Transcript Introduction to Differential Equations

Statistical Inference for the Mean
Objectives: (Chapter 9, DeCoursey)
- To understand the terms: Null Hypothesis,
Rejection Region, and Type I and II errors.
- To understand how to perform hypothesis
testing for a population and a sample (t-test).
Statistical Inference for the Mean
Statistical Inference is a process of using
observations of a sample to estimate the
parameters of the population.
Statistical Inference for the Mean:
To provide answers on whether the sample
mean and the mean of the corresponding
population may be similar enough that the
sample might have come from the
population.
Explanation
Statistical Inference for the Mean
When using the z-test this is done by assuming
the variance of the population is known and not
changed. Another assumption is made:
• Null Hypothesis – Ho: sample came from a
population having stated mean – i.e. notion or
suggestion that nothing has changed
Statistical Inference for the Mean
• If Null Hypothesis is incorrect, an alternative
hypothesis is needed:
• Alternative Hypothesis – Ha: something has
changed – the sample mean does not
correspond to the population mean
Statistical Inference for the Mean
• When comparing data taken from a sample,
z value is calculated:
x
z
/ n
Statistical Inference for the Mean
E.g. When a manufacturing process is operating
properly, the mean length of a steel rod is
known to be 6.175 inches, and lengths are
normally distributed. The standard deviation of
this length is 0.0080 inches. If a sample
consisting of 6 items taken from current
production has a mean length of 6.168 inches,
is there evidence that the mean of the
population is changed? Or the sample is from
different population so that some adjustment of
the process is required?
Statistical Inference for the Mean
Null Hypothesis: the notion or suggestion that
nothing has changed.
E.g. “the population mean is still 6.175 inches.”
If the probability of the observation “mean length
of 6.168 inches” is very small based on the
given population mean and variance, i.e. near
the tails, there is a potential to reject the null
hypothesis.
z
x
x

x
/ n

6.168  6.175
0.008/ 6
 2.14
Statistical Inference for the Mean
z
x
x

x
/ n
 2.14
P r[Z  2.14]
 1  (2.14)
 1  0.9838
 0.0162
P r[Z  2.14]
  (2.14)
 0.0162
-2.14
μ
2.14
We are interested in the probability that the variable takes on
a value of 2.14 standard deviates from the mean in either direction.
(Two-tailed test)
Statistical Inference for the Mean
-2.14
μ
2.14
Pr[Z  2.14 or Z  2.14]
 0.0162 0.0162
 0.0324or 3.2%
Should we reject the Null Hypothesis?
Statistical Inference for the Mean
Then Reject Region has to be defined.
Reject region: the region of a probability
distribution where the probability of the
observation is very small, i.e. near the tails.
-2.14
μ
2.14
Statistical Inference for the Mean
The low probability of the observation suggests
that something has changed. (i.e. the chances
of it occurring randomly are small) and we
must reject the Null Hypothesis.
Reject region = level of significance
e.g. level of significance=5%
-2.14
μ
2.14
When the probability of the
observation called observed level
of significance (3.2%) is Less than
5%, reject the Null Hypothesis;
Otherwise do not reject.
Statistical Inference for the Mean
E.g. When a manufacturing process is operating
properly, the mean length of a steel rod is
known to be 6.175 inches, and lengths are
normally distributed. The standard deviation of
this length is 0.0080 inches. If a sample
consisting of 6 items taken from current
production has a mean length of 6.168 inches,
is there evidence at the 5% level of significance
that some adjustment of the process is
required?
Yes.
Statistical Inference for the Mean
We may decide to take some action on the basis of the
test of significance. But we can never be completely
certain we are taking the right action.
Type I error is to reject the null hypothesis when it is true.
In the case of a mean, this occurs when the null
hypothesis is correct, but an observation or sample mean
is so far from the expected mean by chance that the null
hypothesis is rejected.
The probability of a Type I error is equal to the level of
significance.
To reduce the chance of a type I error:
Small rejection region (small level of significance).
Statistical Inference for the Mean
Type II error is to accept Null Hypothesis when it is False.
If in fact the population mean has changed, the null
hypothesis is false. But the sample mean might still by
chance come close enough to the original sample mean
so that we would accept the null hypothesis, giving a Type
II error.
The probability of a Type II error depends on how much
the population mean changed in comparison to the
standard error of the mean.
Reduce the chance of a type II error: larger rejection
region.
Statistical Inference for the Mean
How to perform a statistical test for the mean when σ of
the population is known?
- State the null hypothesis in terms of a population
parameter, such as mean μ.
- State the alternative hypothesis in terms of the same
population parameter.
- Assume the variance is known as well as the mean.
- Calculate the test statistic z of the observation using the
mean and variance given by the null hypothesis.
- State the level of significance – rejection limit.
- If probability falls outside of the rejection limit, we reject
the Null Hypothesis, which means the measurement is
from a different population.
We are assuming Normal Distribution in this scenario.
Example