To construct a 99% confidence interval for the true population mean dog weight, we will use the z-distribution because the population standard deviation is known. Here are the steps:

1. **Identify the given information:**
- Sample mean ($\bar{x}$) = 50 ounces
- Population standard deviation ($\sigma$) = 7.5 ounces
- Sample size ($n$) = 43
- Confidence level = 99%

2. **Find the critical value (z$_{\alpha/2}$) for a 99% confidence interval:**
- For a 99% confidence level, $\alpha = 1 - 0.99 = 0.01$.
- $\alpha/2 = 0.01/2 = 0.005$.
- Using a z-table or standard normal distribution table, the critical value for $\alpha/2 = 0.005$ is approximately 2.576.

3. **Calculate the standard error of the mean (SE):**
$$
SE = \frac{\sigma}{\sqrt{n}} = \frac{7.5}{\sqrt{43}} \approx \frac{7.5}{6.56} \approx 1.14
$$

4. **Calculate the margin of error (ME):**
$$
ME = z_{\alpha/2} \times SE = 2.576 \times 1.14 \approx 2.94
$$

5. **Construct the confidence interval:**
$$
\text{Lower limit} = \bar{x} - ME = 50 - 2.94 = 47.06
$$
$$
\text{Upper limit} = \bar{x} + ME = 50 + 2.94 = 52.94
$$

Therefore, the 99% confidence interval for the true population mean dog weight is $47.06$ to $52.94$ ounces.

Question

To construct a 99% confidence interval for the true population mean dog weight, we will use the z-distribution because the population standard deviation is known. Here are the steps:

1. **Identify the given information:**
   - Sample mean ($\bar{x}$) = 50 ounces
   - Population standard deviation ($\sigma$) = 7.5 ounces
   - Sample size ($n$) = 43
   - Confidence level = 99%

2. **Find the critical value (z$_{\alpha/2}$) for a 99% confidence interval:**
   - For a 99% confidence level, $\alpha = 1 - 0.99 = 0.01$.
   - $\alpha/2 = 0.01/2 = 0.005$.
   - Using a z-table or standard normal distribution table, the critical value for $\alpha/2 = 0.005$ is approximately 2.576.

3. **Calculate the standard error of the mean (SE):**
   $$
   SE = \frac{\sigma}{\sqrt{n}} = \frac{7.5}{\sqrt{43}} \approx \frac{7.5}{6.56} \approx 1.14
   $$

4. **Calculate the margin of error (ME):**
   $$
   ME = z_{\alpha/2} \times SE = 2.576 \times 1.14 \approx 2.94
   $$

5. **Construct the confidence interval:**
   $$
   \text{Lower limit} = \bar{x} - ME = 50 - 2.94 = 47.06
   $$
   $$
   \text{Upper limit} = \bar{x} + ME = 50 + 2.94 = 52.94
   $$

Therefore, the 99% confidence interval for the true population mean dog weight is $47.06$ to $52.94$ ounces.

Knowee AI · Accepted Answer

To construct a 99% confidence interval for the true population mean dog weight, we will use the z-distribution because the population standard deviation is known. Here are the steps:

1. **Identify the given information:**
   - Sample mean ($\bar{x}$) = 50 ounces
   - Population standard deviation ($\sigma$) = 7.5 ounces
   - Sample size ($n$) = 43
   - Confidence level = 99%

2. **Find the critical value (z$_{\alpha/2}$) for a 99% confidence interval:**
   - For a 99% confidence level, $\alpha = 1 - 0.99 = 0.01$.
   - $\alpha/2 = 0.01/2 = 0.005$.
   - Using a z-table or standard normal distribution table, the critical value for $\alpha/2 = 0.005$ is approximately 2.576.

3. **Calculate the standard error of the mean (SE):**
   $$
   SE = \frac{\sigma}{\sqrt{n}} = \frac{7.5}{\sqrt{43}} \approx \frac{7.5}{6.56} \approx 1.14
   $$

4. **Calculate the margin of error (ME):**
   $$
   ME = z_{\alpha/2} \times SE = 2.576 \times 1.14 \approx 2.94
   $$

5. **Construct the confidence interval:**
   $$
   \text{Lower limit} = \bar{x} - ME = 50 - 2.94 = 47.06
   $$
   $$
   \text{Upper limit} = \bar{x} + ME = 50 + 2.94 = 52.94
   $$

Therefore, the 99% confidence interval for the true population mean dog weight is $47.06$ to $52.94$ ounces.

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