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Let's break down and solve each part of this problem, which involves a normal distribution with a mean of 53 pounds and a standard deviation of 6.4 pounds. We are asked to find percentages based on this normal distribution.

### Part a) Find the percentage of dogs that weigh less than 53 pounds.
- In a normal distribution, 53 pounds is the mean, which corresponds to a Z-score of 0. For a Z-score of 0, the cumulative probability is 50%.
  
Thus, the percentage of dogs that weigh less than 53 pounds is **50%**.

### Part b) Find the percentage of dogs that weigh less than 48 pounds.
- To solve this, we first need to calculate the Z-score for 48 pounds:
  \[
  Z = \frac{X - \mu}{\sigma} = \frac{48 - 53}{6.4} = \frac{-5}{6.4} \approx -0.78
  \]
- Now, using a Z-table or a calculator, the cumulative probability for a Z-score of -0.78 is approximately 0.2177, or 21.77%.

So, the percentage of dogs that weigh less than 48 pounds is **21.8%** (rounded to the nearest tenth).

### Part c) Find the percentage of dogs that weigh more than 48 pounds.
- This is the complement of the result in part b. Since 21.8% of dogs weigh less than 48 pounds, the percentage that weigh more than 48 pounds is:
  \[
  100\% - 21.8\% = 78.2\%
  \]

So, the percentage of dogs that weigh more than 48 pounds is **78.2%**.

Let me know if you need further details or have questions! Here are five related questions to expand on this concept:

1. How would the percentages change if the standard deviation was smaller or larger?
2. What percentage of dogs weigh between 48 and 53 pounds?
3. What is the probability of a dog weighing more than 60 pounds?
4. How do we interpret the Z-score in the context of this problem?
5. Can we use a different method to calculate these percentages other than a Z-table or graphing calculator?

**Tip:** In a normal distribution, about 68% of the data falls within one standard deviation of the mean (34% above and 34% below).

The weights of a certain dog breed are normally distributed with a mean of 53 pounds, and a standard deviation of 6.4 pounds. Find percentages of dogs weighing less than or more than specific values based on this distribution.

Solve a normal distribution problem with a mean of 53 pounds and a standard deviation of 6.4 pounds. Learn how to calculate percentages of dogs weighing less than or more than given weights using Z-scores and cumulative probabilities.

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