Tests of Hypothesis
A statistical hypothesis is an
assertion or conjecture concerning one or more populations. There are two types of statistical
hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis, denoted by Ho, is
defined as a hypothesis of no difference.
It is ordinarily formulated with the purpose of being accepted or
rejected. If the null hypothesis is
rejected, the alternative hypothesis is accepted. The alternative hypothesis denoted by H1.
Alternative
hypotheses are classified as either nondirectional or directional
hypotheses. A nondirectional hypothesis
is one which asserts that one value is different from another. More specifically, it is an assertion that
there is a significant difference between two statistical measures (or that
there are significant differences among three or more summary measures). A directional hypothesis, on the other hand,
is an assertion that one measure is less than (or greater than) another measure
of similar nature. Mathematically a
nondirectional hypothesis make use of the “not equal to” (¹)
sign, while a directional hypothesis involves one of the other relations, “less
than” (<) or “greater than” (>).
Nondirectional hypotheses are also called two-sided hypotheses, and
directional hypotheses also known as one sided-hypotheses.
Consequences of Decisions in Testing Hypotheses
|
DecisionèFact
|
Ho is true
|
Ho is false
|
|
Accept Ho
|
Correct decision
|
TypeII error
|
|
Reject Ho
|
Type I error
|
Correct decision
|
One-tailed and two-tailed tests
A
test of any statistical hypothesis where H1 is directional called a
one-tailed test. A test where H1 is
nondirectional is called a two-tailed test.
The difference between these two types of tests lies in that the first
is characterized by a region of rejection which lies entirely in one end of the
distribution. Whereas the second
involves a critical region which is split into two equal areas placed in each
tail of the distribution.
Steps in Hypothesis Testing
1.
Formulate the null hypothesis (Ho) and the alternative hypothesis(H1).
2.
Specify the level of significance a.
3.
Choose the appropriate test statistic.
4.
Establish the critical region.
5.
Compute for the value of the statistical test.
6.
Make a decision and if possible draw a conclusion.
Critical Value of Z
|
Significance Level
|
.10
|
.05
|
.25
|
.01
|
|
One-tailed
|
1.28
|
1.645
|
1.96
|
2.23
|
|
Two-tailed
|
1.645
|
1.96
|
2.33
|
2.58
|
1.
An electrical company claims that the lives of the light
bulbs it manufactures are normally distributed with a mean of 1000 hours and a
standard deviation of 150 hours. What
can you say about this claim if a random sample of 100 bulbs produced by the
company has a mean life of 980 hours?
Use a .05 level of significance.
2.
An instructor gives his class an achievement test which,
as he knows from years of experience, yields a mean 80. His present class of 40 obtains a mean of n85
and a standard deviation of 8. Can he
claim that his present class is a superior class? Employ alpha .01.
3.
A new production process is being considered to replace
the old process presently used. This ne
process was tested for 8 consecutive hours with the following results: 118,
122, 120, 124, 126, 125, 125, 124. If
the average output per hour using the old process is 120 units. Is the management justified in stating that
the output per hour can be increased with the new process? Use alpha .01
4.
In a time and motion study, it was found that a certain
manual work can be finished at an average time of 40 minutes with a standard
deviation of 8 minutes. A group of 16
students is given a special training and then found to average only 35
minutes. Can we conclude that the
special training can speed up the work using 0.01 level?
5.
A standardized test was administered to thousands of
students with a mean score of 85 and a standard deviation of 8. A
random sample of 50 students were given the same test and showed an average
score of 83.20. Is there evidence to
show that this group has a lower performance than the ones in general at 0.05
level?
6.
The average length of time for students to register for
summer classes at a certain college has been 50 minues. A new registration procedure using modern
computing machines is being tried. If a
random sample of 35 students had an average registration time of 42 minutes
with a standard deviation of 11.9 minutes under the new system, test the
hypothesis that the population mean is now less than 50 minutes using a level
of significance 0.01.
7.
A university librarian suspects that the average number
of books checked out to each student per visit has changed recently. In the past an average of 3.4 books was
checked out. However, a recent sample of
45 students averaged 4.3 books per visit with a standard deviation of 1.5
books. At the 0.05 level of
significance. Has the average checked out changed?
8.
The Computer Center of a well known university has
installed new color video display terminals to replace the monochrome units it
previously used. The 32 graduating BS
Computer Science students trained to use the new machines averaged 7.2 hours
before achieving a satisfactory level of performance. Their sample standard deviation was 16.2 hours.
Long experiences university computer operations on the old monochrome
terminals showed that they averaged 8.1 hours on the machines before their
performance were satisfactory. At the
0.10 significance level, should the head of the computer center conclude that
the new terminals are easier to learn to operate?
9.
In the certain
studies of deception among the students the scores achieve on test given
under conditions in which cheating was possible were compared with scores
achieve by the sample students under strictly supervised condition. In a certain test given under “honest”
conditions the mean score of 25 sample students is 62 and the standard
deviation is 10. A student who took the
test under non-supervised conditions turned in mean score of 87. What conclusion can be drawn out of the given
information at 0.01 level of significance?
10.
An educational researcher found that the average entrance
score of the incoming freshmen in a famous university is 84. A random sample of 24 students from a public
school was then selected has found out that their average score in the entrance
examination is 88 with a standard deviation of 16. Is there any evidence to show that the sample
from public school performed better than the rest in the entrance examination
using the 0.10 level of significance?
11.
A professor in a typing class found that the average
performance of an expert typist is 85 words per minute. A random
sample of 16 students took the typing test and an average speed of 62
words per minute was obtained with a standard deviation of 8. Can
we say that the sample students performance is below the standard at the 0.05
level of significance?
12.
A pharmaceutical firm claims that the average time for a
drug to take effect is 18 minutes with a standard 0f 2 minutes. In a sample of 36 trials, the average time
was 20 minutes. Test the claim against
the alternative that the average time is not equal to 18 minutes, using a 0.01
level of significance.
13.
In a time and motion study, it was found out that the
average time required by workers to complete a certain manual operation was
26.6 minutes with a standard deviation of 3 minutes. A group of 25 workers was randomly chosen to
receive a special training for 2 weeks.
After the training, it was found that their average time was 24
minutes. Can it be concluded that the
special training speeds up the operation? Use alpha 0.05
14.
A ceratain brand of powdered milk is advertised as having
a net weight of 250 grams. If the net
weights of a random sample of 10 cans are 253, 248, 252, 245, 247, 249, 251,
250, 247 and 248 grams, can it be concluded that the average net weight of the
cans is less than the advertised amount? Use alpha 0.01
15.
The mean content of 25 bottles of brand S mango juice is
335 mL with a standard deviation of 9
mL. Is this in line with the
manufacturer’s claim that the bottle contains on the average 360 mL? Use a 0.01
level of significance.
16.
The daily wages a particular industry are normally
distributed with a mean of P66. If a
company in this industry employing 144 workers pays, on the average, P62 with a
standard deviation of P 12.50, can this company be accused of paying inferior
wages at the 0.01 level of significance?
17.
The manager of an appliance store, after noting that the
average daily sales was only 12 units, decided to adopt a new marketing
strategy. Daily sales under this
strategy were recorded for 90 days after which period the average was found to
be 15 units with a standard deviation of 4 units. Does this indicate that the new marketing
strategy increased the daily sales? Employ alpha 0.01. Assume that the distribution of daily sales
are approximately normally distributed.
Where is solution of question 14???
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