer:
**a)** \( 50.0\% \)

### b) Percentage of dogs that weigh less than 45 pounds

To find this, we first calculate the z-score:

\[
z = \frac{X - \mu}{\sigma} = \frac{45 - 49}{5.8} \approx -0.6897
\]

Using a z-table or calculator, we find:

\[
P(Z < -0.6897) \approx 0.2466
\]

Converting to percentage:

\[
P(X < 45) \approx 24.7\%
\]

### Answer:
**b)** \( 24.7\% \)

### c) Percentage of dogs that weigh more than 45 pounds

To find this, we use the complement:

\[
P(X > 45) = 1 - P(X < 45)
\]

From part b, we found:

\[
P(X < 45) \approx 0.2466
\]

So:

\[
P(X > 45) \approx 1 - 0.2466 \approx 0.7534
\]

Converting to percentage:

\[
P(X > 45) \approx 75.3\%
\]

### Answer:
**c)** \( 75.3\% \)

### Summary of Answers:
- **a)** 50.0%
- **b)** 24.7%
- **c)** 75.3%

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you interpret the significance of the mean and standard deviation in this context?
2. What percentage of dogs would weigh between 45 and 53 pounds?
3. How would the percentages change if the mean weight increased?
4. What implications does a larger standard deviation have on the distribution?
5. How can you use this information in practical scenarios, like breeding or health assessments?

**Tip:** When dealing with normal distributions, always check if your values fall within the expected range, and remember that z-scores can help you find probabilities easily!

The weights of a certain dog breed are approximately normally distributed with a mean of 49 pounds, and a standard deviation of 5.8 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.

Explore the normal distribution of dog weights with a mean of 49 pounds and standard deviation of 5.8 pounds. This problem calculates the percentages of dogs weighing less than and more than specific weights, showcasing fundamental concepts in statistics. Ideal for advanced high school students.

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