Comments on Hypothesis Testing
a) The null hypothesis is usually the status quo, what you
suspect has changed.
b) The alternative hypothesis is what you suspect to be
different, three choices (less, not equal, greater).
c) NEVER use sample data to influence choice of null
hypothesis or alternative hypothesis.
d) A high P-value means "cannot reject" the null
hypothesis.
e) A low P-value means "reject" the null hypothesis.
Problem #1:
A university bookstore manager is interested in
determining if there is convincing evidence that the
proportion of students at the university who purchase
one or more of their textbooks online is less than a
reported national figure of 0.20. What hypothesis
should the bookstore manager test?
Tell me H0 and Ha.
Problem #2:
The manager of a large hotel must decide whether to
hire additional front desk staff. He has decided to hire
more staff if there is evidence that the average time
customers must wait in line before being assisted with
check-in is greater than 3 minutes.
Tell me H0 and Ha.
Problem #3:
The principal at a large high school will implement a
proposed after-school tutoring program if there is
evidence that the proportion of students at the school
who would take advantage of such a program is
greater than 0.10.
Tell me H0 and Ha.
Problem #4:
If the conclusion in a hypothesis test is fail to reject
H0, which of the following is an appropriate
conclusion?
a) There is convincing evidence that the null
hypothesis is true.
b) There is convincing evidence that the null
hypothesis is false.
c) There is not convincing evidence that the null
hypothesis is false.
Problem #5:
Which of the following is approximately equal to
the P-value for a test in which the hypotheses are
H0: p = 0.5 versus Ha: p ≠ 0.5 and for which the
value of the test statistic is z = -2?
a) 0.025
b) 0.05
c) 0.95
d) ).975
Problem #7:
To determine whether to produce a new design of a
baseball cap with a team logo printed on the back,
one hundred men who were wearing a baseball cap
were selected at random from those attending a
major league baseball game. On these men, 27
wore the hat with the bill facing backward. This data
was used to test the hypotheses H0: p = 0.33
versus Ha: p < 0.33 where p is the proportion of
baseball cap wearing men at the game who wear
the cap with the bill facing backward. A significance
level of .05 was used, and the P-value for this test
was 0.101. If the new hat design will be produced
unless there is convincing evidence that fewer than one
third wear the hat backward, what is an appropriate
decision based on this P-value?
The choices:
a) Produce the cap, because there is convincing evidence
that less than one third wear the hat backward.
b) Don't produce the cap, because there is convincing
evidence that less than one third wear the hat backward.
c) Produce the cap, because there is not convincing
evidence that less than one third wear the hat backward.
d) Don't produce the cap, because there is not
convincing evidence that less than one third wear the
Text Problem, page 520
Do home baseball teams really have an advantage?
Do they win more at home?
Statistically: do they win significantly more than they
should?
H0: p = 0.50
Ha: p > 0.50
The data from 2006 MLB:
2429 games
1327 won by home team, 54.63%
Independent:
Random:
10% condition:
np, nq > 10
probably not, but pretty close
one full year, a reasonable sample
2006 year < all games over many years
easily verified
The Calculations:
This is a one-tail test, to the right!
P-value
= Pr(p > 0.5463)
convert to z-score
then find area
z = 4.56; Pr(z > 4.56) = normalcdf(4.56, 100) = 0.0000026
If the true proportion of home team wins were 0.50 then the
observed value of 0.5463 would occur less than 3 in a
million samples. With such a small P-value we reject the
null hypothesis and conclude that the true proportion is not
0.50 and that there is a home field advantage.
Text #29, page 530
A company is criticized because only 13 of 43 people in
executive-level positions are women. The company
explains that although this proportion is lower than it might
wish, it's not surprising given only 40% of all its employees
are women.
What do you think? ... think statistically!
Test an appropriate hypothesis and state your conclusion.
Data only for this company, cannot generalize to others.
H0: p = 0.40
Ha: p < 0.40
one tail test, to left
z = -1.31
P-value = 0.0955, about 10%
The p-value is not small, it is large enough to not reject the
null hypothesis.
These data do not show that the proportion of women
executives is less than the 40% of women in the company in
general.
Text #22, page 529
According to the Association of American Medical
Colleges, only 46% of medical school applicants were
admitted to a medical school in the fall of 2006. Upon
hearing this, the trustees of Striving College expressed
concern that only 77 of the 180 students in their class of
2006 who applied to medical school were admitted. The
college president assured the trustees that this was just the
kind of year-to-year fluctuation in fortunes that is to be
expected and that, in fact, the school's success rate was
consistent with the national average. Who is right?
H0: p = 0.46
Ha: p < 0.46
a one-tail test
Check independence, random, 10%, np&nq >10
z = -0.87
P-value = 0.19 = 19%
The high P-value indicates that this is not an unusual result;
it may be just the year-to-year variation, as the president
says.