You measure 30 watermelons' weights, and find they have a mean weight of 67 ounces. Assume the population standard deviation is 14.7 ounces. Based on this, construct a 95% confidence interval for the true population mean watermelon weight.

Give your answers as decimals, to two places

±

ounces

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

\[
\text{CI} = \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\bar{x} = 67\) (the sample mean),
- \(z_{\alpha/2} = 1.96\) (the z-value for a 95% confidence interval),
- \(\sigma = 14.7\) (the population standard deviation),
- \(n = 30\) (the sample size).

### Step 1: Calculate the standard error (SE)

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{14.7}{\sqrt{30}} \approx 2.684
\]

### Step 2: Compute the margin of error (ME)

\[
ME = z_{\alpha/2} \cdot SE = 1.96 \cdot 2.684 \approx 5.26
\]

### Step 3: Calculate the confidence interval

\[
\text{Lower bound} = \bar{x} - ME = 67 - 5.26 = 61.74
\]
\[
\text{Upper bound} = \bar{x} + ME = 67 + 5.26 = 72.26
\]

Thus, the 95% confidence interval for the true population mean watermelon weight is:

\[
67 \pm 5.26 \text{ ounces, or } (61.74, 72.26)
\]

Would you like more details on how these steps were calculated or have any other questions?

### Relative Questions:
1. How does increasing the sample size affect the confidence interval?
2. What happens to the confidence interval if the population standard deviation is unknown?
3. How do we interpret a 95% confidence interval in this context?
4. What would the confidence interval be if we had a 99% confidence level?
5. Can we use a t-distribution for small sample sizes instead of a z-distribution?

### Tip:
Always ensure you're using the appropriate z-value or t-value for the desired confidence level and sample size.

You measure 30 watermelons' weights, and find they have a mean weight of 67 ounces. Assume the population standard deviation is 14.7 ounces. Based on this, construct a 95% confidence interval for the true population mean watermelon weight. Give your answers as decimals, to two places ± ounces.

Learn how to construct a 95% confidence interval for the mean weight of watermelons with a sample size of 30. This solution covers key statistical concepts, including standard error, margin of error, and the central limit theorem. Suitable for students in Grades 10-12.

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