Normal Distribution and Confidence Intervals Review

Comprehensive vocabulary flashcards covering normal distribution, standardization, and confidence intervals based on the lecture notes.

Normal Distribution

A symmetrical, bell-shaped probability distribution where the mean, median, and mode coincide.

Parameters of Normal Distribution

The mean $\mu$ (determines location) and the standard deviation $\sigma$ (determines dispersion or spread).

Standard Normal Distribution

A specific normal distribution where the mean $\mu = 0$ and the standard deviation $\sigma = 1$.

z-Standardization Formula

$z = \frac{x - \bar{x}}{\text{SD}}$ or generic forms like $z = \frac{x - \text{mean}}{\text{standard deviation}}$; specifically in the transcript: $z = \frac{x - \bar{x}}{s}$ or $z = \frac{x - \nu}{\text{o}}$, mathematically represented as $z = \frac{x - \bar{\nu}}{\text{o}}$.

Meaning of a z-value

Indicates the number of standard deviations a specific value is located away from the mean.

z = 0

The value corresponds exactly to the mean.

Positive vs. Negative z-values

A positive z-value indicates the value is above the mean; a negative z-value indicates it is below the mean.

Calculating P(Z > z) for Left-Area Tables

$P(Z > z) = 1 - P(Z \times z)$.

P(Z < 1.25) Calculation

$\Phi(1.25) = 0.8944 = 89.44\, \%$.

P(Z > 1.75) Calculation

$1 - \Phi(1.75) = 1 - 0.9599 = 0.0401 = 4.01\, \%$.

Raw Value (x) vs. z-value

The raw value is the original measurement; the z-value is a standardized value that enables comparisons across different scales.

Purpose of Standardization

To allow values from different groups or distributions to be compared directly.

Interpretation of Identical z-values

It means the individuals performed similarly well or poorly relative to their respective comparison groups.

Rearranging z-formula for x

Determine the matching z-value and calculate $x = \bar{\nu} + z \times \text{o}$.

Critical z-value for Top 10%

$P(X \times x) = 0.90$, which corresponds approximately to $z \times 1.28$.

Critical z-value for Top 1%

$z \times 2.33$ (based on a cumulative probability of $99\, \%$).

Standard Error (SE)

The standard deviation of the sampling distribution of the means, calculated as $SE = \frac{\text{o}}{\sqrt{n}}$.

Confidence Interval (CI)

A range of plausible values for an unknown population parameter calculated from sample data.

95% Confidence Interval Interpretation

With very frequent repetition of the sampling procedure, 95% of the calculated intervals will contain the true population parameter.

Effect of Sample Size (n) on CI

As $n$ increases, the standard error becomes smaller, and the confidence interval becomes narrower (more precise).

Effect of Standard Deviation (o) on CI

As standard deviation ($\text{o}$) increases, the confidence interval becomes wider.

Effect of Confidence Level on CI Width

As the confidence level increases (and $\alpha$ decreases), the confidence interval becomes wider.

z-Confidence Interval Formula

$KI = \bar{x} \times z_{1 - \frac{\text{a}}{2}} \times \frac{\text{o}}{\sqrt{n}}$; used when the population standard deviation $\text{o}$ is known.

t-Confidence Interval Formula

$KI = \bar{x} \times t_{1 - \frac{\text{a}}{2}; df} \times \frac{s}{\sqrt{n}}$; used when $\text{o}$ is unknown and estimated by sample standard deviation $s$.

Degrees of Freedom (df)

Calculated as $df = n - 1$.

Common Critical z-values

95% CI: $z \times 1.96$; 99% CI: $z \times 2.576$; 80% CI: $z \times 1.282$.

Margin of Error (Ez)

$Ez = z_{1 - \frac{\text{a}}{2}} \times SE$.

SE Example (n = 225)

Given $\text{o} = 15$ and $n = 225$, $SE = \frac{15}{\sqrt{225}} = 1$.

SE Example (n = 25)

Given $\text{o} = 15$ and $n = 25$, $SE = \frac{15}{\sqrt{25}} = 3$.

Double-sided Interval Error

Failing to divide the significance level (alpha) by 2 when determining the critical value.

Distribution Selection Error

Using the z-distribution when the population standard deviation $\text{o}$ is unknown, making the t-distribution required.