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Chapter 6

Chapter 6

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Department
Statistics
Course Code
STAB22H3
Professor
Moras

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Chapter 6
Statistical inference draws conclusions about a population or process based on sample data
Two types of statistical Inferences:
-Confidence interval
-Tests of significance
This chapter will only consider scenarios of inference about the mean with the standard deviation
given.
In this setting we can ask questions like:
-What is the average loan debt among undergraduate borrowers?
-What is the average mile per gallon (mpg) for a hybrid car?
- Is the average cholesterol level of undergraduate women at your university below the
national average?
Overview of inference
Purpose of statistical inference is to draw conclusion from data.
Probability allows us to take chance variation into account
Ex 6.2 Effectiveness of a new drug
20 were given the drug 60% had improvement
20 were given the placebo 40% had improvement
Due to probability calculations a difference this large or larger between the results in the two
group would occur one time in five simply because of chance variation.
Confidence interval – used for estimating the value of a population parameter
Tests of significance – assess the evidence for a claim
Both types of inferences are based on sampling distributions of statistics
Probability models are most secure, and inference is most reliable when the data are produced by
a properly randomized design.
Unrealistic assumption: is when the standard deviation ( ) is given.
6.1 Estimating with Confidence
Sample mean (x-bar) can be used for mean (µ) of population since it is unbiased
But each sample mean (x-bar) can differ from sample to sample
An estimate without an indication of its variability is of little value.
Statistical confidence
n = 500
= 100
www.notesolution.com
x = 100/ square root 500 = 4.5
Consider:
-The 68-95-99.7 rule says that the probability is about 0.95 that x-bar will be within 0
points (2 of the population mean score µ.
-To say that x-bar lies within 9 points of µ is the same as saying that µis within 9 points of
x-bar
-So 95% of all samples will capture the true µ in the interval from x-bar – 9 to x-bar + 9
The language of statistical inference uses this fact about what would happen in the long run to
express our confidence in the results of any one sample.
X-bar = 461
95% confidence lies between: 452 and 470
X-bar – 9 = 461 – 9 = 452 and X-bar + 9 = 461 + 9 = 470
2 possibilities of our SRS:
1. The interval between 452 and 470 contains the true µ.
2. The interval between 452 and 470 does not contain the true µ.
95% confidence means that the method used gives correct result 95% of the time.
Confidence Intervals
The interval of # between the values X-bar+9 is called the 95% confidence interval for µ.
X-bar+9 = Estimate + margin of error
Margin of error reflects how accurate we believe our guess is based on the variability of the
estimate, and how confident we are that the procedure will catch the true population mean µ.
2 important things about confident intervals
1. It is an interval of the form (a,b), where a and b are numbers computed from the data.
2. It has a property called a confidence level that gives the probability of producing an
interval that contains the unknown parameter.
Occasionally, 90% and 99% is used, but the 95% rule is mostly used.
C stands for confidence level
A confidence of 95% C= 0.95
Confidence Interval
A level C confidence interval for a parameter is an interval calculated from a sample data be a
method that has a probability C of producing an interval containing the true value of the
parameter.
www.notesolution.com
Ex 6.4 80% confidence intervals
Confidence level tells us what percent will capture µ in the long run.
As number of samples increases, the percent of captures gets closer to the confidence level, 80%.
Confidence interval for a population mean
Sampling distribution of the sample mean x-bar is exactly N(µ, /square root n)
Any normal distribution has probability about 0.95 within + 2 x-bar
C = confidence interval
Z* = # + the µ value
Normal distribution has probability c within + z
Ex: C = 90%, z* = 1.645
C = 95%, z* = 1.96 most important entries
C = 99%, z* = 2.576
Z* values are found at the bottom of table D
For the probability C that X-bar lies between
µ - z* ( /square root n) and µ + z* ( /square root n) 
same as saying that the unknown population mean µ lies between
x-bar – z*( /square root n) and x-bar + z*( /square root n) 
In other words there is a probability C that the interval x-bar = z*/square root n , contains µ.
The estimate of the unknown µ is x-bar, and the margin of error is z*/square root n
Confidence interval for a population mean
Choose an SRS of a size of population having unknown mean µ and known standard deviation .
The margin of error for a level C confidence interval for µ is
m = z* ( /square root n)
Here z* is the value on the standard Normal curve with area C between the critical points –z*
and z*. The level C confidence Interval for µ is
x-bar + m
This interval is exact when the population distribution is Normal and is approximately correct
when n is large in other cases.
Ex 6.4 Average debt of undergraduate borrowers
Given
n = 1280
x-bar = $18900, = $49000
C = 0.95 use table d to find z*
www.notesolution.com

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Description
Chapter 6 Statistical inference draws conclusions about a population or process based on sample data Two types of statistical Inferences: - Confidence interval - Tests of significance This chapter will only consider scenarios of inference about the mean with the standard deviation given. In this setting we can ask questions like: - What is the average loan debt among undergraduate borrowers? - What is the average mile per gallon (mpg) for a hybrid car? - Is the average cholesterol level of undergraduate women at your university below the national average? Overview of inference Purpose of statistical inference is to draw conclusion from data. Probability allows us to take chance variation into account Ex 6.2 Effectiveness of a new drug 20 were given the drug 60% had improvement 20 were given the placebo 40% had improvement Due to probability calculations a difference this large or larger between the results in the two group would occur one time in five simply because of chance variation. Confidence interval used for estimating the value of a population parameter Tests of significance assess the evidence for a claim Both types of inferences are based on sampling distributions of statistics Probability models are most secure, and inference is most reliable when the data are produced by a properly randomized design. Unrealistic assumption: is when the standard deviation () is given. 6.1 Estimating with Confidence Sample mean (x-bar) can be used for mean () of population since it is unbiased But each sample mean (x-bar) can differ from sample to sample An estimate without an indication of its variability is of little value. Statistical confidence n = 500 = 100 www.notesolution.comx = 100 square root 500 = 4.5 Consider: - The 68-95-99.7 rule says that the probability is about 0.95 that x-bar will be within 0 points (2 of the population mean score . - To say that x-bar lies within 9 points of is the same as saying that is within 9 points of x-bar - So 95% of all samples will capture the true in the interval from x-bar 9 to x-bar + 9 The language of statistical inference uses this fact about what would happen in the long run to express our confidence in the results of any one sample. X-bar = 461 95% confidence lies between: 452 and 470 X-bar 9 = 461 9 = 452 and X-bar + 9 = 461 + 9 = 470 2 possibilities of our SRS: 1. The interval between 452 and 470 contains the true . 2. The interval between 452 and 470 does not contain the true . 95% confidence means that the method used gives correct result 95% of the time. Confidence Intervals The interval of # between the values X-bar+9 is called the 95% confidence interval for . X-bar+9 = Estimate + margin of error Margin of error reflects how accurate we believe our guess is based on the variability of the estimate, and how confident we are that the procedure will catch the true population mean . 2 important things about confident intervals 1. It is an interval of the form (a,b), where a and b are numbers computed from the data. 2. It has a property called a confidence level that gives the probability of producing an interval that contains the unknown parameter. Occasionally, 90% and 99% is used, but the 95% rule is mostly used. C stands for confidence level A confidence of 95% C= 0.95 Confidence Interval A level C confidence interval for a parameter is an interval calculated from a sample data be a method that has a probability C of producing an interval containing the true value of the parameter. www.notesolution.com
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