(CO 4) In a sample of 15 small candles, the weight is found to be 3.72 ounces with a standard deviation of 0.963 ounces. What would be the 87% confidence interval for the size of the candles?
Group of answer choices

(3.369, 4.071)

(2.757, 4.683)

(1.794, 5.646)

(3.320, 4.120)

To calculate the 87% confidence interval for the mean weight of the candles, we will use the following formula for the confidence interval of the mean:

\[
\text{CI} = \bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(t_{\alpha/2}\) is the critical value from the t-distribution.
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.

Given:
- \(\bar{x} = 3.72\) ounces
- \(s = 0.963\) ounces
- \(n = 15\)
- Confidence level = 87%

First, we need to find the critical value \(t_{\alpha/2}\) for 87% confidence level with \(n-1 = 14\) degrees of freedom.

The confidence level is 87%, so \(\alpha = 1 - 0.87 = 0.13\), and \(\alpha/2 = 0.065\).

Using the t-distribution table or a calculator for 14 degrees of freedom:

\[
t_{0.065, 14} \approx 1.511
\]

Next, we calculate the margin of error:

\[
\text{Margin of Error} = t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right) = 1.511 \left(\frac{0.963}{\sqrt{15}}\right)
\]

\[
\text{Margin of Error} = 1.511 \left(\frac{0.963}{3.873}\right) \approx 1.511 \times 0.249 \approx 0.376
\]

Now, we calculate the confidence interval:

\[
\text{CI} = \bar{x} \pm \text{Margin of Error} = 3.72 \pm 0.376
\]

\[
\text{CI} = (3.344, 4.096)
\]

Comparing this to the given choices, the closest match is:

(3.320, 4.120)

Thus, the correct answer is:

\[
\boxed{(3.320, 4.120)}
\]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the formula for calculating the sample mean?
2. How do you interpret the standard deviation in a dataset?
3. What is the difference between a z-score and a t-score?
4. How do degrees of freedom affect the critical value in a t-distribution?
5. Why do we use the t-distribution instead of the normal distribution for small sample sizes?
6. What is the formula for calculating the standard error of the mean?
7. How does changing the confidence level affect the width of the confidence interval?
8. What are some practical applications of confidence intervals in real-world scenarios?

**Tip:** Remember, the choice of using a t-distribution over a normal distribution is due to the sample size. For small samples (n < 30), the t-distribution is more appropriate as it accounts for the additional variability.

Learn how to calculate an 87% confidence interval for the mean weight of a sample of 15 small candles with a step-by-step solution. Understand the use of the t-distribution, sample mean, sample standard deviation, and degrees of freedom in statistical inference. Suitable for college-level students and professionals.

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