Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 10:
Estimating with Confidence
Copyright © 2008 by W. H. Freeman & Company
Ex.
Suppose a sample of 50 men had a mean score of 109 on an intelligence test.
• We can estimate that the population mean m, is approximately 109.
• x bar is normally distributed.
• The mean of the sampling distribution is equal to m, the unknown population
mean.
• The standard deviation of x bar for an SRS of 50 given the population
standard deviation s = 15 is 15/(50)0.5 = 2.1
• The 68 – 95 – 99.7 rule states that about 95% of all possible sample means x
bar will be within 2 standard deviations of the population mean m.
•
In 95% of all possible samples the unknown m, lies between x bar + or – 4.2
•
We are 95% confident that m lies between 109 + 4.2; that is (104.8 , 113.2)
•
There are only two possibilities:
1.
The interval between 104.8 and 113.2 contains the true population mean
m.
2.
Our SRS was one of the few samples for which x bar is not within 4.2
points of the true m. Only 5% of all samples give such inaccurate results.
 The method we used gives the correct result 95% of the time.
Applet showing confidence intervals:
http://onlinestatbook.com/stat_sim/conf_interval/index.html
Suppose you want to construct an 80% confidence interval
m
 Confidence level is usually
chosen as > 0.90
Confidence
level
Tail area
Z*
80%
0.1
1.282
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576
estimate
Margin of
error
Ex.
A questionnaire of 160 hotel managers asked how long they had been with their current
company. The average time was reported as 11.78 years. Give a 99% confidence
interval for the mean number of years that the entire population of managers have been
with there current company. Assume the standard deviation of the population is s =
3.2 years.
11.78 + 2.576(3.2/√160) = 11.78 + 0.652 = (11.128, 12.432)
We are 99% confident that the true population mean lies between 11.128 and 12.432.
The method we used will give the correct result 99% of the time.
• Ideally, we would like;
1) high confidence; method almost always
gives the right result.
and
2) small margin of error; population
parameter estimated very precisely.
Margin of error decreases when;
1)
z* gets smaller; but this makes confidence level smaller
2)
s is small – sample drawn from less spread population.
3)
n, sample size is large. Quadrupling the sample size cuts margin of error in half.
How to choose a sample size for a desired margin of error.
Ex.
How many observations must be made to produce results accurate to within + 0.005
with 95% confidence? Assume s = 0.0068.
z* s/√n < 0.005 => (1.960 * 0.0068/0.005)2 < n => 7.1 < n ; choose n greater than
or equal to 8
You must round up to next integer
It is incorrect to say that the probability is 95% that
the true mean lies within a certain interval.
We can say that we are 95% confident that the mean
lies within a certain interval or ; The method we used
to calculate the interval gives the correct result in
95% of all possible sample of a particular size.
Tests of Significance
• Significance tests assess the evidence provided by the data in favor of some claim
about the population.
• Significance tests begin by stating a hypothesis about a population parameter.
• The null hypothesis Ho, is always stated as an equivalence.
Ho : m = mo
• The alternative hypothesis Ha, can be stated in one of three ways.
Ha : m ≠ mo
m < mo
m > mo
Ex.
A car manufacturer claims that one of their car models gets 33mpg. A random
sample of 30 cars is selected and the mean gas mileage of this sample x-bar is
calculated to be 31 mpg. Can we refute the claim of the automaker? Assume
s = 3.5 mpg.
Ho: m = 33 mpg
Ha: m < 33 mpg
x - bar = 31 mpg, sample std. = 3.5/√30 = 0.639
33
3.5/√30 =
0.639
- 0.639
33
31
32.361
33
P( z < -3.12) =
0.00087
0.00087
31
33
• X-bar = 31 is way out on the normal curve. So far out that a result this small
almost never occurs by chance if the true m = 33 mpg.
• This is good evidence that the automakers claim should be rejected in favor of the
alternate hypothesis, m < 33 mpg
• Generally P-values < 0.05 are considered small enough to reject the Ho. It is
statistically significant.
Significance level
• We compare the P – value with a fixed value that we regard as decisive.
• The decisive value of P is called the significance level. Symbol => a
• Choosing a = 0.05 require that the data give evidence against Ho so extreme that it
would happen in no more than 5% of the possible samples if Ho is true. a = 0.01 require
that the data give evidence against Ho so extreme that it would happen in no more than
1% of the possible samples if Ho is true.
If P-value
is low,
reject the
HO
• If the P – value is as small or smaller than a, we say that the data are
statistically significant at level a = _____. The null hypothesis should be
rejected in favor of the alternate hypothesis.
One sided
test
Two sided
test
{
Choosing an a level in significance tests
• If Ho represents an assumption that people you must convince have
believed for a long period of time, strong evidence (small a), is needed to
persuade them.
• If the consequences of rejecting Ho are drastic; ie expensive, finality. You
may want strong evidence, (small a).
• May be more useful to report the P-value so each individual may decide
for themselves.
• Even though significance levels of 0.10, 0.05 and 0.01 have been used
traditionally. The border between what levels are significant is not black
and white. Not much difference between P-values of 0.049 and 0.051.
• No significance level is sacred.
Inference as decision
Type I and Type II errors
• If we reject Ho (accept Ha) when Ho is really true, this is a
Type I error.
• If we reject Ha (accept Ho) when Ha is really true, this is
Type II error.
Ho True
Ha True
Reject Ho
Type I
Error
Correct
Decision
Reject Ha
Correct
Decision
Type II
error
Significance and Type I error
• The significance level a of any fixed level significance test is equal
to the probability of making a Type I error.
• the value of a is the probability that the test will reject the null
hypothesis Ho when Ho is really true.
Power of the test
• The probability that a fixed level a significance test will reject Ho
when Ha is true is called the power of the test.
• Increasing sample size n, increases the power of the test.
• Increasing the significance level a, increases the power of the test.