Transcript 9.1 Notes - morgansmathmarvels

9.1 Notes
Introduction to
Hypothesis Testing
In hypothesis testing there are 2 hypothesis for each problem, the null hypothesis
and the alternate hypothesis.
Null Hypothesis (H0) – A working hypothesis about the population.
*Always uses =
i.e. A car dealer claims that the avg. mpg for a certain model is 47.
H0: μ = 47 mpg
Alternate Hypothesis (H1/HA) – Any hypothesis that differs from the null.
*Always uses <, >, or ≠.
• < indicates a left-tailed test,
Critical Region
• > indicates a right-tailed test
Critical Region
• ≠ indicates a two-tailed test
Critical Regions
Critical Regions
i.e. We believe the dealer is exaggerating the mpg claim.
H1: μ < 47 mpg
Ex. 1 A company manufactures ball bearings for precision machines. The
average diameter of a certain type of ball bearing should be 6.0 mm. To
check that the average diameter is correct, the company formulates a
statistical test.
a) What should be the used for H0?
H0: μ = 6.0 mm
b) What should be used for H1?
Hint: an error either way (too large or too
small) would be serious
H1: μ ≠ 6.0 mm
Two-tailed
Ex. 2 A package delivery service claims it takes an average of 24 hours to send
a package from New York to San Francisco. An independent consumer
agency is doing a study to test the truth of this claim. Several complaints
have led the agency to suspect that the delivery time is longer than 24 hours.
a) What should be the used for H0?
H0: μ = 24 hours
b) What should be used for H1?
H1: μ > 24 hours
right-tailed
In hypothesis testing there are two possible outcomes, reject the null or fail to
reject the null.
Reject the Null
There is enough evidence in the data to imply that the null is false and the
alternate is true.
Fail to Reject the Null
There is not enough evidence to justify rejecting the null.
Relate to court process
Neither one of these results are error free.
Types of Errors
Type I – We reject the null when in fact the null is true
Type II – We fail to reject the null when in fact the null is false.
In order to reduce Type I error, Type II error increases and vice-versa.
Level of Significance (α) – The probability with which we are willing to risk a
type I error (reject the null when if fact it is true). Is determined before data is
gathered. Used throughout much of the remaining portion of the course.
Power of a Test (1 – β) – The probability with which the null is correctly rejected
when in fact it is false. Note: β is probability of making a type II error. Hard to
calculate and is not related to much in this level of statistics.
Some Generalities about α and 1 – β
1. As α increases then 1 – β also increases.
2. Even though an increase in α results in an increase in 1 – β, it also results in
a higher probability that we reject the null when in fact it is true.
Most people would prefer to accept the null when in fact it is false than to accept
the alternate when in fact it is false.
Assignment
p. 411 #1-8
Basic Components of a Statistical Test
1. Null Hypothesis H0 , Alternate Hypothesis H1 , and a preset level of
significance α
If the evidence (sample data) against the H0 is strong enough, we
reject the H0 and adopt the H1 . The level of significance α is the
probability of rejecting H0 when it is in fact true.
2. Test Statistic and Sampling Distribution
(For now we will be focusing mainly on normal and t-student
distributions).
3. P-value
This is the probability of obtaining a test statistic from the sampling
distribution that is as extreme as, or more extreme than the sample
test statistic computed from the data under the assumption that H0 is
true.
4. Test Conclusion
If P-value < α, we reject H0 and say that the data are significant at
level α. If P-value > α, we do not reject H0.
5. Interpretation of the test results
Give a simple explanation of your conclusions in context of the
application.
Ex. 3 The Environmental Protection Agency has been studying Miller Creek
regarding ammonia nitrogen concentration. For many years, the concentration
has been 2.3 mg/l. However, a new golf course and housing developments are
raising concern that the concentration may have changed because of lawn
fertilizer. A change either way in ammonia nitrogen concentration can affect
plant and animal life in and around the creek. Let x be a random variable
representing ammonia nitrogen concentration (in mg/l). Based on recent
studies of Miller Creek, we may assume that x has a normal distribution with
σ = 0.30. Recently, a random sample of eight water tests from the creek gave
the following x values.
2.1
Test at α = 0.01
2.5
2.2
2.8
3.0
2.2
2.4
2.9
a) What is the level of significance? State the null and alternate hypothesis.
Will you use a left-tailed, right-tailed, or two-tailed test?
b) What sampling distribution will you use? Explain the rationale for your
choice of sampling distribution. What is the value of the sample test statistic?
c) Find (or estimate) the P-value. Sketch the sampling distribution and show
the area corresponding to the P-value.
d) Based on your answers in parts (a) to (c), will you reject or fail to reject the
null hypothesis? Are the data significant at level α?
e) State your conclusion in the context of the application.
Assignment
P. 413 #9-14