Suppose you use simple random sampling to select and measure 50 dogs' weights, and find they have a mean weight of 72 ounces. Assume the population standard deviation is 10.2 ounces. Based on this, construct a 95% confidence interval for the true population mean dog weight.

Give your answers as decimals, to two placesounces

To construct a 95% confidence interval for the true population mean weight, we use the formula:

\[
\text{Confidence Interval} = \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right)
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(z^*\) is the z-score corresponding to the desired confidence level (1.96 for 95% confidence).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.

Given:
- Sample mean, \(\bar{x} = 72\) ounces.
- Population standard deviation, \(\sigma = 10.2\) ounces.
- Sample size, \(n = 50\).

### Step 1: Calculate the standard error (SE)
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{10.2}{\sqrt{50}} \approx \frac{10.2}{7.07} \approx 1.44
\]

### Step 2: Find the margin of error (ME)
\[
ME = z^* \times SE = 1.96 \times 1.44 \approx 2.82
\]

### Step 3: Construct the confidence interval
\[
\text{Lower limit} = 72 - 2.82 = 69.18
\]
\[
\text{Upper limit} = 72 + 2.82 = 74.82
\]

The 95% confidence interval for the true population mean dog weight is \(69.18\) to \(74.82\) ounces.

Would you like more details on the calculations, or have any questions?

Here are five related questions to expand on this topic:
1. How would the confidence interval change if the sample size were increased?
2. What would happen to the interval if a 99% confidence level were used instead?
3. How is the confidence interval affected by the population standard deviation?
4. Why do we use the z-score of 1.96 for a 95% confidence interval?
5. What is the difference between using the t-distribution and the z-distribution for constructing confidence intervals?

**Tip:** The width of a confidence interval decreases as the sample size increases, which leads to a more precise estimate of the population mean.

Suppose you use simple random sampling to select and measure 50 dogs' weights, and find they have a mean weight of 72 ounces. Assume the population standard deviation is 10.2 ounces. Based on this, construct a 95% confidence interval for the true population mean dog weight.

Learn how to calculate a 95% confidence interval for the true mean dog weight, given a sample mean of 72 ounces, a population standard deviation of 10.2 ounces, and a sample size of 50. This problem involves key statistical concepts such as confidence intervals, normal distribution, and sampling techniques. Suitable for high school statistics students.

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