We specify the interval using the grouping operator, c(), e.g. c(14,20).
What is the percent chance that the average gross per movie in 2024 will be between 14 million and 20 million (mean (\(\mu\)) = 17.77 and sd (\(\sigma\))= 2.41)?
Probability (Percent Chance) outside of an Interval
What is the percent chance that the average gross per movie in 2024 will be more than 20 million or less than 14 million (mean (\(\mu\)) = 17.77 and sd (\(\sigma\)) = 2.41)?
HINTS:
Area under the whole normal curve is 1.
Percent chance that the new average gross is OUTSIDE of that interval is 100 - Percent Chance from Question 4.
Effects of Changes to the Mean (\(\mu\)) or SD (\(\sigma\))
The mean (\(\mu\)) specifies to location of the normal distribution on the number line.
The standard deviation (\(\sigma\)) specifies the width of the normal distribution.
On the next slide are three hypothetical alternatives showing
A change in mean (\(\mu\)) from $17.77 million to $14.00 million
A change in the mean (\(\mu\)) shifts the distribution’s location.
A change in standard deviation (\(\sigma\)) from $2.41 to $6.0
A larger standard deviation (\(\sigma\)) means the distribution is wider .
A change in standard deviation (\(\sigma\)) from $2.41 to $1.2
A smaller standard deviation (\(\sigma\)) means the distribution is more narrow.
Changes in Mean or SD Effect % Chance
Original Mean and Standard Deviation
Decrease in Mean
Increase in Standard Deviation
Decrease in Standard Deviation
💥Lecture 6 In-class Exercises - Q6 💥
Session ID: MAS261f24
How is the percent chance within an interval affected if the standard deviation increases or decreases
Use the previous code and experiment with different values for sd.
We specify the interval using the grouping operator, c(), e.g. c(14,20).
What is the percent chance that the average gross per movie in 2024 will be between 14 million and 20 million (mean (\(\mu\)) = 17.77 and sd = 4.7)?
Z score and Why It Is Important
In our data example, each observation is the average gross for single year.
For a single observation, X, e.g. the average gross for 2024, the Z score is calculated as follows
\(Z = \frac{X-\mu}{\sigma}\), observation minus mean divided by standard deviation.
If we know the Z score, we can also find X: \(X = (Z\times\sigma) + \mu\)
Converting our data to Z scores, converts the data to the Standard Normal Distribution
The Standard Normal Distribution has
a mean, \(\mu\), of 0
a standard deviation, \(\sigma\), of 1.
Probability (Percent Chance) questions like those covered today previously required converting all values to Z scores and using a table like this.
Calculations and Interpretations of Z-scores
Today we can bypass the Z-table BUT it is still helpful to know what the Z-score tells us.
Z indicates how many standard deviations an observed value is away from the mean (\(\mu\)).
If we know X then
\(Z = \frac{X-\mu}{\sigma}\) is how many standard deviations our X value is from the mean (\(\mu\)).
If we know the Z score, i.e. how many standard deviations a value is away from the mean (\(\mu\)), then
\(X = (Z\times\sigma) + \mu\) will tell us what the original data value is.
Examples of Z score calculations from Today
Recall that in the original data, the mean (\(\mu\)) = 17.77 and the standard deviation (\(\sigma\)) is 2.41
We examined the probability that X (next year’s average) is less than 15 is P(X < 15)
If we encounter a new average gross in a subsequent year of 12.95, that value is 2 standard deviations below the population mean \(\mu\).
In the next lecture we’ll learn that knowing Z tells up how likely an observed value is to occur.
💥Lecture 6 In-class Exercises - Q7 💥
Session ID: MAS261f24
Recall that in the UPDATED average movie gross data, the population mean (\(\mu\)) = 14 and the standard deviation (\(\sigma\)) is 2.41.
What would average gross return (X) be if we were 3 standard deviations ABOVE population mean (\(\mu\)) = 14?
In the next class we will talk about just how unlikely it is to observe a value far from the population mean.
Key Points from Today
Normal Distribution is symmetric and bell-shaped
Width is determined by the population standard deviation, \(\sigma\).
Location is determined by the population mean (\(\mu\)).
we can find the probability of seeing a new value or one farther from the mean (\(\mu\)).
we can also find the probability (percent chance) of X being in a range or interval.
Probabilities (Percent Chances) will change if the mean (\(\mu\)) or standard deviation \(\sigma\) changes based on new information about the population.
We can also convert our observed value X to a Z score and we can convert a Z score to and X value.
Z tells the number of standard deviations (\(\sigma\)) X is above or below the mean \(\mu\).
To submit an Engagement Question or Comment about material from Lecture 6: Submit it by midnight today (day of lecture).
