The weights of a certain dog breed are approximately normally distributed with a mean of 50 pounds, and a standard deviation of 5.6 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 50 pounds. %
b) Find the percentage of dogs of this breed that weigh less than 48 pounds. %
c) Find the percentage of dogs of this breed that weigh more than 48 pounds. %

To answer these questions, we'll use the properties of the normal distribution and calculate Z-scores to determine the percentages of dogs within the specified weight ranges. The weights of the dogs are normally distributed with:

- **Mean (μ)** = 50 pounds
- **Standard Deviation (σ)** = 5.6 pounds

### a) Percentage of dogs that weigh less than 50 pounds

Since the mean weight of the dogs is 50 pounds, and in a normal distribution, half of the values lie below the mean:

\[
P(X < 50) = 0.5 \quad \text{(or 50%)}
\]

**Answer: 50.0%**

### b) Percentage of dogs that weigh less than 48 pounds

First, calculate the Z-score for 48 pounds:

\[
Z = \frac{X - \mu}{\sigma} = \frac{48 - 50}{5.6} = \frac{-2}{5.6} \approx -0.3571
\]

Using a standard normal distribution table (or graphing calculator) to find the cumulative probability for \(Z = -0.3571\):

\[
P(Z < -0.3571) \approx 0.3607
\]

So, the percentage of dogs that weigh less than 48 pounds is:

\[
P(X < 48) = 0.3607 \times 100 = 36.1\%
\]

**Answer: 36.1%**

### c) Percentage of dogs that weigh more than 48 pounds

We can use the complement rule, since:

\[
P(X > 48) = 1 - P(X < 48)
\]

From part (b), we have:

\[
P(X < 48) \approx 0.3607
\]

Therefore:

\[
P(X > 48) = 1 - 0.3607 = 0.6393
\]

Convert this to a percentage:

\[
P(X > 48) = 0.6393 \times 100 = 63.9\%
\]

**Answer: 63.9%**

### Summary of Results

- **Percentage of dogs that weigh less than 50 pounds**: 50.0%
- **Percentage of dogs that weigh less than 48 pounds**: 36.1%
- **Percentage of dogs that weigh more than 48 pounds**: 63.9%

Would you like more details on the calculations or have any questions?

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Here are some additional questions to consider:

1. What percentage of dogs weigh more than 55 pounds?
2. What percentage of dogs weigh between 45 and 55 pounds?
3. How many dogs weigh less than 42 pounds if there are 1000 dogs?
4. What weight corresponds to the 90th percentile of this breed?
5. If a dog weighs 60 pounds, what percentage of dogs are lighter than this dog?

**Tip:** When dealing with percentages in normal distributions, remember that the cumulative distribution function (CDF) tells you the probability that a value is less than or equal to a given point.

Explore how to analyze the weights of a dog breed with a normal distribution, using mean and standard deviation. Calculate percentages for weights less than 50 pounds, less than 48 pounds, and more than 48 pounds. Learn about Z-scores and cumulative probabilities.

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