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) –
i.e. A car dealer claims that the avg. mpg for a certain model is 47.
Alternate Hypothesis (H1/HA) –
• < indicates
• > indicates
• ≠ indicates
i.e. We believe the dealer is exaggerating the mpg claim.
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?
b) What should be used for H1?
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?
b) What should be used for H1?
In hypothesis testing there are two possible outcomes, reject the null or fail to
reject the null.
Reject the Null
Fail to Reject the Null
Relate to court process
Neither one of these results are error free.
Types of Errors
Type I –
Type II –
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. 412 #1-8
Basic Components of a Statistical Test
1. _____ Hypothesis H0 , ____________ Hypothesis H1 , and a preset _____
____________________ α
If the evidence (sample data) against the H0 is strong enough, we
________________________. The level of significance α is the
probability of ______________________________________.
2. Test Statistic and Sampling Distribution
(For now we will be focusing mainly on _________ and ___________
distributions).
3. P-value
This is the probability of obtaining a test statistic from the sampling
distribution that is _____________, or _________________than the
sample test statistic computed from the data under the assumption
that H0 is true.
4. Test Conclusion
If P-value ________, we reject H0 and say that the data are
significant at level α. If P-value ________, 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 Rosie is an aging sheep dog in Montana who gets regular check-ups
from her owner, the local veterinarian. Let x be a random variable that
represents Rosie’s resting heart rate (in beats per minute). From past
experience, the vet knows that x has a normal distribution with σ = 12. The vet
checked the Merck Veterinary Manual and found that for dogs of this breed, μ =
115 beats per minute. Over the past six weeks, Rosie’s heart rate
(beats/minute) measured
93
109
110
89
112
117
The vet is concerned that Rosie’s heart rate is falling below normal. Do the
data indicate that this is the case? Test at α = 0.05
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.
Ex. 4 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