1.       The weights of a large number of miniature poodles are approximately normally distributed with mean of 8.5 kilograms and a standard deviation of 1.2  kilogram.   If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights 

(50points)

a)       Over 10 kilograms;

b)      Of at most 8.6 kilograms;

c)       Between 7.5 and 9.5 kilograms inclusive.

Given that the weights of the poodles are normally distributed with:

- Mean \( \mu = 8.5 \) kg
- Standard deviation \( \sigma = 1.2 \) kg

We need to calculate the probability (or fraction) of poodles falling in the specified weight ranges.

To solve these problems, we convert the raw weights to standard normal \( Z \)-scores using the formula:

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

where:
- \( X \) is the weight.
- \( \mu \) is the mean.
- \( \sigma \) is the standard deviation.

### (a) Probability of poodles weighing over 10 kilograms:

First, calculate the \( Z \)-score for 10 kg:

\[
Z = \frac{10 - 8.5}{1.2} = \frac{1.5}{1.2} = 1.25
\]

The problem asks for the fraction of poodles weighing more than 10 kg, which corresponds to \( P(Z > 1.25) \).

Using standard normal distribution tables or a calculator:

\[
P(Z > 1.25) = 1 - P(Z \leq 1.25) = 1 - 0.8944 = 0.1056
\]

So, approximately **10.56%** of the poodles weigh more than 10 kg.

### (b) Probability of poodles weighing at most 8.6 kilograms:

First, calculate the \( Z \)-score for 8.6 kg:

\[
Z = \frac{8.6 - 8.5}{1.2} = \frac{0.1}{1.2} \approx 0.0833
\]

The problem asks for the fraction of poodles weighing at most 8.6 kg, which corresponds to \( P(Z \leq 0.0833) \).

Using standard normal distribution tables or a calculator:

\[
P(Z \leq 0.0833) \approx 0.5330
\]

So, approximately **53.30%** of the poodles weigh at most 8.6 kg.

### (c) Probability of poodles weighing between 7.5 and 9.5 kilograms inclusive:

1. Calculate the \( Z \)-score for 7.5 kg:

\[
Z = \frac{7.5 - 8.5}{1.2} = \frac{-1.0}{1.2} \approx -0.8333
\]

2. Calculate the \( Z \)-score for 9.5 kg:

\[
Z = \frac{9.5 - 8.5}{1.2} = \frac{1.0}{1.2} \approx 0.8333
\]

The problem asks for the fraction of poodles weighing between 7.5 and 9.5 kg, which corresponds to \( P(-0.8333 \leq Z \leq 0.8333) \).

Using standard normal distribution tables or a calculator:

\[
P(Z \leq 0.8333) \approx 0.7977
\]

\[
P(Z \leq -0.8333) \approx 0.2023
\]

So,

\[
P(-0.8333 \leq Z \leq 0.8333) = 0.7977 - 0.2023 = 0.5954
\]

Approximately **59.54%** of the poodles weigh between 7.5 and 9.5 kg inclusive.

---

### Summary of Results:

a) Fraction of poodles weighing over 10 kg: **10.56%**

b) Fraction of poodles weighing at most 8.6 kg: **53.30%**

c) Fraction of poodles weighing between 7.5 and 9.5 kg: **59.54%**

---

Would you like more details or have any questions?

Here are some related questions:

1. What is the probability that a poodle weighs less than 7 kg?
2. What is the probability that a poodle weighs between 9 kg and 11 kg?
3. How would the probabilities change if the standard deviation was 2 kg instead of 1.2 kg?
4. What is the range of weights that corresponds to the middle 80% of poodles?
5. If the mean weight increased to 9 kg, how would the fractions change for the same ranges?

**Tip:** For normal distribution problems, always use the \( Z \)-score to standardize your data and easily find probabilities using a table or calculator.

Explore how to calculate probabilities for normally distributed miniature poodle weights with a mean of 8.5 kg and a standard deviation of 1.2 kg. Learn how to determine the fraction of poodles with weights over 10 kg, at most 8.6 kg, and between 7.5 and 9.5 kg inclusive.

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