Probability Distributions and the Normal Approximation

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Mean (Expected Value): Where n is the number of trials and p is the probability of success.

• Standard Deviation:

• Shape: The binomial distribution is approximately bell-shaped (normal) if and .

Example: For , :

• Mean:

• Standard Deviation:

• Since and , the distribution is bell-shaped.

Normal Approximation to the Binomial

When the binomial distribution is approximately normal, probabilities can be estimated using the normal distribution.

• Draw a normal curve centered at with spread .

• Use continuity correction when approximating discrete binomial probabilities with the normal distribution.

Calculating Binomial Probabilities

• At least k successes:

• At most k successes:

Example: Probability that at least 75 papers are missing a name in , :

Normal Distribution and Z-Scores

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Any normal variable can be standardized:

• Z-score:

Z-scores measure how many standard deviations a value is from the mean.

Finding Probabilities and Percentiles

• Probability above a value:

• Probability below a value:

• Percentiles: To find the value corresponding to the th percentile, use where is the z-score for the percentile.

Example: For , , probability of scoring above 70:

Unusual Values

• A value is considered unusual if its probability is less than 0.05.

• Example: If , then 55 is an unusual score.

Middle 50% of a Normal Distribution

• Find z-scores for the 25th and 75th percentiles (, ).

• Calculate , .

Applications: Waiting Times and Probability Plots

Normal Probability Plots

A normal probability plot is used to assess if a data set is approximately normally distributed. If the points fall roughly along a straight line, the data are considered normal.

Example: Waiting Times

• Given min, min.

• Probability a person waits between 5 and 8 minutes:

• Top 5% of wait times:

• min

Sampling Distributions

Sampling Distribution of the Mean

The sampling distribution of the sample mean describes the distribution of sample means from all possible samples of a given size from a population.

• Mean:

• Standard Deviation (Standard Error):

• Shape: If the population is normal or , the sampling distribution is approximately normal (Central Limit Theorem).

Sampling Distribution of a Proportion

• Mean:

• Standard Deviation:

Example: Energy Needs in Pregnancy

• Population mean kcal/day, kcal/day.

• Sample size .

• Standard error:

• Probability a randomly selected woman has energy need over 2625 kcal/day:

• Probability sample mean of 20 women exceeds 2625 kcal/day:

  • (unusual, since )

Summary Table: Key Formulas

Additional info:

• Continuity correction is often used when approximating binomial probabilities with the normal distribution, by adjusting the discrete x-value by 0.5.

• Normal probability plots are a graphical tool for assessing normality; strong linearity suggests normality.