Statistics Review: Probability, Binomial Distributions, Normal Distributions, and Bayes' Rule

Review of Key Statistics Topics

Finding Z-scores (Section 7.2)

Z-scores are standardized values that indicate how many standard deviations a data point is from the mean of a normal distribution. They are essential for calculating probabilities and percentiles in normally distributed data.

• Definition: The z-score of a value $x$ is given by $z = \frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

• Application: Z-scores allow us to use the standard normal table to find probabilities for any normal distribution.

• Example: If the mean incubation time for eggs is 21 days with a standard deviation of 1 day, the z-score for an egg that hatches in 19 days is $z = \frac{19 - 21}{1} = -2$.

Finding Percentiles (Section 7.2)

Percentiles indicate the relative standing of a value within a data set. In a normal distribution, percentiles can be found using z-scores and the standard normal table.

• Definition: The p-th percentile is the value below which p% of the data fall.

• Finding a Percentile: To find the value corresponding to a given percentile, use the z-score from the standard normal table and solve for $x$ in $x = \mu + z\sigma$.

• Example: To find the 17th percentile for egg incubation times (mean = 21, SD = 1), find the z-score for 0.17 (approximately -0.95), then $x = 21 + (-0.95)(1) = 20.05$ days.

Normal Approximation to the Binomial (Section 7.4)

When the sample size is large, the binomial distribution can be approximated by the normal distribution, making calculations easier.

• Conditions: The approximation is appropriate when $np \geq 10$ and $n(1-p) \geq 10$.

• Continuity Correction: When using the normal approximation, add or subtract 0.5 to the discrete x-value (continuity correction).

• Formula: For a binomial variable $X$ with parameters $n$ and $p$, $X \sim N(np, \sqrt{np(1-p)})$ approximately.

• Example: In a survey of 500 students where 31% are expected to lie, $np = 155$, $n(1-p) = 345$; both are greater than 10, so normal approximation is valid.

Binomial Random Variable (Section 6.2)

The binomial random variable counts the number of successes in a fixed number of independent trials, each with the same probability of success.

• Definition: A binomial experiment has n independent trials, each with probability p of success.

• Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean and Standard Deviation: $\mu = np$, $\sigma = \sqrt{np(1-p)}$

• Example: For 15 flights with 80% on-time rate, $n = 15$, $p = 0.8$.

Sample Table: Binomial Probabilities

Number of On-Time Flights (k)	Probability $P(X = k)$
10	Calculated using $P(X=10) = \binom{15}{10} (0.8)^{10} (0.2)^5$
Fewer than 10	Sum $P(X=k)$ for $k=0$ to $9$
At least 10	Sum $P(X=k)$ for $k=10$ to $15$
Between 8 and 10	Sum $P(X=8)$, $P(X=9)$, $P(X=10)$

Bayes' Rule (Section 5.8)

Bayes' Rule allows us to update probabilities based on new evidence. It is especially useful in diagnostic testing and decision-making under uncertainty.

• Formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

• Application: Used to find the probability that a person is innocent or guilty given a positive or negative test result.

• Example: If 95% of suspects are guilty, and the polygraph is 90% accurate for guilty and 99% accurate for innocent, Bayes' Rule can be used to find the probability a suspect is innocent given a positive test.

Additional info: These topics are foundational for understanding probability, distributions, and statistical inference in college-level statistics courses. Mastery of these concepts is essential for interpreting data and making informed decisions based on statistical evidence.