STATS

Understanding Z-Scores

  • A z-score represents how many standard deviations a value is from the mean.

  • Example z-score given: 0.22 (not 22%).

  • To convert a z-score to risk probability, use a probability calculator.

Using the Probability Calculator

  • Begin by finding z-scores for the queries (a, b, c).

  • Part A:

    • Z-score calculated: 0.22.

    • Direction: Right tail (greater than).

    • Result: Probability is 0.4129 (or 41.29%).

  • Part B:

    • Z-score calculated: -0.5.

    • Direction: Left tail (less than).

    • Result: Probability is 0.3085.

  • Part C:

    • Z-score calculated: -2.5.

    • Direction: Continuing with Left tail calculation.

Probability Calculations

  • Utilize a z-score calculator to find probabilities for values above or below a specific number.

  • Example: 65% chance of x being above a particular value leads to a z-score of -0.3853.

Sample Means and Standard Error

  • Standard Error (SE) is defined as the standard deviation divided by the square root of the sample size (n).

  • For n=25, if the population standard deviation (σ) is 10:

    • SE = σ / √n = 10 / √25 = 2.

  • Key formula for z-scores concerning sample means:

    • Z = (x̄ - μ) / (σ / √n)

    • Where x̄ is the sample mean, μ is the population mean, and n is the sample size.

Importance of Sample Size/n

  • Increasing sample size (n) generally results in more accurate estimates, squeezing the sampling distribution towards the mean of the population.

  • For larger samples, extreme values (outliers) are less likely to skew the results due to averaging out effects.

Calculation Steps for Sample Probabilities

  • For a sample size of 25, to find the probability that a sample mean is above or below certain values, contextually shift from individual to sample statistics.

  • Sample mean probabilities are derived through similar calculations as individual observations.

  • Always ensure calculations and context align (population mean vs. sample mean).

Exam Preparation

  • Review methods for calculating z scores, probability values, and differences between population vs sampling distributions.

  • Focus on correctly understanding sample sizes and continuous distributions during practice problems for greater proficiency.