I will post R/RStudio files on Posit Cloud that you can access in provided links.
I will also provide demo videos that show how to access files and complete exercises.
NOTE: The free Posit Cloud account is limited to 25 hours per month.
I will demo how to download completed work so that you can use this allotment efficiently.
For those who want to go further with R/RStudio:
After Test 1, I will provide videos on how to download the software (R/RStudio/Quarto) and lecture files to your computer.
💥Lecture 7 In-class Exercises - Q1 💥
Session ID: MAS261f24
The mean number of customers at a local cafe on a Monday morning is 39 with a standard deviation of 3
What is the percent chance that they will have 45 or more customers next Monday morning?
Use the vdist_norm_prob command to help you answer this question.
Review of Histograms of Different Distributions
Histograms are an effective tool for examining the distribution of the data.
LEFT SKEWED
Tail pulled out to LEFT
Low outliers
e.g. Human Lifespan
NORMAL/SYMMETRIC
Data appear in a symmetric bell-shaped curve
No graphic evidence of outliers
e.g. Test scores
RIGHT SKEWED
Tail pulled out to RIGHT
High outliers
e.g. Movie Gross values
Hypothetical Histogram
Most of the data falls in the middle intervals
Distribution is symmetric, and bell-shaped with no outliers.
Histogram overlayed with Density Curve
Recall the sum of the proportion of data in each interval equals 1
Area under the curve ALSO sums to 1
Normal Density Curve
We “smooth out” the histogram to a curve.
Area under the curve equals 1
We use this distribution to find the probability (percent chance) that a certain data value occurs.
Normal Distribution
In lecture 6 we talked about the normal distribution
It is symmetric and bell-shaped.
It’s location is determined by the population mean, \(\mu\)
It’s width is determined by the population standard devation, \(\sigma\)
Regardless of the values of \(\mu\) and \(\sigma\), the normal distribution has a consistent shape
That shape is well known and provides information about all normally distributed populations.
Also recall that all normally distributed populations can be converted to the standard normal distribution \(Z\)
Z is normally distributed with mean, \(\mu\) of 0 and SD, \(\sigma\) of 1.
If X is from a normal population, \(Z = \frac{X-\mu}{\sigma}\)
Normal Distribution - Empirical Rule
Part 1: 68% of all Normal populations fall within one standard deviation of their mean (illustrated using Z distribution).
Normal Distribution - Empirical Rule
Part 2: 95% of all Normal populations fall within two standard deviations of their mean (illustrated using Z distribution).
Normal Distribution - Empirical Rule
Part 3: 99.7% of all Normal population falls within three standard deviations of their mean (illustrated using Z distribution).
True probability is 0.9973, so I think this package rounds up.
Normal Distribution - Empirical Rule
Also Referred to as the 68-95-99.7 Rule
Summarizing the Emperical Rule in Words
68% of all values are within 1 std. dev, \(\sigma\), of the pop. mean, \(\mu\)
95% of all values are within 2 std. dev, \(2\times \sigma\), of the pop. mean, \(\mu\)
99.7% of all values are within 3 std. dev, \(3\times \sigma\), of the pop. mean, \(\mu\)
How is the 68-95-99.7 Rule Useful?
R, Excel, Other software, Normal Tables, Apps for phone or PC, etc. can ALL be used find probabilities from a normal distribution.
BUT
Internalizing the Empirical Rule allows you to understand the probability of seeing observed data intuitively WITHOUT using a computer or phone.
Learning these rules and how to use them allows you to immediately evaluate data to determine
Is the observation reasonable
Is it unlikely but not too surprizing
Is it so unlikely that it may be due to an error in data collection or
Is it so unlikely it might cause us to reevaluate are assumptions about the population distribution.
Example: Trading on the NYSE
Historic data indicates that the first 30 minutes of New York Stock Exchange (NYSE) trading volume (millions of shares) is normally distributed with
a mean of 200 million shares, \(\mu = 200\)
a standard deviation of 26 million shares, \(\sigma = 26\).
Answering Questions using the Empirical Rule
Use the Empirical Rule to find the probability that the trading volume will be in the range of 174 to 226 million shares.
A good approach is to convert range endpoints to Z-scores:
(174-200)/26# -1 means 1 sd below the mean
[1] -1
(226-200)/26# 1 means 1 sd above the mean
[1] 1
174 to 226 is (\(\mu \pm \sigma\)) mean +/- 1 SD
Recall the Rule:
68% of population within \(\mu \pm \sigma\)
95% of population within \(\mu \pm 2\sigma\)
99.7% of population within \(\mu \pm 3\sigma\)
Probability that trading will be between 174 and 226 is 68%
💥Lecture 7 In-class Exercises - Q2 💥
Session ID: MAS261f24
Use the Empirical Rule to find the probability that the NYSE morning trading volume will be in the range of 148 to 200 million shares.
Convert endpoints to Z scores
Hint: Normal distribution is SYMETRIC
💥Lecture 7 In-class Exercises - Q3 💥
Session ID: MAS261f24
Use the Empirical Rule to find the probability that the NYSE morning trading volume will be in the range of 200 to 278 million shares.
Convert endpoints to Z scores
Hint: Normal distribution is SYMETRIC
Interpreting a Z score using the Empirical Rule
If Z is between -1 and 1, observed value is VERY LIKELY.
If Z is between -1 and -2 or between 1 and 2, observed value is NOT AT ALL UNLIKELY, BUT MAY NOT BE TOO COMMON (especially as Z gets closer to 2 or -2).
If Z value is between -2 and -3 or between 2 and 3, observed value is UNLIKELY, BUT NOT TOO SUPRISING (until Z gets closer to 3 or -3).
If Z value is less than -3 or greater than 3, observed value is EXTREMELY UNLIKELY and could be due to error if \(\vert{Z}\vert\) is very large.
💥Lecture 7 In-class Exercises - Q4 💥
Session ID: MAS261f24
If trading in the first half hour is at 250 million shares, how should we interpret that?
Step 1. Convert 250 to Z score
Step 2. Use the guidance on the previous slide (based on the Empirical Rule) to interpret that Z score.
Finding X (observed value) from a Percentile
Sometimes what we want to know is what value would put us in
the top 10%
the bottom 5%
etc.
For example, how high would trading have to be to put it in the top 5% for sales
To answer this question we use a similar command to one we already know,
vdist_normal_perc
perc stands for percentile.
We (the user) specify the percentile and the output shows the value needed to achieve that percentile.
Finding X from a Percentile - NYSE
How high would trading have to be to put it in the top 5%?
