Estimating a Population Mean

1
SECTION 10.2

• Estimating a Population Mean

2
Whats the difference between what we did in
Section 10.1 and what we are beginning in Section
10.2?

• In reality, the standard deviation s of the
  population is unknown, so the procedures from
  last section are not useful. However, the
  understanding of the logic of the procedures will
  continue to be of use.
• In order to be more realistic, s is estimated
  from the data collected using s

3
Conditions for Inference about a Population Mean

1. The data is an SRS from the population
2. Observations from the population have a normal
   distribution with an unknown mean (?) and unknown
   standard deviation (s)
3. Independence is assumed for the individual
   observations when calculating a confidence
   interval. When we are sampling without
   replacement from a finite population, it is
   sufficient to verify that the population is at
   least 10 times the sample size.

4
CAUTION

• Be sure to check that the conditions for
  constructing a confidence interval for the
  population mean are satisfied before you perform
  any calculations.

5
ROBUSTNESS

• ROBUST Confidence levels do not change when
  certain assumptions are violated
• Fortunately for us, the t-procedures are robust
  in certain situations.
• Therefore . . .

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This is when we use the t-procedures

• Its more important for the data to be
• an SRS from a population than the population has
  a normal distribution
• If n is less than 15, the data must be normal to
  use t-procedures
• If n is at least 15, the t-procedures can be used
  except if there are outliers or strong skewness
• If n30, t-procedures can be used even in the
• presence of strong skewness, but outliers must
  still be examined
• Essentially, as long as there are no significant
  departures from Normality (especially outliers)
  then the t procedures still work quite well.

7
Standard Error

• In this setting, each sample is a part of a
  sampling distribution that is a normal
  distribution with a mean equal to the
  populations mean
• Since we do not know s, we will replace the
  standard deviation formula of with this
  formula
• This is called the standard error of the
  sample mean

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Degrees of Freedom

• Commonly listed as df
• Equal to n-1
• When a t-distribution has k degrees of freedom,
  we will write this as t(k)
• When the actual df does not appear in Table C,
  use the greatest df available that is less than
  your desired df
• This guarantees a wider confidence interval than
  needed to justify a given confidence level

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Density Curves for t Distributions

• Bell-shaped and symmetric
• Greater spread than a normal curve
• As degrees of freedom (or sample size) increases,
  the t density curves appear more like a normal
  curve

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Confidence Intervals

• t
• t is the upper (1-C)/2 critical value for the
  t(n-1) distribution
• We find t using the table or our calculator
• tinvT(area to left of t, df)
• We interpret these the same way we did in the
  last chapter.
• This interval is exactly correct when the
  population distribution is Normal and is
  approximately correct for large n in other cases.

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INFERENCE TOOLBOX (p 631)
DO YOU REMEMBER WHAT THE STEPS ARE???
Steps for constructing a CONFIDENCE INTERVAL

• 1PARAMETERIdentify the population of interest
  and the parameter you want to draw a conclusion
  about.
• 2CONDITIONSChoose the appropriate inference
  procedure. VERIFY conditions (SRS, Normality,
  Independence) before using it.
• 3CALCULATIONSIf the conditions are met, carry
  out the inference procedure.
• 4INTERPRETATIONInterpret your results in the
  context of the problem. CONCLUSION, CONNECTION,
  CONTEXT(meaning that our conclusion about the
  parameter connects to our work in part 3 and
  includes appropriate context)

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Example GOT MILK?
A milk processor monitors the number of bacteria
per milliliter in raw milk received for
processing. A random sample of 10 one-milliliter
specimens from milk supplied by one producer give
the following data 5370, 4890, 5100, 4500,
5260, 5150, 4900, 4760, 4700, 4870 Construct a
90 confidence interval.

• --We want to estimate ? the mean number of
  bacteria per milliliter in all of the milk from
  this supplier
• --Since we dont know s, we should construct a
  one-sample t interval for ?.
• We must be confident that the data are an SRS
  from the producers milk. We must learn how the
  sample was chosen to see if it can be regarded as
  an SRS (we are only told that it is a random
  sample).
• A boxplot and a Normal probability plot of the
  data show no outliers and no strong skewness.
  This gives us little reason to doubt the
  Normality of the population from which this
  sample was drawn. In practice, we would probably
  rely on the fact that past measurements of this
  type have been roughly Normal.
• Since these measurements came from a random
  sample of specimens, they should be independent
  (assuming that there were many, at least 100,
  one-milliliter specimens available at the milk
  processing facility).

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Example GOT MILK? Cont.

• --Entering these data into a calculator gives
• 4950 and s268.45. So a 90 confidence
  interval for the mean bacteria count per
  milliliter in this producers milk is
• --We can say that we are 90 confident that the
  actual mean number of bacteria per milliliter of
  milk from this supplier is between 4794.4 and
  5105.6 because we used a method that yields
  intervals such that 90 of all these intervals
  will capture the true mean desired.

df 10-1 9
14
Paired t Procedures

• Recall, matched pairs studies are a form of block
  design in which just two treatments are being
  compared
• Also, experiments are rarely done on randomly
  selected subjects. Random selection allows us to
  generalize results to a larger population, but
  random assignment of treatments to subjects
  allows us to compare treatments.
• Be careful to distinguish a matched pairs setting
  from a two-sample setting.
• The real key is independence.
• TREAT THE DIFFERENCES from a matched pairs study
  as a single sample.

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TECHNOLOGY

• As always, you will be allowed unrestricted use
  of your calculator on quizzes and tests (as well
  as the actual AP Exam). For this reason, ALWAYS
  be certain to write down the values of key
  numbers that are being used (means, standard
  deviations, degrees of freedom, significance
  levels, etc.) along with results of the
  calculator procedures in order to receive full
  credit.
• The calculator information is available in your
  book on pages 661-662.
• We are now using the T Interval instead of the Z
  Interval
• Plug in exactly what you are asked for