Transcript Unit 6 – confidence intervals

CONFIDENCE INTERVALS
When we take a sample and calculate
the mean of a sample it will be used
to estimate the population parameter.
CONFIDENCE INTERVALS
In order to determine the number of points a new high
blood pressure medicine reduces a patient’s systolic
measurement . A sample of 6 patients were given the
medicine and their blood pressure was measured after
the prescribe period of time. The number of points the
blood pressure was reduce is listed below.
12
5
13
0
-1
Calculate the Mean
19
CONFIDENCE INTERVALS
The pharmaceutical company knows the results
of the high blood pressure medicine will
• React differently on another group of
patients.
• When they report the results, they want to
use an interval.
• They said we are 95% confidence the new
blood pressure medicine will reduce
patient’s blood pressure between 2 and 14
points.
MARGIN OF ERROR
The distance from the sample mean and
the lower and upper limits.
Mean = 8
Lower limit = 2
8–2=6
Upper limit = 14
14 – 8 = 6
Margin of Error = MOE = 6
HOW TO CALCULATE THE MARGIN OF ERROR
We know how to calculate the mean.
Today we will discuss how to calculate the
Margin of error.
LARGE SAMPLE CONFIDENCE INTERVALS FOR MEANS
LARGE SAMPLE CONFIDENCE INTERVALS FOR MEANS
LARGE SAMPLE CONFIDENCE INTERVALS FOR MEANS
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PAGE 319 51
A publisher wants to estimate the mean
length of time (in minutes) all adults
spend reading newspapers. To determine
this estimate, the publisher takes a
random sample of 15 people and obtains
the following results.
11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
From past studies, the publisher assumes
σ is 1.5 minutes and that the population of
times is normally distributed.
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11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
Calculate the mean
We will calculate a 99% Confidence Interval
Subtract 99% from 1
Divide by 2
Find the Z value from the Table 4.
What is the population standard deviation σ?
PAGE 319 51
11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
Sample Mean = 9.07 Z = 2.575
σ = 1.5
What is the sample size n?
Calculate the square root of n?
MOE = Z *σ/√n =
PAGE 319 51
11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
Sample Mean = 9.07 Z = 2.575
σ = 1.5 MOE = .997 ≈ 1.00
Lower Limit = Sample Mean – MOE
Upper Limit = Sample Mean + MOE
We are 99% confident that the mean length of
time an adult reads the paper is between …
CONFIDENCE INTERVAL
What happens if
• Sample size n < 30
• Population standard deviation is
unknown.
The Margin of Error formula must be
changed to
MOE = t*s/√n
T DISTRIBUTION
0.2
The t distribution was developed for small
samples number (n < 30). It has characteristic
similar to the normal distribution.
• Symmetric, Mean = Median = Mode
• Bell Shaped
• Extends forever to the left and the right.
0.15
0.1
P(x)
0.05
0
-∞
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
P(x)
9
∞
PAGE #11
CONFIDENCE INTERVAL – SMALL SAMPLE SIZE
PAGE 330 #14
CONFIDENCE INTERVAL