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Engineering Data Analysis
Notes # 14
Steps in Hypothesis Testing – Traditional Method
Suppose that a certain airline company requires the manufacturer of its aircraft to use
rivets whose mean shearing strength exceeds 120 lbs. Each rivet manufacturer that wants
to sell rivets to the aircraft manufacturer must demonstrate that its rivets meet the
required specification, namely, that the mean shearing strength of all the manufacturer’s
rivets, μ, be greater than 120 lbs.
In this illustration, the rivet supplier is interested in demonstrating that the mean shearing
strength of its rivets is greater than 120 (μ > 120). The statistical procedure used to make
this determination is called a hypothesis test.
There are four steps to a conducting a hypothesis test:
STEP 1: Formulate the null and alternative hypothesis
Hypothesis - A statement that something is true
There are 2 hypotheses:
Null Hypothesis - The hypothesis that we will test. Generally this is a statement that a
population parameter has a specific value. The null hypothesis is so
named because it is the starting point for the investigation. The
phrase “there is no difference” is often used in its interpretation.
Symbol:
Null Hypothesis Clue:
Alternative Hypothesis - A statement about the same population parameter that is used
in the null hypothesis. Generally, this is a statement that
specifies that the population parameter has a value different,
in some way, from the value given in the null hypothesis. The
rejection of the null hypothesis will imply the acceptance of
this alternative hypothesis.
Symbol:
Alt. Hypothesis Clue:
There are four possible outcomes in a Hypothesis Test:
Table 1
Null Hypothesis
Decision True False
Fail to reject H 0 Type A Type II
Correct Decision Error
Reject H 0 Type I Type B
Error Correct Decision
Table 2
Probability with Which Error Occurs
Error Type Error Probability
Rejection of a true hypothesis I
Failure to reject a false null hypothesis II β
Illustration
The null hypothesis “the airplane is safe” is being tested.
a) Carefully state the meaning of the four possible outcomes in Table 1.
b) Describe the seriousness of the type I error and the type II error.
STEP 2: Determine the Test Criteria
The test criteria consist of:
specifying a level of significance,
determining a test statistic
determining the critical region(s)
determining the critical value(s)
Level of significance - the probability of committing the type I error, .
Conclusion Rule
If the decision is “reject H 0 ” then the conclusion should be worded “There is sufficient
evidence at the level of significance to show that... (the meaning of the alternative
hypothesis).”
If the decision is “fail to reject H 0 ” then the conclusion should be worded something like
“There is not sufficient evidence at the level of significance to show that... (the
meaning of the alternative hypothesis).”
The z-Test for a Mean
First, practice formulating the null and alternative hypotheses:
Illustration 1
An ecologist would like to show that Quiapo has an air pollution problem. Specifically,
she would like to show that the mean level of carbon monoxide along the Recto-Quiapo
area air is higher than 4.9 parts per million. State the null and alternative hypotheses.
Illustration 2
In trying to promote the city, the Engineering Office in charge would be more likely to
want to conclude that the mean level of carbon monoxide in the City of Manila is less
than 4.9 parts per million. State the null and alternative hypotheses related to this
viewpoint.
Illustration 9
The “mean level of carbon monoxide in Quiapo is not 4.9 parts per million.” State the
null and alternative hypotheses that correspond to this statement.
Sign in the
Alternative Hypothesis < ≠ >
Critical Region One Region Two Regions One Region
Left Side One on Each Side Right Side
One-Tailed Test Two-Tailed Test One-Tailed Test
NOTES
THE CLASSICAL HYPOTHESIS TEST
Step 1: State the null hypothesis ( H 0 ) and the alternative hypothesis ( a
H ).
Step 2 : Determine the test criteria:
a. The level of significance, , to be used
b. The test statistic to be used
c. The critical region(s)
d. The critical value(s)
Step 3: Collect and present the sample evidence.
a. Collect the sample information.
b. Calculate the value of the observed test statistic.
Step 4: Determine the results.
a. Compare the calculated value of the test statistic to the critical value(s) from
Step 2.
b. Make a decision about H 0_._
c. State the conclusion about Ha.
Hypothesis Testing – t - Test for a Mean
When the population standard deviation is unknown and the sample size is less than 30,
the z - score is inappropriate for testing hypotheses. We have to use the t - test.
n
= df = n – 1
Ex. 1 A manufacturer of sports equipment has developed a new synthetic fishing line
that he claims has a mean breaking strength of 8 kilograms with a standard
deviation of 0.5 kilogram. Test the hypothesis that μ = 8 kilograms against the
alternative that μ < 8 kilograms if a random sample of 16 lines is tested and found
to have a mean breaking strength of 7.7 kilograms. Use α = 0.05 level of
significance.
Ex. 2 A machine is designed to fill jars with 16 ounces of coffee. A consumer suspects
that the machine is not filling the jars completely. The consumer believes it is
less than 16 ounces of coffee. A sample of 8 jars has a mean of 15. 5 ounces and a
standard deviation of 0.3 ounces. At =0.10, test the consumer’s claim.
Ex. 3 A physician claims that joggers’ maximal volume oxygen uptake is greater than
the average of all adults. A sample of 15 joggers has a mean of 43.6 mL per
kilogram and a standard deviation of 6 ml/kg. If the average of all adults is 36.
ml/kg, test the physician’s claim at =0.01.
Ex. 4 A researcher collected a random sample of 25 energy bars from a number of
different stores to represent the population of energy bars available to the general
consumer. The labels on the bars claim that each bar contains 20 grams of protein.
Do the data present sufficient evidence that the mean weight of protein exceed 20 grams?
Test at the 0.01 level of significance.
