1. Inference: Conclusion with Confidence
Strand IV
Inference: Conclusion with Confidence
Statistical Inference: drawing conclusions about a population based upon the data collected from a sample of that population.
To determine if our conclusion are correct/reasonable, we can use the concepts of probabilities to express the strength of our conclusions.
In chapter 10, we will discuss confidence intervals which will allow us to estimate the value of a population parameter.

2. Section 10.1: Confidence Intervals
A range of values in which the population value is likely to occur
Usually given in the format:
Estimate ( 𝑥 𝑜𝑟 𝑝 ) ± margin of error
Confidence Level: the probability that the interval will
capture the true parameter value in repeated samples (the
probability that our estimation method is successful)

3. Example: The Sampling Distribution of the mean score ( ) of an SRS of 50 Big City University freshmen on an IQ test
If we wanted to be 95% confident that we could estimate the mean IQ score for the population, we would use a margin of error equal to approximately 2 standard deviations from the estimated mean.

4. Conditions for Constructing a Confidence Interval for m
These conditions MUST be checked and validated first…
SRS: the data came from a simple random sample of the population of interest
Normality: the sampling distribution of is approximately normal (Use CLT to prove this )
Independence: individual observations are independent when sampling without replacement and
population ≥ 10 * sample size
This also assumes that you know the value of s.

5. What would the critical values for 90%, 95%, and 99% be?
Investigation:
Let’s say that you want to create a confidence interval with an 80% confidence level for a SRS that is approximately normal and independent. Using z-scores can help us to find a confidence interval about the unknown mean (m). What would those z-score values be?
z-score values that are used to create cutoff values for a confidence interval are called critical values (the positive value is represented as z*)
What would the critical values for 90%, 95%, and 99% be?

6. The critical values are the number of standard deviations from the mean that will give you the confidence level you want. Since we know the 𝜎 for the population we can now find the confidence interval using a calculated sample mean ( ).
IQ score example:
For, 𝜇 𝑥 =112 𝑎𝑛𝑑 𝜎=15 , what would be the confidence interval at an 80% confident level?

7. Interpretations of Confidence Intervals and Confidence Levels
This means that we are 80% confident that the TRUE mean IQ score of ALL freshmen at Big City University falls
between and
Confidence Level:
Being “80% confident” means that 80% of the time, repeated samples of the same size will produce an interval that contains the TRUE mean.

8. Make the s value smaller Make the n (sample size) larger
Often a high confidence level (95 or 99%), means that your interval must be very large (high margin of error). Ultimately, we would like to create a confidence interval with a high confidence level and very small margin of error.
How can we control that???
Make the z* value smaller that causes you to accept a lower confidence level.
Make the s value smaller
this does make it easier to get a more accurate m, but is difficult to do
Make the n (sample size) larger
dividing by a larger number makes
smaller and in turn the margin of error smaller
Best Option!

9. So how do we find the desired sample size?
Let m = the desired margin of error
So,
Now solve for n.
Pg 640 #20b
The sample size must be at least 2653 people