Do Now

1
Do Now

• If I just sampled 1000 people and I found their
  average body temperature to be 98.2F, what else
  do I need to construct a confidence interval for
  the average body temperature of all human beings?

2
Things to Watch Out For When Working With Sample
Means

• Beware of observations that are not independent.
• The CLT depends crucially on the assumption of
  independence.
• You cant check this with your datayou have to
  think about how the data were gathered.
• Watch out for small samples from skewed
  populations.
• The more skewed the distribution, the larger the
  sample size we need for the CLT to work.

3
Confidence Intervals for Sample Means

• Just as we did before, we will base both our
  confidence interval on the sampling distribution
  model.
• The Central Limit Theorem told us that the
  sampling distribution model for means is Normal
  with mean µ and standard deviation

4
Confidence Intervals for Sample Means

• All we need is a random sample of quantitative
  data.
• And the true population standard deviation, s.
• Well, thats a problem

5
Confidence Intervals for Sample Means

• Proportions have a link between the proportion
  value and the standard deviation of the sample
  proportion.
• This is not the case with meansknowing the
  sample mean tells us nothing about the standard
  deviation of the sample proportion
• Well do the best we can estimate the population
  parameter s with the sample statistic s.

6
Confidence Intervals for Sample Means

• We now have extra variation in our standard error
  from s, the sample standard deviation (since the
  sample standard deviation will change from sample
  to sample).
• If we dont account for this extra variation, we
  will provide an incorrect confidence interval
  (which depends on an accurate standard error)

7
Confidence Intervals for Sample Means

• And, the shape of the sampling model changesthe
  model is no longer Normal. So, what is the
  sampling model?
• William S. Gosset, an employee of the Guinness
  Brewery in Dublin, Ireland, worked long and hard
  to find out what the sampling model was.

8
Gossets t

• The sampling model that Gosset found has been
  known as Students t.
• The Students t-models form a whole family of
  related distributions that depend on a parameter
  known as degrees of freedom.
• We often denote degrees of freedom as df, and the
  model as tdf.

9
The Sampling Distribution Model for Means

• When the conditions are met, the standardized
  sample mean
• follows a Students t-model with n 1 degrees
  of freedom.
• We estimate the standard error with

10
The Sampling Distribution Model for Means

• When Gosset corrected the model for the extra
  uncertainty, the margin of error got bigger.
• Your confidence intervals will be just a bit
  wider and your P-values just a bit larger than
  they were with the Normal model.
• By using the t-model, youve compensated for the
  extra variability in precisely the right way.

11
The Sampling Distribution Model for Means

• Students t-models are unimodal, symmetric, and
  bell shaped, just like the Normal.
• But t-models with only a few degrees of freedom
  have much fatter tails than the Normal.
• http//www.stat.tamu.edu/jhardin/applets/signed/T
  .html

12
The Sampling Distribution Model for Means

• As the degrees of freedom increase, the t-models
  look more and more like the Normal (when n equals
  30, its hard to tell the two apart).
• In fact, the t-model with infinite degrees of
  freedom is exactly Normal.

13
Finding t-Values By Hand

• The Students t-model is different for each value
  of degrees of freedom.
• Because of this, Statistics books usually have
  one table of t-model critical values for selected
  confidence levels.

14
Finding t-Values By Hand (cont.)

• What if we dont have the degrees of freedom in
  our t-table?
• Option 1 Interpolate the value using the two
  surrounding values. (Example find the critical
  t-value for 38 d.f. that cuts off the top and
  bottom 5)
• Option 2 Just use the STAT -gt TESTS -gt TInterval
  to generate the confidence interval. Just make
  sure that you always write the formula for the
  C.I. and plug in as much as you can.
• You can take the confidence interval, divide it
  by 2, and divide it by the S.E.

15
One-Sample t-Interval

• When the conditions are met, we are ready to find
  the confidence interval for the population mean,
  µ.
• The confidence interval is
• where the standard error of the mean is
• The critical value depends on the particular
  confidence level, C, that you specify and on the
  number of degrees of freedom, n 1, which we get
  from the sample size.

16
Finding t-Values By Hand (cont.)

• Do example on p.449 using the calculator to find
  the t-interval.
• Try the 1-proportion z-interval for the last
  problem on the test.
• Remember, its okay to use these as long as you
  write out the formula and plug the numbers in.

17
Assumptions and Conditions

• Gosset found the t-model by simulation.
• Years later, when Sir Ronald A. Fisher showed
  mathematically that Gosset was right, he needed
  to make some assumptions to make the proof work.
• We will use these assumptions when working with
  Students t.

18
Assumptions and Conditions (cont.)

• Independence Assumption
• Randomization Condition The data arise from a
  random sample or suitably randomized experiment.
  Randomly sampled data (particularly from an SRS)
  are ideal.
• 10 Condition When a sample is drawn without
  replacement, the sample should be no more than
  10 of the population.

19
Assumptions and Conditions (cont.)

• Normal Population Assumption
• We can never be certain that the data are from a
  population that follows a Normal model, but we
  can check the
• Nearly Normal Condition The data come from a
  distribution that is unimodal and symmetric.
• Check this condition by making a histogram or
  Normal probability plot.

20
Make a Picture, Make a Picture, Make a Picture

• Pictures tell us far more about our data set than
  a list of the data ever could.
• The only reasonable way to check the Nearly
  Normal Condition is with graphs of the data.
• Make a histogram of the data and verify that its
  distribution is unimodal and symmetric with no
  outliers.
• You may also want to make a Normal probability
  plot to see that its reasonably straight.

21
Assumptions and Conditions (cont.)

• Nearly Normal Condition
• The smaller the sample size (n lt 15 or so), the
  more closely the data should follow a Normal
  model.
• For moderate sample sizes (n between 15 and 40 or
  so), the t works well as long as the data are
  unimodal and reasonably symmetric.
• For larger sample sizes, the t methods are safe
  to use even if the data are skewed.

22
Sample Size

• To find the sample size needed for a particular
  confidence level with a particular margin of
  error (ME), solve this equation for n
• The problem with using the equation above is that
  we dont know most of the values (and t is a
  function of the sample size!) We can overcome
  this
• We can use s from a small pilot study.
• We can use z in place of the necessary t value.

23
Sample Size

• Example Say you want a M.E. of 0.5F and you
  think the standard deviation of body temps is
  0.75F . What sample size should you use?
• First, use a critical z-score to find what sample
  size would create that margin of error (with a
  95 confidence).
• Now, use that sample size to determine the d.f.
  for the critical t value.
• Last, calculate the sample size required using
  the critical t.

24
Homework

• Add the following problems to p. 462-464 10, 11,
  23 p. 323 33, p. 343 26, p. 362-365 2, 4, 6,
  8, 13, 17, 22, 29, 31, 33 p. 378-381 1, 3, 4,
  8, 10, 12, 14, 18, 25 (This will be due on Monday
  and worth 15 points)
• Correct tests for half the points back as per the
  test correction policies (due Monday)
• Make sure you have given your AP exam payment to
  Mrs. Cohan in the Guidance Office by Friday!