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Part 1

A small animal veterinarian reviews her records for the day and notes that she has seen eight dogs and eight cats with the following weights​ (in pounds).

​Dogs:

1515​,

2222​,

3838​,

4444​,

5656​,

6161​,

7575​,

102102

​Cats:

44​,

44​,

66​,

99​,

1212​,

1616​,

1717​,

2121

a. Before analyzing these data​ sets, make a conjecture about which set has the larger​ mean, median, and standard deviation. Explain your reasoning.

b. Compute the mean and standard deviation of each set.

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Part 1

a. Make correct conjectures below about which set has the larger​ mean, median, and standard deviation. Choose the correct answer below.

A.

The​ mean, median, and standard deviation are all higher for dogs because most of the weights are​ larger, so the average​ value, middle​ value, and spread must be larger.

B.

The​ mean, median, and standard deviation are all higher for cats because there is less variation in the​ weights, so the average​ value, middle​ value, and spread must be larger.

C.

The mean and median are higher for cats because there is less variation in the weights. The standard deviation is higher for dogs because there is more variation in the weights.

D.

The mean and median are higher for dogs because most of the weights are​ larger, so the average value and middle value must be larger. The standard deviation is higher for cats because there is more variation in the weights.

### Part 1a: Conjecture

The conjecture involves predicting which set (dogs or cats) has the larger mean, median, and standard deviation before calculating these statistics.

- **Mean and Median**: 
  The dogs' weights range from 15 to 102 pounds, which are much larger compared to the cats' weights, which range from 4 to 21 pounds. Since the dog weights are higher overall, we expect both the **mean** (average) and the **median** (middle value) to be higher for dogs.
  
- **Standard Deviation**: 
  Standard deviation measures the spread of the data. The weights of the dogs are more spread out, ranging from 15 to 102 pounds, while the cats' weights are less spread out, from 4 to 21 pounds. Since the dogs' weights vary more, we would expect the standard deviation to be higher for the dogs.

Therefore, the correct answer is likely:

**Answer D**:  
- The **mean** and **median** are higher for dogs because most of the weights are larger.
- The **standard deviation** is higher for dogs because there is more variation in their weights.

### Part 1b: Calculating the Mean and Standard Deviation

To confirm, let's calculate the mean and standard deviation for both the dogs' and cats' weights.

#### Dogs' Weights:
15, 22, 38, 44, 56, 61, 75, 102

1. **Mean for dogs**:
\[
\text{Mean}_{\text{dogs}} = \frac{15 + 22 + 38 + 44 + 56 + 61 + 75 + 102}{8} = \frac{413}{8} = 51.625
\]

2. **Standard Deviation for dogs**:
   First, we calculate the variance by finding the squared differences from the mean, then taking the square root:
\[
\text{Variance}_{\text{dogs}} = \frac{(15 - 51.625)^2 + (22 - 51.625)^2 + \dots + (102 - 51.625)^2}{8} 
\]

Finally, take the square root to get the standard deviation.

#### Cats' Weights:
4, 4, 6, 9, 12, 16, 17, 21

1. **Mean for cats**:
\[
\text{Mean}_{\text{cats}} = \frac{4 + 4 + 6 + 9 + 12 + 16 + 17 + 21}{8} = \frac{89}{8} = 11.125
\]

2. **Standard Deviation for cats**:
   Again, calculate the variance and take the square root to get the standard deviation for the cats' weights.

Let me know if you want the full standard deviation calculations!

### Next Steps and Additional Questions
1. Do you want detailed calculations for the standard deviations of both sets?
2. Would you like to see the medians computed step-by-step?
3. Are there other descriptive statistics you'd like to explore (e.g., range or interquartile range)?
4. Should we compare the variation using another measure like variance?
5. Are you interested in a graphical representation of these data sets, like a box plot or histogram?

**Tip**: When working with data sets, always pay attention to the range and spread of the data to infer potential outcomes before calculating actual statistics.

A small animal veterinarian reviews her records for the day and notes that she has seen eight dogs and eight cats with the following weights (in pounds). Dogs: 15, 22, 38, 44, 56, 61, 75, 102. Cats: 4, 4, 6, 9, 12, 16, 17, 21. a. Before analyzing these data sets, make a conjecture about which set has the larger mean, median, and standard deviation. Explain your reasoning. b. Compute the mean and standard deviation of each set.

Learn how to calculate the mean, median, and standard deviation for the weights of eight dogs and eight cats. This problem involves data analysis, including key concepts like central tendency and spread, making it ideal for students in Grades 9-12.

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