Estimation and Inference

NoteDownload Files

Download PDF | Download Word | Download PowerPoint

Learning Module 7: Estimation and Inference

---

Standard Error of the Sample Mean (Known Population Variance)

\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \tag{1} \]

Where:

• \(\sigma_{\bar{X}}\): standard error of the sample mean
• \(\sigma\): population standard deviation
• \(n\): sample size

View Markdown Source

## Standard Error of the Sample Mean (Known Population Variance)

$$
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \tag{1}
$$

Where:

* $\sigma_{\bar{X}}$: standard error of the sample mean
* $\sigma$: population standard deviation
* $n$: sample size

---

Standard Error of the Sample Mean (Unknown Population Variance)

\[ s_{\bar{X}} = \frac{s}{\sqrt{n}} \tag{2} \]

Where:

• \(s_{\bar{X}}\): estimated standard error of the sample mean
• \(s\): sample standard deviation
• \(n\): sample size

View Markdown Source

## Standard Error of the Sample Mean (Unknown Population Variance)

$$
s_{\bar{X}} = \frac{s}{\sqrt{n}} \tag{2}
$$

Where:

* $s_{\bar{X}}$: estimated standard error of the sample mean
* $s$: sample standard deviation
* $n$: sample size

---

Sample Variance

\[ s^2 = \frac{\sum_{i=1}^{n}(X_i - \bar{X})^2}{n - 1} \tag{3} \]

Where:

• \(s^2\): sample variance
• \(X_i\): \(i\)th observation in the sample
• \(\bar{X}\): sample mean
• \(n\): sample size

View Markdown Source

## Sample Variance

$$
s^2 = \frac{\sum_{i=1}^{n}(X_i - \bar{X})^2}{n - 1} \tag{3}
$$

Where:

* $s^2$: sample variance
* $X_i$: $i$th observation in the sample
* $\bar{X}$: sample mean
* $n$: sample size

---

Model-free resampling or non-parametric resampling

\[ s_{\bar{X}} = \sqrt{\frac{1}{B - 1} \sum_{b=1}^{B} \left( \hat{\theta}_b - \bar{\theta} \right)^2} \tag{4} \]

Where:

• \(s_{\bar{X}}\): the estimate of the standard error of the sample mean
• \(B\): the number of resamples drawn from the original sample
• \(\hat{\theta}_b\): the mean of a resample, and
• \(\bar{\theta}\): the mean across all the resample means

View Markdown Source

## Model-free resampling or non-parametric resampling

$$
s_{\bar{X}} = \sqrt{\frac{1}{B - 1} \sum_{b=1}^{B} \left( \hat{\theta}_b - \bar{\theta} \right)^2} \tag{4}
$$

Where:

* $s_{\bar{X}}$: the estimate of the standard error of the sample mean
* $B$: the number of resamples drawn from the original sample
* $\hat{\theta}_b$: the mean of a resample, and
* $\bar{\theta}$: the mean across all the resample means

---