Normal Distributions, Z-Scores, and Sampling Distributions: Study Notes

Normal Distributions and Z-Scores

Definition and Properties of the Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve.

• Mean (μ): The center of the distribution.

• Standard deviation (σ): Measures the spread or dispersion of the distribution.

• Standard normal distribution: A normal distribution with mean 0 and standard deviation 1.

Example: Heights of adult men are often normally distributed.

Z-Score

The z-score indicates how many standard deviations an observation is from the mean. It is used to standardize values for comparison.

• Formula:

$z = \frac{X - \mu}{\sigma}$

• Interpretation: A z-score of 2 means the value is 2 standard deviations above the mean.

Example: If a test score is 85, the mean is 80, and the standard deviation is 5, then $z = \frac{85-80}{5} = 1$.

Comparing Normal Curves

Normal curves with larger standard deviations are more spread out (less peaked), while those with smaller standard deviations are more concentrated around the mean.

• Dispersed curve: Higher standard deviation.

• Peaked curve: Lower standard deviation.

Example: Comparing two curves, one with σ = 10 and one with σ = 1, the curve with σ = 10 is more spread out.

Areas Under the Normal Curve

Finding Probabilities and Areas

Probabilities for normal distributions are found by calculating the area under the curve for a given interval. This is often done using z-tables or statistical software.

• Area to the left of z: $P(Z < z)$

• Area to the right of z: $P(Z > z)$

• Area between two z-values: $P(z_1 < Z < z_2)$

Example: The area under the standard normal curve to the left of $z = 1.5$ is approximately 0.9332.

Percentiles and Quartiles

Percentiles indicate the relative standing of a value within a data set. The third quartile (Q3) is the 75th percentile.

• Finding z for a percentile: Use the z-table to find the z-score corresponding to the desired cumulative area.

Example: The z-score for the third quartile (75th percentile) is approximately 0.67.

Applications of the Normal Distribution

Solving Real-World Problems

Many real-world phenomena, such as heights, test scores, and manufacturing tolerances, can be modeled using the normal distribution.

• Example: If the mean number of suitcases lost per week is 15.5 with a standard deviation of 3.6, the probability of losing less than 20 suitcases can be found by calculating the z-score and using the normal table.

Finding Probabilities for Intervals

To find the probability that a value falls within a certain interval, calculate the z-scores for the endpoints and find the area between them.

• Formula:

$P(a < X < b) = P\left(\frac{a-\mu}{\sigma} < Z < \frac{b-\mu}{\sigma}\right)$

Finding Values for Given Percentiles

To find the value corresponding to a given percentile, use the inverse of the z-score formula.

• Formula:

$X = \mu + z\sigma$

• Example: If the mean tire lifespan is 47,500 miles and the standard deviation is 3,000 miles, the mileage for the top 10% can be found using the z-score for the 90th percentile.

Sampling Distributions

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of sample means over repeated sampling from the same population.

• Central Limit Theorem: As sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the population's distribution.

• Standard error of the mean: Measures the variability of sample means.

Formula:

$\text{Standard Error} = \frac{\sigma}{\sqrt{n}}$

• Example: If σ = 20 and n = 16, then $\text{Standard Error} = \frac{20}{4} = 5$.

Properties of Sampling Distributions

• Larger sample sizes result in less variability in the sample mean.

• The mean of the sampling distribution equals the population mean.

• The standard deviation of the sampling distribution (standard error) decreases as sample size increases.

Summary Table: Key Concepts

Concept	Definition	Formula	Example
Z-score	Number of standard deviations from the mean	$z = \frac{X - \mu}{\sigma}$	Score of 85, mean 80, σ = 5: $z = 1$
Standard Error	Standard deviation of sample mean	$\frac{\sigma}{\sqrt{n}}$	σ = 20, n = 16: SE = 5
Percentile	Relative standing in data set	Use z-table	75th percentile: z ≈ 0.67
Area under curve	Probability for interval	Use z-table	P(Z < 1.5) ≈ 0.9332

Additional info:

• Some questions involve interpreting graphs of normal curves with different means and standard deviations.

• Real-world applications include airline luggage loss, teacher salaries, and heights of men.

• Sampling distribution concepts are illustrated with diagrams showing variability reduction as sample size increases.