01:00
Topic 4 - The Normal Distribution and Sampling Variability
Let’s start with an example
In a study on driving reaction time, researchers record the time (in seconds) that it takes study participants to react to the brake lights of a vehicle traveling in front of them.
A histogram of the recorded data is shown below.
Describe the shape, center, and spread of the distribution.
Normal Distribution
- When to use:
Useful when modeling a continuous random variable that has a bell-shaped distribution.
- Parameters of the distribution:
\(\mu\): mean - determines the center of the distribution
\(\sigma\): standard deviation - determines the spread of the distribution
- Probability Density Function:
\[f(x) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] for \(x\) in \((-\infty, \infty)\)
Expectation: \(E(X) = \mu\)
Variance:\(Var(X) = \sigma^2\)
Example: Driving Reaction Time
Standard Normal Distribution
The Standard Normal Distribution is a Normal distribution with mean \(\mu=0\) and standard deviation \(\sigma=1\).
\[Z \sim N(0, 1)\]
Standardizing Any Normal Random Variable
For a Normal random variable, \(X\), a z-score represents the number of standard deviations any observation \(x\) is from the mean.
\[z = \frac{x-\mu}{\sigma}\]
Example: Driving Reaction Time
The driving reaction time study suggests that reaction times to break lights are Normally distributed with a mean of 1.25 seconds and standard deviation of 0.46.
What is the probability of a randomly selecting a driver with a reaction time less than 1 second?
\[P(X < 1) = \int \limits_{-\infty}^{1} \frac{1}{\sqrt{2\pi (0.46^2)}}e^{-\frac{(x-1.25)^2}{2(0.46^2)}} dx\]
We cannot solve this analytically - we must use R!
Sampling Distributions Simulation
Please do the following:
Answer the one question in the google form that can be accessed in any of the following ways:
Typing the following URL into your browser. The URL is case sensitive. https://beav.es/W4_survey
Find the Week 4 Survey link on Canvas in the Quick Links module
Scan the QR code
Example
As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.
The information above describes the outcome of a single sample. Suppose the true proportion of defective chips at this factory is \(p=0.1\).
Participation Question 📊
The value \(p=0.1\) is the
a. population parameter
b. sample statistic
Answer the question at
PollEv.com/erinhowardstats
Sampling Variability
Even when robust sampling schemes are used, different samples will yield different point estimates.
Population
Sample 1 
Sample 2 
Sample 3 
\(\hat{\theta}_1\) \(\hat{\theta}_2\) \(\hat{\theta}_3\) \(\hat{\theta}\) represents a generic point estimate.
Distributions of Inference
Population Distribution
Distribution of the entire collection of interest.
SamplED Distribution
Distribution of \(n\) observations obtained from a single sample.
SamplING Distribution
Distribution of a sample statistic, such as \(\overline{x}\) or \(\hat{p}\), from repeated samples of size \(n\) from the population.
Example
As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.
The information above describes the outcome of a single sample. Suppose the true proportion of defective chips at this factory is \(p=0.1\).
Participation Question 📊
Suppose we were able to collect every possible sample of size 212 from the population of chips produced.
The distribution of the sample proportions of defective chips is the
a. population distribution.
b. sampled distribution.
c. sampling distribution.
Answer the question at
PollEv.com/erinhowardstats
Sample Size & Sampling Variability
The variability of the point estimate is called the standard error.
The standard error is the standard deviation of the sampling distribution.
As \(n\) increases, the standard error of the point estimate decreases.
Central Limit Theorem
Sampling Distribution of Sample Proportion, \(\hat{p}\)
Central Limit Theorem
Sampling Distribution of Sample Mean, \(\overline{x}\)
Example
As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.
The information above describes the outcome of a single sample. Suppose the true proportion of defective chips at this factory is \(p=0.1\).
Participation Question 📊
Suppose we were able to collect every possible sample of size 212 from the population of chips produced.
The distribution of the sample proportion of defective chips is
Answer the question at
PollEv.com/erinhowardstats
Example
As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.
The information above describes the outcome of a single sample. Suppose the true proportion of defective chips at this factory is \(p=0.1\).
Participation Question 📊
In the previous participation question, we stated that the distribution of \(\hat{p}\) was Normal, but we didn’t check the sample size conditions before doing so! Are the sample size conditions met in this scenario in order to apply the central limit theorem?
Answer the question at
PollEv.com/erinhowardstats