Central Limit Theorem
2 conditions:
1. ___________________________________________
2. ____________________________________________
What distribution will sample means have if these 2 conditions are met?_________________
Can you use CLT to find information relating to a single variable? yes / no
Can you use CLT to find information relating to a sample mean? yes / no
Will all sample means be the same? yes / no
Why? ___________________
When a population is normal, do you need to use CLT? yes / no
Using a normal distribution to find information
*What is the mean of your sample mean distribution? ___________
*What is the standard deviation of your sample mean distribution?______
*To find the probability of obtaining a sample mean, A, find a z-score to correlate with A using
the formula…
___________________
*Use the z-score table to find the probability related to that z-score.
Confidence Intervals
2 conditions
1. _______________________________________________________________
2. _______________________________________________________________
Equation:
Using a z*value
Using a t* value
_____________________
Math Symbol
______________________
Definition
z*
z-value related to α
t*
t-distribution value related to α and
degrees of freedom
confidence level
α
__
X
se(X) or SE(X)
df
How to find it
Estimated mean
Estimated standard error/Standard
Error
Degrees of freedom
*assume 95% confidence intervals for the next questions.
Interpretation sentence: (using blanks to fill in based on context)
Also…
____of the confidence intervals made this way would contain the true population mean.
Therefore…
The probability that your confidence interval does NOT contain the true population mean
is___
When do you use…
t* values
z* values
What do you need to know to find the value
in the t-table?
What do you need to know to find the value
in the z-table?
How could you make a confidence interval narrower or wider?
Narrower
Wider
Margin of error =
Hypothesis Testing
The null hypothesis represents
_________________________________
The alternative hypothesis represents __________________________________
The 2 options when dealing
with null and alternative
hypotheses
When do you choose
this option?
P-value < α
P-value > α
Sentence to interpret
Central Limit Theorem
The life expectancy in America is non - normal with an average of 72years and a standard deviation of 15
years.
O
What is the probability that a randomly selected American will live to be 80 years old.
O
What is the probability that you will have obtain a random sample of 45 people with a sample
mean of 98 years.
Confidence Intervals
A box of Lucky Charms promises an average 15 oz of cereal per box. You want to test this claim, so you
tested a sample of 40 boxes. The average amount of cereal in these 40 boxes was 13 oz with an
estimated s = 8.8oz.
O
Create a 95% confidence interval for your estimated average.
O
Do you think the company is holding up to their promise?
Interpret this Interval:
In trying to predict average body temperature of Americans, I got a 95% confidence interval of…
( 96.9 99.3)
_________________________________________________________________________
Hypothesis Testing
The average college student pays $20,335 per year in tuition. In order to see if Iowa State costs are less
than the national average, a random sample of 100 ISU students is taken
O
What is 𝐻𝑜
O
What is 𝐻𝐴
Your sample of 100 students pays an average of $18,856 and has a sample standard deviation of $8,750.
When you run a hypothesis test you obtain the following output…
t-stat
p-value ≠ t
P-value < t
P-value > t
-2.83
0.006
0.003
0.997
What can you conclude about ISU tuition costs?
Putting it all together
The Situation
You are DISH Network. You are trying to decide whether to offer a new sports channel. You are willing
to offer it if it seems to have a customer ranking of at least 7.5 (out of 10).
The Basics
What is the population?
What is your parameter?
What are you trying to prove?
What is your Null hypothesis and alternative hypothesis?
Let’s prove it!
Your sample yields an average of a 6.8 rating. Your sample standard deviation is 2.15.
Create a 95% confidence interval for the population parameter.
You perform a hypothesis test with 𝛼 = 0.05 on your hypothesis based on your sample and these are
the test results…
𝑋̅ = 6.8
t-stat = -3.64
p-value < t = 0.043 p-value > t = 0.957 p-value ≠ t = 0.086
Conclusion
Are you going to offer this new sports channel?
- Using confidence interval
-
Using your hypothesis test