Welcome back, class! In Chapter 8, we are shifting gears from discrete variables to Continuous Probability Distributions. This is a pivotal chapter in statistics because it introduces the Normal Distribution (the "Bell Curve"), which is the foundation for much of the inferential statistics we will do later in the course.
8.1 The Uniform Distribution
We start with the simplest continuous distribution: the Uniform Distribution. Imagine a situation where every outcome in a range is equally likely. The graph looks like a rectangle, which makes calculating probabilities straightforward—it's just calculating the area of a rectangle ($Area = base \times height$).
Key Formulas:
- The Probability Density Function: $$f(x) = \frac{1}{b-a}$$ for $$a \leq x \leq b$$
- Mean: $$\mu = \frac{a+b}{2}$$
- Standard Deviation: $$\sigma = \frac{b-a}{\sqrt{12}}$$
Example from class: We looked at Polar Bear Frozen Foods. If the weight of French fries is uniformly distributed between 57 and 63 ounces, the mean is simply 60 ounces. To find the probability of a package weighing less than 59 ounces, we calculate the area of the rectangle up to 59.
8.2 & 8.3 The Normal and Standard Normal Distributions
Next, we move to the star of the show: the Normal Distribution. This symmetric, bell-shaped curve appears everywhere in nature and industry. To compare different normal distributions (like test scores vs. heights), we standardize them using Z-scores.
The Standard Normal Distribution always has a mean ($ \mu $) of 0 and a standard deviation ($ \sigma $) of 1.
The Z-Score Formula:
$$z = \frac{x - \mu}{\sigma}$$This formula tells us how many standard deviations a value $x$ is away from the mean. Positive Z-scores are above the mean; negative ones are below.
8.4 Applications of the Normal Distribution
We solve two main types of problems in this section:
- Finding Probability: Given a value $x$, we convert it to a Z-score and use the standard normal table (or technology) to find the area (probability) under the curve.
- Finding Percentiles (Inverse Normal): Given a probability or percentage (e.g., "the top 10%"), we find the corresponding Z-score and work backward to solve for $x$.
Class Example: We calculated the 90th percentile for body temperatures. With $\mu = 98.6$ and $\sigma = 0.73$, we found the Z-score for 0.9000 area ($z \approx 1.28$) and solved for $x$, resulting in roughly $99.53^\circ F$.
8.6 Normal Approximation to the Binomial
Finally, we learned a powerful shortcut. When calculating Binomial probabilities with a large number of trials ($n$), the formulas can get tedious. Fortunately, as long as $np \geq 5$ and $nq \geq 5$, the Binomial distribution looks very much like a Normal distribution.
Steps to Approximate:
- Find the Mean: $$\mu = np$$
- Find the Standard Deviation: $$\sigma = \sqrt{np(1-p)}$$
- Apply Continuity Correction: Since we are going from discrete (bars) to continuous (smooth curve), we must adjust our $x$ values by 0.5.
For example, to find the probability of exactly 5 successes ($P(x=5)$), we find the area between 4.5 and 5.5 under the normal curve.
Keep practicing those Z-score conversions and sketch your curves! Visualizing the shaded area is the best way to ensure your answer makes sense. See you in the next lecture!