You measure 22 watermelons' weights, and find they have a mean weight of 60 ounces. Assume the population standard deviation is 6.9 ounces. Based on this, construct a 90% confidence interval for the true population mean watermelon weight.

Give your answers as decimals, to two places

 ±  ounces

To construct a 90% confidence interval for the true population mean weight of watermelons, we'll use the following formula for a confidence interval with a known population standard deviation:

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

where:
- \(\bar{x}\) is the sample mean (60 ounces),
- \(Z\) is the Z-score corresponding to a 90% confidence level,
- \(\sigma\) is the population standard deviation (6.9 ounces),
- \(n\) is the sample size (22).

### Step 1: Find the Z-score for a 90% Confidence Level
For a 90% confidence level, the Z-score is approximately **1.645**.

### Step 2: Calculate the Margin of Error (ME)
The margin of error (ME) is given by:

\[
ME = Z \left(\frac{\sigma}{\sqrt{n}}\right) = 1.645 \times \left(\frac{6.9}{\sqrt{22}}\right)
\]

First, calculate \(\frac{\sigma}{\sqrt{n}}\):

\[
\frac{6.9}{\sqrt{22}} \approx 1.47
\]

Now, multiply by the Z-score:

\[
ME = 1.645 \times 1.47 \approx 2.42
\]

### Step 3: Construct the Confidence Interval
Now, we can construct the confidence interval:

\[
\text{Confidence Interval} = 60 \pm 2.42
\]

So, the 90% confidence interval for the true population mean weight is:

\[
(60 - 2.42, 60 + 2.42) = (57.58, 62.42)
\]

### Answer
The 90% confidence interval for the true population mean weight of the watermelons is:

**60 ± 2.42 ounces**

---

Would you like further details, or have any questions?

#### Related Questions:
1. How would the confidence interval change with a 95% confidence level?
2. How does increasing the sample size impact the confidence interval width?
3. How would a higher population standard deviation affect the confidence interval?
4. Why do we use the Z-score instead of the t-score in this calculation?
5. What assumptions are made in this confidence interval calculation?

#### Tip:
Remember, a smaller margin of error results in a narrower confidence interval, which can be achieved with a larger sample size or a lower confidence level.

You measure 22 watermelons' weights, and find they have a mean weight of 60 ounces. Assume the population standard deviation is 6.9 ounces. Based on this, construct a 90% confidence interval for the true population mean watermelon weight. Give your answers as decimals, to two places ± ounces.

Learn how to calculate a 90% confidence interval for the mean weight of watermelons. Using a sample mean of 60 ounces, a population standard deviation of 6.9 ounces, and a sample size of 22, this solution demonstrates step-by-step calculations to determine the confidence interval for the population mean weight.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation