Find probabilities using the normal distribution

Learning Objectives Describe the key properties of a normal distribution curve, including its symmetry and the role of the mean and standard deviation. Convert any raw data point (x-value) from a normal distribution into a standardized z-score. Use a standard normal distribution table (z-table) or a calculator to find the cumulative probability associated with a given z-score. Calculate the probability that a randomly selected value from a normal distribution falls below a specific value, P(X < a). Calculate the probability that a randomly selected value falls above a specific value, P(X > b), using the complement rule. Calculate the probability that a randomly selected value falls between two specific values, P(a < X < b). Apply the normal distribution to solve...

TermDefinitionExample Normal DistributionA continuous probability distribution for a random variable, represented by a symmetric, bell-shaped curve. The total area under the curve is equal to 1 (or 100%).The distribution of adult human heights in a large population. Most people are of average height (the peak of the bell), with fewer people being extremely tall or extremely short. Mean (μ)The average of the data set, which is located at the exact center of a normal distribution curve. It defines the line of symmetry.If the mean score on a test is 80, the peak of the bell curve for the test scores will be at 80. Standard Deviation (σ)A measure of how spread out the data points are from the mean. A smaller standard deviation results in a taller, narrower curve, while a larger one results in...

The Z-score Formula z = \frac{x - \mu}{\sigma} Use this formula to standardize any data point 'x' from a normal distribution with mean 'μ' and standard deviation 'σ'. This converts your specific distribution into the standard normal distribution (μ=0, σ=1), allowing you to use a z-table. Probability for 'Less Than' P(X < x) = P(Z < z) The probability of a value being less than 'x' is found by converting 'x' to a z-score and looking up the corresponding cumulative area in a standard normal (z) table. The table directly gives you this value. Probability for 'Greater Than' (Complement Rule) P(X > x) = 1 - P(X < x) The probability of a value being greater than 'x' is 1 minus the p...