Examples of
Inference
Example Problem
Example: Runners

You measure the weights of 24 male
runners. You do not actually choose an
SRS, but you are willing to assume that
these runners are a random sample from
the population of male runners in your
town. Here are their weights in kilograms:
67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9
60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8
66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7
Example: Runners

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step
1: First, state what you want to know in terms
of the Parameter and determine what the question is
asking
 We wish to estimate the true mean weight, μ, of male
runners in our town
Example: Runners

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step
2: Second, examine the Assumptions and check
the conditions.
 Independence:
Randomization condition: We assumed that the 24
runners represented a random sample of the town.
 10% condition: We safely assume that there are more
than 240 runners in this town
 If there are not more than 240 runners or independence
is violated, then our results may not be valid.

Example: Runners

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step
2: Second, examine the Assumptions and check
the conditions.
 Normality:
Since the sample is small, we cannot apply the CLT.
 We do not know if the population is normal.
 We need to examine the sample data
 Since the sample looks fairly normal, we can assume that
the sampling model will look approximately normal.

Example: Runners

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step
3: Third, Name the inference, do the work, and
state the Interval
 We will construct a 95% 1-sample t-Interval for
means:
 4 . 8078 
61 . 792  2 . 069 
  ( 59 . 762 , 63 . 822 )
24 


Where did 2.069 come from?
 It is t* for 95% Confidence and 23 degrees of
freedom from TABLE B
Example: Runners

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step
4: Fourth, last but not least, state your
Conclusion in context of the problem
 We are 95% confident that the true mean weight of
male runners in the town is between 59.762kg and
63.822kg. However, some of our assumptions may
not be valid. If the assumptions are violated, then our
results may not be legitimate.
Example: Runners Part 2



Suppose that the average weight of a runner in
the city has traditionally been 64kg. The runners
that you gathered in your sample were on a diet
to enhance their metabolism. The diet claims
that it will make the runners lighter and develop
a leaner muscle mass. Did the diet work? Use a
95% confidence level.
If our confidence is 95%, what is α? 0.05
Is this one or two tailed? This is one tailed
Example: Runners Part 3
 Step
1: Identify population Parameter, state
the null and alternative Hypotheses,
determine what you are trying to do (and
determine what the question is asking).
H 0 :   64 kg
H A :   64 kg
 H0:
The mean weights of the runners will be 64kg
 HA: The mean weights of the runners will be less
than 64kg
Example: Runners
 Step
2: Verify the Assumptions by checking
the conditions
 Independence:
 Randomization
condition: We assumed that the 24
runners represented a random sample of the town.
 10% condition: We assume that there are more than
240 runners in this town
 If there are not more than 240 runners or
independence is violated, then our results may
not be valid.
Example: Runners
 Step
2: Verify the Assumptions by checking
the conditions:
 Normality
 Since
the sample is small, we cannot apply the CLT.
 So, we examine the sample data
 Since
the sample data is roughly normal, the
sampling distribution will be approximately
normal.
Example: Runners
 Step
3: If the conditions are met, Name the
inference procedure, state the Test statistic,
and Obtain the p-value:
 We will perform a 1-sample t-test
t
y
s
n
t
61 . 792  64
4 . 8078
24
t   2 . 25
p  value  . 0171
Example: Runners
 Step
4: Make a decision (reject or fail to reject
H0). State your conclusion in context of the
problem using p-value.
 The
p-value is small enough, p-value = 0.0171, that
we reject the null hypothesis in favor of the
alternative at a 95%confidence level. There is
sufficient evidence to say that the mean weight of the
runners is less than 64kg. This implies that the diet
worked.
 However, our independence assumption may have
been violated. If so, our results may not be valid.
Assignment
Chapter 23
Lesson:
Inference for Means
Read:
Chapter 23
Problems:
1 - 31 (odd)