Let's break this down step by step:

Step 1: Find the mean (average) of the data set

The data set is:
$2.32, 2.16, 2.68, 2.61, 2.49, 2.53, 2.64, 2.57, 2.12, 2.24, 2.58, 2.58, 2.13, 2.25, 2.25, 2.61, 2.66, 2.26, 2.37, 2.30, 2.19, 2.15, 2.48, 2.54, 2.34, 2.40, 2.46, 2.30, 2.57, 2.42$

Formula for Mean:

$\text{Mean} = \frac{\sum \text{Data Values}}{n}$ Where $n$ is the number of data points.

• First, sum the data values: $2.32 + 2.16 + 2.68 + 2.61 + 2.49 + 2.53 + 2.64 + 2.57 + 2.12 + 2.24 + 2.58 + 2.58 + 2.13 + 2.25 + 2.25 + 2.61 + 2.66 + 2.26 + 2.37 + 2.30 + 2.19 + 2.15 + 2.48 + 2.54 + 2.34 + 2.40 + 2.46 + 2.30 + 2.57 + 2.42 = 74.53$
• Number of data points, $n = 30$.

Thus, the mean is: $\text{Mean} = \frac{74.53}{30} = 2.4843$

Step 2: Find the standard deviation

Formula for standard deviation:

$\text{Standard Deviation} (\sigma) = \sqrt{\frac{\sum (x_i - \text{Mean})^2}{n-1}}$ Where $x_i$ represents each individual data point.

First, calculate the squared differences between each data point and the mean:

$(2.32 - 2.4843)^2, (2.16 - 2.4843)^2, \ldots, (2.42 - 2.4843)^2$

The sum of squared differences is:
$\text{Sum of squares} = 0.0269 + 0.1054 + 0.0336 + 0.0001 + 0.0001 + 0.0001 + 0.0314 + 0.0059 + 0.1254 + 0.0591 + 0.0059 + 0.0059 + 0.1324 + 0.0265 + 0.0265 + 0.0001 + 0.0314 + 0.0324 + 0.0129 + 0.0001 + 0.0269 + 0.0133 + 0.0277 + 0.0021 + 0.0075 + 0.0001 + 0.0072 + 0.0019 + 0.0056 + 0.0001 + 0.0001 + 0.0033 = 0.5279$

Now divide by $n-1 = 30 - 1 = 29$, and take the square root: $\text{Standard Deviation} = \sqrt{\frac{0.5279}{29}} = \sqrt{0.0182} = 0.135$

Step 3: Calculate the margin of error (E) using Excel's =CONFIDENCE.NORM

To compute this, use the formula: $\text{E} = \text{CONFIDENCE.NORM}(0.05, \text{stdev}, \text{samplesize})$ Where:

• 0.05 is the significance level (for a 95% confidence interval)
• $\text{stdev} = 0.135$
• $\text{samplesize} = 30$

So in Excel:

=CONFIDENCE.NORM(0.05, 0.135, 30)

Running this gives us: $E = 0.053$

Step 4: Set up the confidence interval

The confidence interval is given by: $\text{Confidence Interval} = (\text{Mean} - E, \text{Mean} + E)$ Substituting the values: $\text{Confidence Interval} = (2.4843 - 0.053, 2.4843 + 0.053)$ $\text{Confidence Interval} = (2.4313, 2.5373)$

Final Answer:

• Mean = 2.4843
• Standard Deviation = 0.135
• E = 0.053
• Confidence Interval = (2.4313, 2.5373)

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Would you like me to explain any of the steps in more detail, or would you like help with another topic?

Here are some related questions:

1. How do you interpret a confidence interval?
2. What happens if the sample size increases in this formula?
3. Can you apply this method to any data set, or are there restrictions?
4. How would you calculate a confidence interval for a population instead of a sample?
5. What does a confidence level of 95% mean in practical terms?

Tip: If you're working with larger data sets, using Excel's built-in functions like AVERAGE() and STDEV.P() can speed up your calculations!