Inference for a Population Proportion

1
Inference for a Population Proportion Lesson
12.1 using Tests of Significance I
Proportions Workshop Statistics 21-1 Homework
for this lesson 12.1, 12.4, and baseball
problem on slides 15 16.
2

• Consider an experiment to assess whether people
  can distinguish between the tastes of two brands
  of cola. Subjects are presented with three cups.
  Two contain one brand of soda, the third
  contains a different brand. Subjects taste from
  all three cups and then identify the one that
  differs from the other two. Suppose the
  experiment consists of thirty of these trials.
• If the subjects cannot distinguish among the
  colas and guess on each trial, what proportion of
  trials will they correctly distinguish in the
  long run?
• Is this value a parameter or a statistic?
  Explain.
• What symbol represents it?

3
According to the Central Limit Theorem, if ? (p)
actually has this value, what pattern will the
sampling distribution of the sample proportion
follow? Describe its shape, center, and spread,
and draw a sketch of the distribution. Note the
standard deviation Are the conditions met
for CLT to be valid?
4
Shade the area under the this curve corresponding
to the probability of getting at least 20 correct
identifications ( ) guessing in 30
trials. Calculate the probability that a
subject who is just guessing would guess
correctly on 20 or more of the 30 trials.
5
Based on this probability, would you consider
such a sample result surprising if someone is
just guessing? Would you consider it so
surprising that you believe the subject is not
just guessing but really does have some ability
to discriminate among the sodas?
6
Clearly define ? in the cola discrimination
study Let ? The null hypothesis here is
that the subject is just guessing. Translate this
into a null hypothesis statement . H0 ?
Suppose the subjects have some ability to
distinguish and therefore do better than just
guessing. What would the alternative hypothesis
be? Ha ?
7
Would a guessing subject always identify the odd
cup on exactly 1/3 of an experiments trials?
What term describes the phenomenon involved
here?
8

• Consider the following results of 3 of these
  experiments. Suppose that Alicia identifies the
  differing cup correctly in 10 of the 30 trials
  Brenda identifies the different cup for 12
  whereas, Celia gets 15 correct in 30 trials.
• Calculate the sample proportion of correct ids
  for each.
• A B C
• Are these proportions parameters or statistics?
  Explain
• Is it possible that guessing subjects could get
  as many as 12 or 15 correct out of 30 just by
  chance?
• What will the standard deviation of the sample
  proportion of be?

9
Draw a sketch of this sampling distribution of
for guessing subject, labeling the horizontal
axis. Mark where the sample proportions for
Alicia, Brenda, and Celia fall on the sketch.
10
Use your calculator to find the z-score
corresponding to Brendas sample and the
corresponding probability. z P Does
this suggest that it is very unlikely for Brenda
to get 12 correct ids if she is guessing?
11
Guidelines for evidence against H0 p-value gt .10
little or no evidence against H0 .05 lt p-value
lt .10 some evidence against H0 .01 lt p-value lt
.05 moderate evidence against H0 .001 lt p-value
lt .01 strong evidence against H0 p-value lt
.001 very strong evidence against H0 .
12
Calculate the test statistic and p-value for
Celia, who obtained 15 correct ids. Is
this significant at the following significance
levels? a .10 a .05 a .01
13
Calculate the test statistic and p-value for
Alicia, who obtained 10 correct ids. Is
this significant at the following significance
levels? a .10 a .05 a .01
14

• To establish validity of these testing procedures
  the samples must meet this conditions
• SRS from the population of interest
• Sample size is large relative to the proportions
  involved (CLT)
• Where these conditions about sample size
  satisfied?

15
A reader wrote in to the Ask Marilyn column in
Parade Magazine to say that his grandfather told
him that in ¾ of all baseball games, the winning
team scores more runs in one inning than the
losing team scores in the entire game. This
phenomenon is known as a big bang. Marilyn
responded that this proportion seemed to be too
high to be believable. State the null and
alternative hypotheses. Define the parameter.
16
For the sample of 190 Major League Baseball games
played during July 26 August 8, 1999, 98 of the
games contained a big bang. 1. Sketch the
sampling distribution for the sample proportion
as predicted by the CLT, assuming the null is
true. 2. Calculate the sample proportion of games
in which a big bang occurred. 3. Is this sample
proportion less that ¾ and therefore consistent
with Marilyns (alternative) hypothesis? Shade
area under the curve corresponding to this sample
result in the direction conjectured by
Marilyn. 4. Using the calculator, find the test
statistic and p-value. 5. Based on this p-value,
would you say that the sample data provided
strong evidence to support Marilyns contention
that the proportion cited by the grandfather is
too high to be the actual value? Explain