Introduction to Statistics

1
Introduction to Statistics

• Lecture Notes
• March 3, 2009
• Dr. Sinn

2
Estimation Chapter 14

• Wish
• To estimate the population mean
• Process
• Calculate the Sample Mean
• Predict Population Mean
• Give Confidence Level

3
Defining Symbols
4
Graphically Pop. Vs. Sample
Estimated
Known
Population µ mean s std dev
Sample mean s std dev
5
Z- vs. T-Test Comparisons

• Theory
• If we know s (population std. dev.)
• Use Z-Test or Z-Interval
• Examples
• SAT ACT
• Incomes (IRS)
• If we dont know s
• Use T-Test or T-Interval
• Estimate s by using s
• Example Biology
• Medical Research (blood pressure, heart rate,
  etc)
• Practice
• Always use t-tests or t-intervals

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Assumptions

• We may only use z-tests and t-tests when the
  following assumptions are met
• Independence
• All measure variables are independent for each
  subject studied
• Can be a problem in educational research (Why?)
• Normality
• The distribution of each sample approximately
  bell-shaped
• Homogeneity of the Variances
• All standard deviations of the samples and
  population are approximately equal

7
Verifying the Assumptions

• Normality
• This is the most important assumption to check
• Use data analysis techniques from 1st part of the
  course
• Check for bell-shaped distributions
• Check for outliers
• Outliers badly skew results for small data sets
• Homogeneity
• Nearly impossible to check (ignored in basic
  stats)
• Independence
• Verify in the research methodolgy

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Verifying the Normality Assumption

• We must verify the sample data is normally
  distributed.
• For n 25, No Worries!
• Law of large numbers takes over
• We can assume normality
• If n lt 25, Must Perform 2 Checks
• Histogram
• Should see roughly normal (bell-shaped)
  distribution
• Box-and-Whisker Plot (outliers on)
• No outliers detected
• Other methods can be used
• Stem-plots, z-scores
• For this course, use Histogram and Box Plot

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Example 1

• Dan wants to determine the average distance he
  drives the ball with a new driver. He uses a
  range-finder to discover the distance of 12
  drives (in yards).

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Example 1 Questions

• Since sample size is less than 25, test the data.
  Is a Z- or T-Test appropriate?
• Histogram
• Box-and-Whisker Plot
• Construct a 98 confidence interval for the
  driving distance.
• Interpret the output.

11
Flow Chart Conf. Intervals
Almost never used!!
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Accuracy of Estimate

• Narrow intervals Better estimates
• How do get better?
• Increase n
• Larger sample size
• Decrease confidence
• Example Reduce from 95 to 90 confidence.
• Idea Wider strike-zone means more strikes.
• Only 1st option improves the estimate. Why?

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Example 2

• A Gallup poll conducted in 1999 asked 1031
  randomly selected Americans how often they
  bathed. Results indicated an N(6.9,2.8)
  distribution.
• Should we use a Z- or T-Interval?
• Construct a 99 confidence interval.
• Interpret the interval.

14
Clicker Question 1

• A Gallup poll conducted in 1999 asked 1031
  randomly selected Americans how often they
  bathed. Results indicated an N(6.9,2.8)
  distribution. We should use
• a Z-Interval
• or T-Interval
• Neither

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Clicker Question 2

• A Gallup poll conducted in 1999 asked 1031
  randomly selected Americans how often they
  bathed. Results indicated an N(6.9,2.8)
  distribution. Construct a 95 confidence interval
  based on these findings.
• ( 6.7291 , 7.0709 )
• ( 6.7289 , 7.0711 )
• ( 6.6968 , 7.1032 )
• ( 6.7566 , 7.0434 )

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Example 3

• Motorola wishes to estimate the average talk
  time before batteries are recharged. For a
  random sample of 40 phones, the sample mean was
  315 minutes. The standard deviation (based on
  prior research) is estimated to be 24 minutes.
• Use a T-Interval to construct a 95 confidence
  interval and determine the point estimate.
• Note about which test to use in this case
• Interpret the interval.

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Clicker Question 3

• What is the effect of changing the confidence
  level from 90 to 95?
• Two confidence intervals were generated based
  upon the same data set (sample size exactly the
  same for both calculations), one using a 90
  level of confidence, the other using a 95 level.
  The one that was generated using the 95 level
  must have been
• ( 5.1 , 6.5 )
• ( 5.0 , 6.6 )
• Not enough information

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Thats All, Folks!!

• This is all there is to Confidence Intervals
• Key Points
• Z vs. T
• Same as Hypothesis Testing (coming next!)
• Population vs. Sample
• s vs. s
• Estimation
• Point Estimate vs. Interval