Normal Probability Distributions

Density Curves and Continuous Probability Distributions

In statistics, a density curve is a smooth curve that represents the distribution of a continuous variable. The area under the curve for a given interval represents the probability of the variable falling within that interval. The function describing the density curve is called the probability density function (p.d.f.).

• Continuous Probability Distribution: Satisfies three conditions:

  1. The random variable is continuous.

  2. The probability density function f(x) ≥ 0 for all x.

  3. The total area under the p.d.f. curve is 1.

Uniform Distribution

A uniform distribution is a continuous probability distribution where all outcomes are equally likely within a certain interval [a, b]. The graph of a uniform distribution is rectangular.

• Probability Density Function:

• Area Calculation: Area = height × width

Example: Waiting Time at Subway

• Waiting time is uniformly distributed between 0 and 5 minutes.

• Height of the distribution:

• Probability of waiting more than 2 minutes:

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped, symmetric curve. It is defined by its mean (μ) and standard deviation (σ).

• Symmetric about the mean μ.

• The spread depends on σ; larger σ means a flatter, more spread-out curve.

• Probability density function:

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Values from any normal distribution can be standardized using the z-score formula.

• Standard normal curve is used to find probabilities and percentiles.

• Z-score formula:

Finding Probabilities Using the Normal Distribution

Probabilities are found by calculating the area under the normal curve for a given range of values.

• Use statistical software (e.g., StatCrunch) or tables to find probabilities for given z or x values.

• For probability between two values: , calculate the area between a and b.

Example: Toyota Car Battery Lifetime

• Mean = 45 months, σ = 8 months

• Probability battery fails within 36 months:

• Probability battery lasts between 36 and 50 months:

• Probability battery lasts 55 months or longer:

Visualizing Probabilities on the Normal Curve

Shaded regions on the normal curve represent the probability for a given range.

Percentiles and Critical Values

Percentiles and critical values are important for hypothesis testing and confidence intervals.

• Given a probability (area), find the corresponding z-score or x value.

• 95th percentile (P95):

• Critical value : The z-score for which the area to the right is α.

Example: Bone Density Score

• Find the score corresponding to the 95th percentile:

• Find scores separating the bottom and top 2.5%: and

Sampling Distributions and Estimators

Sampling Distribution of the Sample Mean

The sampling distribution of a statistic is the probability distribution of that statistic based on all possible samples from the population.

• The mean of the sample means equals the population mean:

• The standard deviation of the sample means:

Example: Heights of Students

Sampling Distribution of Sample Proportion

The sampling distribution of sample proportion describes the distribution of sample proportions from all possible samples.

• Sample proportion:

• Population proportion:

• Mean:

• Standard deviation:

• Shape is normal if and

Central Limit Theorem

Statement and Applications

The Central Limit Theorem (CLT) states that if the sample size is large enough (n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• If population is normal:

• If n ≥ 30:

• For individual values:

• For sample means:

Example: Water Taxi Capacity

• Mean weight of men: 182.9 lb, σ = 40.8 lb

• Probability individual man weighs > 140 lb:

• Probability average weight of 25 men > 140 lb:

• Conclusion: Capacity is not safe; probability of overloading is 1.

Example: SAT Scores

• Mean = 1511, σ = 312

• Probability individual score > 1600:

• Probability average score of 36 students > 1600:

• 98th percentile for average score of 45 students:

Summary Table: Key Formulas