A sample of 29 shopping bags are selected and weighed as customers exit a supermarket. The histogram of 𝑦𝑦, the weights of the bags, looks normal. The sample mean is equal to 1.24 kg and the sample standard deviation is equal to 0.26 kg.

Compute a 95% confidence interval for the population mean weight of bags.

Give your final answer with at least 33 dp.

Lower bound of the 95% CI

Upper bound of the 95% CI 

To compute the 95% confidence interval for the population mean weight of shopping bags, we use the formula for the confidence interval for a mean when the population standard deviation is unknown:

\[
\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}
\]

Where:
- \(\bar{y}\) is the sample mean.
- \(t\) is the critical value from the \(t\)-distribution corresponding to a 95% confidence level and \(n-1\) degrees of freedom.
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.

### Step 1: Identify the values
- Sample mean \(\bar{y} = 1.24\) kg
- Sample standard deviation \(s = 0.26\) kg
- Sample size \(n = 29\)

### Step 2: Find the \(t\)-value
For a 95% confidence level and \(n-1 = 28\) degrees of freedom, the \(t\)-value is approximately \(t_{0.025, 28} = 2.048\).

### Step 3: Calculate the standard error (SE)

\[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{0.26}{\sqrt{29}} \approx 0.0483 \text{ kg}
\]

### Step 4: Compute the margin of error (ME)

\[
\text{ME} = t \times \text{SE} = 2.048 \times 0.0483 \approx 0.0989 \text{ kg}
\]

### Step 5: Calculate the confidence interval

\[
\text{Lower bound} = \bar{y} - \text{ME} = 1.24 - 0.0989 \approx 1.1411 \text{ kg}
\]
\[
\text{Upper bound} = \bar{y} + \text{ME} = 1.24 + 0.0989 \approx 1.3389 \text{ kg}
\]

Thus, the 95% confidence interval for the population mean weight of shopping bags is:

\[
\text{Lower bound: } 1.141 \text{ kg} \quad \text{(rounded to 3 decimal places)}
\]
\[
\text{Upper bound: } 1.339 \text{ kg} \quad \text{(rounded to 3 decimal places)}
\]

**Lower bound of the 95% CI:** 1.141  
**Upper bound of the 95% CI:** 1.339

Would you like more details or have any questions?

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Here are 5 related questions you might explore:

1. How would the confidence interval change if the sample size increased to 50?
2. What would the confidence interval be if the sample mean was 1.30 kg instead of 1.24 kg?
3. How does the shape of the histogram affect the validity of the confidence interval?
4. What would the confidence interval be at a 99% confidence level?
5. How would the confidence interval change if the sample standard deviation was 0.30 kg instead of 0.26 kg?

**Tip:** Always ensure the sample size is large enough or the data is approximately normal when using the \( t \)-distribution for confidence intervals.

Learn how to calculate a 95% confidence interval for the population mean weight of shopping bags using sample data. Given a sample mean of 1.24 kg and a sample standard deviation of 0.26 kg from a normal distribution, compute the lower and upper bounds with a precision of at least 33 decimal places.

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