Part 1a. Conjecture about the mean, median, and standard deviation:

Let's analyze the data sets before calculating the actual values.

• Dogs: The weights range from 12 to 102 pounds, with a significant spread between the minimum and maximum values. This suggests a larger mean and median compared to cats. Also, the large variation in dog weights suggests a higher standard deviation.
• Cats: The weights range from 4 to 23 pounds, with less variation compared to the dog weights. The cat weights are generally smaller, so the mean and median are likely smaller. The relatively small spread of the data suggests a lower standard deviation.

Therefore, based on this reasoning, we can conclude:

• The mean and median are likely higher for dogs because the weights are generally larger.
• The standard deviation is also likely higher for dogs because the variation (spread of the data) is greater.

The best conjecture is: C. 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.

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Part 1b. Compute the mean and standard deviation:

We can now calculate the mean and standard deviation for each data set (dogs and cats).

Dogs:
Weights: 12, 21, 39, 44, 52, 61, 74, 102

• Mean: $\frac{12 + 21 + 39 + 44 + 52 + 61 + 74 + 102}{8}$
• Standard deviation: We will use the formula: $\text{SD} = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$

Cats:
Weights: 4, 5, 8, 12, 15, 19, 23, 23

• Mean: $\frac{4 + 5 + 8 + 12 + 15 + 19 + 23 + 23}{8}$
• Standard deviation: Using the same formula.

Let me calculate these values.### Computed Values:

• Dogs:

  • Mean: 50.63 pounds
  • Standard deviation: 28.89 pounds

• Cats:

  • Mean: 13.63 pounds
  • Standard deviation: 7.63 pounds

Conclusion:

• The mean and standard deviation for dogs are significantly higher than for cats.
• This confirms the conjecture in option C: The mean, median, and standard deviation are all higher for dogs.

Would you like a more detailed breakdown or explanation of these calculations?

Related Questions:

1. How does the range of data affect the standard deviation?
2. What are other measures of central tendency besides mean and median?
3. How can outliers influence the mean and standard deviation?
4. What is the significance of using the sample standard deviation (with $n-1$)?
5. How do you interpret the difference in variability between two data sets?

Tip:

Always use the sample standard deviation (with $n-1$) when working with a sample of data, rather than the entire population, to avoid underestimating the variability.