Inference about a Population Mean

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Chapter 17

• Inference about a Population Mean

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s not known
In practice, we do not usually know population
standard deviation s Therefore, we cannot
calculate sx-bar Instead, we calculate this
standard error of the mean
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t Procedures
Because s is now known, we do NOT use z
statistics. Instead, we use this t statistic
T procedures are based on Students t
distribution
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Students t Distributions

• A family of distributions
• Each family member has different degrees of
  freedom (df)
• More area in their tails than Normal
  distributions (fatter tails)
• As df increases, s becomes a better estimate of s
  and the t distributions becomes more Normal
• t with more than 30 df ? very similar to z

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t Distributions
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Table C t Table
Table entries t critical values Rows df
Columns probability levels
Familiarize yourself with the t table in the
Tables and Formulas for Moore handout
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Using Table C
Question What t critical value should I use for
95 confidence when df 7?
Answer t 2.365
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Confidence Interval for µ
t is the critical value with df n-1 and C
level of confidence Lookup in Table C
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Example
Statement What is the population mean µ birth
weight of the SIDS population?
Data We take an SRS of n 10 from the
population of SIDS babies and retrieve their
birth certificates. This was their birth weights
(grams) 2998, 3740, 2031, 2804, 2454, 2780,
2203, 3803, 3948, 2144
Plan We will calculate the sample mean and
standard deviation. We will then calculate and
interpret the 95 CI for µ.
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Example (Solution)
We are 95 confident population mean µ is between
2375 and 3406 gms.
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One-Sample t Test (Hypotheses)

• Draw simple random sample of size n from a large
  population having unknown mean µ
• Test null hypothesis H0 µ µ0 where µ0
  stated value for the population mean
• µ0 changes from problem to problem
• µ0 is NOT based on the data
• µ0 IS based on the research question
• The alternative hypothesis is
• Ha µ gt µ0 (one-sided looking for a larger value)
  OR
• Ha µ lt µ0 (one-sided looking for a smaller
  value) OR
• Ha µ ? µ0 (two-sided)

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One-Sample t Test
One-sample t statistic
P-value tail beyond tstat (use Table C)
2/19/2012
Inference about µ
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P-value Interpretation

• P-value (interpretation) Smaller-and-smaller
  P-values indicate stronger-and-stronger evidence
  against H0
• Conventions
• .10 lt P lt 1.0 ? evidence against H0 not
  significant
• .05 lt P .10 ? evidence against H0 marginally
  signif.
• .01 lt P .05 ? evidence against H0 significant
• P .01 ? evidence against H0 highly significant

Basics of Significance Testing
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Example Weight Gain
Statement We want to know whether there is good
evidence for weight change in a particular
population. We take an SRS on n 10 from this
population and find the following changes in
weight (lbs).
2.0, 0.4, 0.7, 2.0, -0.4, 2.2, -1.3, 1.2, 1.1, 2.3
Calculate
Do data provide significant evidence for a weight
change?
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Example Weight Gain (Hypotheses)

• Under null hypothesis, no weight gain in
  population H0 µ 0 Note µ0 0 in this
  particular example
• One-sided alternative, weight gain in population.
  Ha µ gt 0
• Two-sided alternative hypothesis, weight
  changeHa µ ? 0

2/19/2012
Inference about µ
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Example (Test Statistic)
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Example (P-value)

• Table C, row for 9 df
• t statistic (2.70) is between t 2.398 (P
  0.02) and t 2.821 (P 0.01)
• One-sided P-value is between .01 and .02 .01 lt
  P lt .02

2/19/2012
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Inference about µ
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Two-tailed P-value

• For two-sided Ha, P-value 2 one-sided P
• In our example, the one-tailed P-value was
  between .01 and .02
• Thus, the two-tailed P value is between .02 and
  .04

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Interpretation

• Interpret P-value in context of claim made by H0
• In our example, H0 µ 0 (no weight gain)
• Two-tailed P-value between .02 and .04
• Conclude significant evidence against H0

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Paired Samples
Responses in matched pairs
Parameter µ now represents the population mean
difference
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Example Matched Pairs

• Pollution levels in two regions (A B) on 8
  successive days
• Do regions differ significantly?
• Subtract B from A last column
• Analyze differences

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Example Matched Pairs
Hypotheses H0 µ 0 (note µ0 0,
representing no mean difference) Ha µ gt 0
(one-sided) Ha µ ? 0 (two-sided) Test Statistic

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Illustration (cont.)

• P-value
• Table C ? 7 df row
• t statistic is greater than largest value in
  table t 5.408 (upper p 0.0005).
• Thus, one-tailed P lt 0.0005
• Two-tailed P 2 one-tailed P-value P lt 0.001
  
• Conclude highly significant evidence against H0

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95 Confidence Interval for µ
Air pollution data n 8, x-bar 1.0113, s
0.1960 df 8 ? 1 7 For 95 confidence, use t
2.365 (Table C)
95 confidence population mean difference µ is
between 0.847 and 1.175
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Interpreting the Confidence Interval

• The confidence interval seeks population mean
  difference µ (IMPORTANT)
• Recall the meaning of confidence, i.e., the
  ability of the interval to capture µ upon
  repetition
• Recall from the prior chapter that the confidence
  interval can be used to address a null hypothesis

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Normality Assumption

• t procedures require Normality, but they are
  robust when n is large
• Sample size less than 15 Use t procedures if
  data are symmetric, have a single peak with no
  outliers. If data are highly skewed, avoid t.
• Sample size at least 15 Use t procedures except
  in the presence of strong skewness.
• Large samples Use t procedures even for skewed
  distributions when the sample is large (n 40)

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Can we use a t procedure?
Moderately sized dataset (n 20) w/strong skew.
t procedures cannot be trusted
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Word lengths in Shakespeares plays (n 1000)
The data has a strong positive skew but since the
sample is large, we can use t procedures.
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Can we use t?
The distribution has no clear violations of
Normality. Therefore, we trust the t procedure.
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