Transcript Confidence Intervals

VOCABULARY
Confidence Interval – a range of values
based on a sample that we can say the
population mean will fall with a given %
certainty. The correct mean will be
captured in that % of sample intervals.
NECESSARY INFORMATION
1. A sample whose size is n ≥ 30 (preferred for our zscore calculation).
2. The mean of the sample.
3. The standard deviation of the population.
4. The confidence percentage that we want to
calculate.
PROBLEM 1
The admissions director at Wheeler Magnet has a
novel idea. She wants to use the PSAT scores of
current students as a marketing tool for other Cobb
students. She has enough money to evaluate 50
students’ scores. The director picks a SRS (simple
random sample) of 50 students to evaluate their
scores. The mean PSAT score for the sample is 𝒙 =
156. What can the director say about the mean
score μ of the population with 95% confidence?
PROBLEM 1 (CONT’D)
We are going to think about how this problem would work if
we took many 50 student samples of students. In that
situation, we could expect through the Central Limit
Theorem that the distribution of the sample means would
approach a normal curve. From this theorem, we know
that:
1. The mean of the sampling distribution 𝒙 is the same as the
unknown mean of the entire population, μ.
2. The standard deviation of the sample would be equal to
𝝈
𝒏
PROBLEM 1 (CONT’D)
To find the standard deviation of the sample, we need to know the
standard deviation of PSAT scores for the population. For the
most recent test, it was about 33.
Step 1: Calculate the standard deviation of the sample using the
standard deviation that is given to us.
PROBLEM 1 (CONT’D)
Now we want to decide how accurate we want to be. Using the
Empirical rule of 68-95-99.7, we could estimate how accurate
our interval was in including the actual population mean based
on how many standard deviations we include.
Step 2: Use the confidence interval you want to determine a margin
of error. In this situation, let’s look for a 95% confidence interval.
PROBLEM 1 (CONT’D)
Step 3: Use your sample mean and your margin of error to write an
interval to your desired accuracy.
PROBLEM 1 (CONT’D)
What does this mean? There are two
possibilities.
1.The population mean falls within this
interval.
2.This was one of the 5% of samples for
which 𝒙 was not within about 2 standard
deviations of the mean.
PROBLEM 2
If we wanted an 80% confidence interval, what z-score would we
use?
PROBLEM 3
If in our original problem, we wanted to improve our confidence to
99.7% without changing our interval, what sample size would we
need?
HOMEWORK
Worksheet on Confidence
Intervals