https://file.aiquickdraw.com/mathbot/file/172856906802418i2qgel.jpg

Tapi aku pake rumus s nya itu 1/n dikali akar n kali sigma x1 pangkat 2 di kurang (sigma x1) pangkat 2 per n -1

To solve this problem, you want to calculate the correct average diameter of the solid sphere and its uncertainty based on the given measurements. The formula for calculating the uncertainty (standard deviation of the mean) is typically derived from statistical principles. Let's break this down step by step.

### Step 1: Calculate the Mean Diameter
Given measurements of the sphere's diameter:
- 8.15 cm
- 8.10 cm
- 8.00 cm
- 8.15 cm
- 8.00 cm

The mean \(\bar{x}\) is given by:

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where \(x_i\) are the measurements, and \(n = 5\) (the number of measurements).

\[
\bar{x} = \frac{8.15 + 8.10 + 8.00 + 8.15 + 8.00}{5} = \frac{40.40}{5} = 8.08 \, \text{cm}
\]

### Step 2: Calculate the Standard Deviation
The formula for the standard deviation \(s\) is:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

First, compute \((x_i - \bar{x})^2\) for each measurement:
- For \(x_1 = 8.15\), \((8.15 - 8.08)^2 = 0.0049\)
- For \(x_2 = 8.10\), \((8.10 - 8.08)^2 = 0.0004\)
- For \(x_3 = 8.00\), \((8.00 - 8.08)^2 = 0.0064\)
- For \(x_4 = 8.15\), \((8.15 - 8.08)^2 = 0.0049\)
- For \(x_5 = 8.00\), \((8.00 - 8.08)^2 = 0.0064\)

Now, sum the squared deviations:

\[
\sum (x_i - \bar{x})^2 = 0.0049 + 0.0004 + 0.0064 + 0.0049 + 0.0064 = 0.023
\]

Then calculate the standard deviation:

\[
s = \sqrt{\frac{0.023}{5-1}} = \sqrt{\frac{0.023}{4}} = \sqrt{0.00575} = 0.0758
\]

### Step 3: Calculate the Uncertainty (Standard Error of the Mean)
The uncertainty \(\Delta x\) (standard error of the mean) is given by:

\[
\Delta x = \frac{s}{\sqrt{n}} = \frac{0.0758}{\sqrt{5}} = \frac{0.0758}{2.236} \approx 0.034 \, \text{cm}
\]

### Step 4: Final Answer
The correct way to write the diameter with its uncertainty is:

\[
(8.08 \pm 0.034) \, \text{cm}
\]

This corresponds to option **e**.

---

Do you have any questions or need further details? Here are some related questions:

1. How would the uncertainty change if more measurements were taken?
2. Why do we use \(n-1\) in the standard deviation formula?
3. How can you interpret the significance of the uncertainty value?
4. What are other ways to measure uncertainty in experimental data?
5. What if one of the measurements was significantly different, how would that impact the results?

**Tip:** Always double-check your calculations for the mean and standard deviation, as small errors can impact the final uncertainty result.

Pengukuran diameter bola pejal menggunakan jangka sorong dilakukan secara berulang dan dihasilkan data sebagai berikut. Penulisan diameter bola pejal beserta ketidakpastiannya yang benar adalah ....

Learn how to calculate the mean and uncertainty of a solid sphere's diameter using repeated measurements and statistical methods. This problem involves calculating the mean, standard deviation, and uncertainty, ideal for Grades 10-12 students.

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