To solve the given dataset step by step, we will follow the formulae for the required values:

• ΣX: Sum of the data points.
• M: Mean (average).
• s²: Sample variance.
• s: Sample standard deviation.
• sM: Standard error of the mean.
• t: t-statistic.

The dataset is:
$X = \{13, 7, 6, 12, 0, 4\}$

Step 1: ΣX (Sum of the data points)

$\Sigma X = 13 + 7 + 6 + 12 + 0 + 4 = 42$

Step 2: M (Mean)

The mean is calculated as: $M = \frac{\Sigma X}{n} = \frac{42}{6} = 7$ where $n = 6$ is the number of data points.

Step 3: s² (Sample variance)

Sample variance is given by: $s^2 = \frac{\Sigma (X_i - M)^2}{n - 1}$ First, calculate each $X_i - M$: $\{13 - 7, 7 - 7, 6 - 7, 12 - 7, 0 - 7, 4 - 7\} = \{6, 0, -1, 5, -7, -3\}$ Now square each value: $\{6^2, 0^2, (-1)^2, 5^2, (-7)^2, (-3)^2\} = \{36, 0, 1, 25, 49, 9\}$ Now sum them: $\Sigma (X_i - M)^2 = 36 + 0 + 1 + 25 + 49 + 9 = 120$ Finally, divide by $n - 1$: $s^2 = \frac{120}{6 - 1} = \frac{120}{5} = 24$

Step 4: s (Sample standard deviation)

Sample standard deviation is the square root of the variance: $s = \sqrt{s^2} = \sqrt{24} \approx 4.899$

Step 5: sM (Standard error of the mean)

Standard error of the mean is calculated as: $s_M = \frac{s}{\sqrt{n}} = \frac{4.899}{\sqrt{6}} \approx \frac{4.899}{2.449} \approx 2$

Step 6: t (t-statistic)

The t-statistic (for testing whether the mean is significantly different from 0) is: $t = \frac{M}{s_M} = \frac{7}{2} = 3.5$

Final Results:

• ΣX = 42
• M = 7
• s² = 24
• s ≈ 4.899
• sM ≈ 2
• t ≈ 3.5

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Would you like further details or have any questions?

5 Relative Questions:

1. How is the sample variance affected if the dataset is doubled?
2. Can the sample standard deviation ever be zero for a non-constant dataset?
3. What is the difference between population variance and sample variance?
4. How does the size of the sample impact the standard error of the mean?
5. How is the t-statistic used to determine significance in hypothesis testing?

Tip:

Always remember that sample variance uses $n - 1$ in the denominator to account for the degrees of freedom, ensuring an unbiased estimate of population variance.