Last time we have discussed hypothesis test for
two population proportions. And we mentioned that
the assumptions are for the first sample. n times
pi should be at least 5, and also n times 1 minus
pi is also at least 5. The same for the second
sample, n 2 times pi 2 is at least 5, as well as n
times 1 minus pi 2 is also at least 5. Also, we
discussed that the point estimate for the
difference of Pi 1 minus Pi 2 is given by P1 minus
P2. That means this difference is unbiased point
estimate of Pi 1 minus Pi 2. Similarly, P2 minus
P1 is the point estimate of the difference Pi 2
minus Pi 1.
We also discussed that the bold estimate for the
overall proportion is given by this equation. So B
dash is called the bold estimate for the overall
proportion. X1 and X2 are the number of items of
interest. And the two samples that you have in one
and two, where in one and two are the sample sizes
for the first and the second sample respectively.
The appropriate statistic in this course is given
by this equation. Z-score or Z-statistic is the
point estimate of the difference pi 1 minus pi 2
minus the hypothesized value under if 0, I mean if
0 is true, most of the time this term equals 0,
divided by this quantity is called the standard
error of the estimate, which is square root of B
dash 1 minus B dash times 1 over N1 plus 1 over
N2. So this is your Z statistic. The critical
regions. I'm sorry, first, the appropriate null
and alternative hypothesis are given by three
cases we have. Either two-tailed test or one
-tailed and it has either upper or lower tail. So
for example, for lower-tailed test, We are going
to test to see if a proportion 1 is smaller than a
proportion 2. This one can be written as pi 1
smaller than pi 2 under H1, or the difference
between these two population proportions is
negative, is smaller than 0. So either you may
write the alternative as pi 1 smaller than pi 2,
or the difference, which is pi 1 minus pi 2
smaller than 0. For sure, the null hypothesis is
the opposite of the alternative hypothesis. So if
this is one by one smaller than by two, so the
opposite by one is greater than or equal to two.
Similarly, but the opposite side here, we are
talking about the upper tail of probability. So
under the alternative hypothesis, by one is
greater than by two. Or it could be written as by
one minus by two is positive, that means greater
than zero. While for the two-tailed test, for the
alternative hypothesis, we have Y1 does not equal
Y2. In this case, we are saying there is no
difference under H0, and there is a difference.
should be under each one. Difference means either
greater than or smaller than. So we have this not
equal sign. So by one does not equal by two. Or it
could be written as by one minus by two is not
equal to zero. It's the same as the one we have
discussed when we are talking about comparison of
two population means. We just replaced these by's
by mus. Finally, the rejection regions are given
by three different charts here for the lower tail
test. We reject the null hypothesis if the value
of the test statistic fall in the rejection
region, which is in the left side. So that means
we reject zero if this statistic is smaller than
negative zero. That's for lower tail test. On the
other hand, for other tailed tests, your rejection
region is the right side, so you reject the null
hypothesis if this statistic is greater than Z
alpha. In addition, for two-tailed tests, there
are two rejection regions. One is on the right
side, the other on the left side. Here, alpha is
split into two halves, alpha over two to the
right, similarly alpha over two to the left side.
Here, we reject the null hypothesis if your Z
statistic falls in the rejection region here, that
means z is smaller than negative z alpha over 2 or
z is greater than z alpha over 2. Now this one, I
mean the rejection regions are the same for either
one sample t-test or two sample t-test, either for
the population proportion or the population mean.
We have the same rejection regions. Sometimes we
replace z by t. It depends if we are talking about
small samples and sigmas unknown. So that's the
basic concepts about testing or hypothesis testing
for the comparison between two population
proportions. And we stopped at this point. I will
give three examples, three examples for testing
about two population proportions. The first one is
given here. It says that, is there a significant
difference between the proportion of men and the
proportion of women who will vote yes on a
proposition?
In this case, we are talking about a proportion.
So this problem tests for a proportion. We have
two proportions here because we have two samples
for two population spheres, men and women. So
there are two populations. So we are talking about
two population proportions. Now, we have to state
carefully now an alternative hypothesis. So for
example, let's say that phi 1 is the population
proportion, proportion of men who will vote for a
proposition A for example, for vote yes, for vote
yes for proposition A.
is the same but of men, of women, I'm sorry. So
the first one for men and the other of
women. Now, in a random, so in this case, we are
talking about difference between two population
proportions, so by one equals by two.
Your alternate hypothesis should be, since the
problem talks about, is there a significant
difference? Difference means two tails. So it
should be pi 1 does not equal pi 2. Pi 1 does not
equal pi 2. So there's still one state null and
alternate hypothesis. Now, in a random sample of
36 out of 72 men, And 31 of 50 women indicated
they would vote yes. So for example, if X1
represents number of men who would vote yes, that
means X1 equals 36 in
172. So that's for men. Now for women. 31 out of
50. So 50 is the sample size for the second
sample. Now it's ask about this test about the
difference between the two population proportion
at 5% level of significance. So alpha is given to
be 5%. So that's all the information you have in
order to answer this question. So based on this
statement, we state null and alternative
hypothesis. Now based on this information, we can
solve the problem by using three different
approaches. Critical value approach, B value, and
confidence interval approach. Because we can use
confidence interval approach because we are
talking about two-tailed test. So let's start with
the basic one, critical value approach. So
approach A.
Now since we are talking about two-tailed test,
your critical value should be plus or minus z
alpha over 2. And since alpha is 5% so the
critical values are z
plus or minus 0.25 which is 196. Or you may use
the standard normal table in order to find the
critical values. Or just if you remember that
values from previous time. So the critical regions
are above 196 or smaller than negative 196. I have
to compute the Z statistic. Now Z statistic is
given by this equation. Z stat equals B1 minus B2.
minus Pi 1 minus Pi 2. This quantity divided by P
dash 1 minus P dash multiplied by 1 over N1 plus 1
over N1. Here we have to find B1, B2. So B1 equals
X1 over N1. X1 is given.
to that means 50%. Similarly,
B2 is A equals X2 over into X to the third power
over 50, so that's 60%. Also, we have to compute
the bold estimate of the overall proportion of B
dash
What are the sample sizes we have? X1 and X2. 36
plus 31. Over 72 plus 7. 72 plus 7. So that means
67 over 152.549.
120.
So simple calculations give B1 and B2, as well as
B dash. Now, plug these values on the Z-state
formula, we get the value that is this. So first,
state null and alternative hypothesis, pi 1 minus
pi 2 equals 0. That means the two populations are
equal. We are going to test this one against Pi 1
minus Pi 2 is not zero. That means there is a
significant difference between proportions. Now
for men, we got proportion of 50%. That's for the
similar proportion. And similar proportion for
women who will vote yes for position A is 62%. The
pooled estimate for the overall proportion equals
0.549. Now, based on this information, we can
calculate the Z statistic. Straightforward
calculation, you will end with this result. So, Z
start negative 1.31.
So, we have to compute this one before either
before using any of the approaches we have.
If we are going to use their critical value
approach, we have to find Z alpha over 2 which is
1 more than 6. Now the question is, is this value
falling the rejection regions right or left? it's
clear that this value, negative 1.31, lies in the
non-rejection region, so we don't reject a null
hypothesis. So my decision is don't reject H0. My
conclusion is there is not significant evidence of
a difference in proportions who will vote yes
between men and women. Even it seems to me that
there is a difference between Similar proportions,
50% and 62%. Still, this difference is not
significant in order to say that there is
significant difference between the proportions of
men and women. So based on the critical value
approach. We end with this result, which is we
don't reject null hypotheses. That means the
information you have is not sufficient in order to
support alternative hypotheses. So your managerial
conclusion should be there is not significant
difference in proportion and proportions who will
vote yes between men and women. That's for using
critical value approach. Before continue, we have
to discuss the confidence interval for the
difference pi 1 minus pi 2. The confidence
interval, as we mentioned before, can be
constructed by point estimate, plus or minus
critical value times the standard error of the
point estimate. In this case, the point estimate
for pi 1 minus pi 2 is b1 minus b2. So that's your
point estimate, plus or minus z alpha over 2. Now
from the information from chapter 8, the standard
error of the difference, b1 minus pi 2, is given
by this equation. B1 times 1 minus B1, so B1 and
its complement, divided by the first sample size,
plus the second sample proportion times its
complement divided by the sample size of the
second sample. So that's your confidence interval.
So it looks similar to the one we have discussed
for the mu 1 minus mu 2. And that one we had x1
bar minus x2 bar plus or minus z or t, it depends
on the sample sizes, times s square b times 1 over
n1 plus 1 over n2. Anyway, the confidence interval
for pi 1 minus pi 2 is given by this equation. Now
let's see how can we use the other two approaches
in order to test if there is significant
difference between the proportions of men and
women. I'm sure you don't have this slide for
computing B value and confidence interval.
Now since we are talking about two-thirds, your B
value should be the probability of Z greater than
1.31 and smaller than negative 1.31. So my B value
in this case equals Z greater than 1.31 plus Z
smaller than negative 1.31. Since we are talking
about two tail tests, so there are two rejection
regions. My Z statistic is 1.31, so it should be
here 1.31 to the right, and negative. Now, what's
the probability that the Z statistic will fall in
the rejection regions, right or left? So we have
to add. B of Z greater than 1.31 and B of Z
smaller than negative. Now the two areas to the
right of 1.31 and to the left of negative 1.31 are
equal because of symmetry. So just compute one and
multiply that by two, you will get the B value. So
two times. Now by using the concept in chapter
six, easily you can compute either this one or the
other one. The other one directly from the
negative z-score table. The other one you should
have the complement 1 minus, because it's smaller
than 1.1. And either way you will get this result.
Now my p-value is around 19%. Always we reject the
null hypothesis. if your B value is smaller than
alpha that always we reject null hypothesis if my
B value is smaller than alpha alpha is given 5%
since B value equals
19% which is much bigger than Much greater than
5%, so we don't reject our analysis. So my
decision is we don't reject at zero. So the same
conclusion as we reached by using critical
penalty. So again, by using B value, we have to
compute the probability that your Z statistic
falls in the rejection regions. I end with this
result, my B value is around 19%. As we mentioned
before, we reject null hypothesis if my B value is
smaller than alpha. Now, my B value in this case
is much, much bigger than 5%, so my decision is
don't reject null hypothesis. Any questions?
The other approach, the third one, confidence
interval approach.
Now, for the confidence interval approach, we have
this equation, b1 minus b2. Again, the point
estimate, plus or minus z square root b1 times 1
minus b1 divided by a1. B2 times 1 minus B2
divided by N2.
Now we have B1 and B2, so 0.5 minus 0.62. That's
your calculations from previous information we
have. Plus or minus Z alpha over 2, the critical
value again is 1.96 times Square root of P1.5
times 1 minus 0.5 divided by N1 plus P2 62 percent
times 1 minus P2 divided by N2. 0.5 minus 62
percent is negative 12 percent plus or minus the
margin of error. This amount is again as we
mentioned before is the margin of error is 0.177.
Now simple calculation will end with this result
that is The difference between the two proportions
lie between negative 0.296 and 0.057. That means
we are 95% confident that the difference between
the proportions of men who will vote yes for a
position A and men equals negative 0.297 up to 0
.057. Now the question is since we are testing
it's zero by one minus by two equals zero the
question is does this interval contain zero or
capture zero? Now since we start here from
negative and end with positive, I mean the lower
bound is negative 0.297 and the upper bound is 0
.057. So zero inside the interval, I mean the
confidence interval contains zero in this case, so
we don't reject the null hypothesis because maybe
the difference equals zero. So since this interval
does contain the hypothesis difference zero, so we
don't reject null hypothesis at 5% level. So the
same conclusion as we got before by using critical
value approach and de-value approach. So either
one will end with the same decision. Either reject
or fail to reject, it depends on the test itself.
That's all. Do you have any question? Any
question? So again, there are three different
approaches in order to solve this problem. One is
critical value approach, the standard one. The
other two are the value approach and confidence
interval. One more time, confidence interval is
only valid for
two-tailed test. Because the confidence interval
we have is just for two-tailed test, so it could
be used only for testing about two-tailed test.
As we mentioned before, I'm going to skip
hypothesis for variances as well as ANOVA test. So
that's all for chapter ten.
But now I'm going to do some of the practice
problems.
Chapter 10. To practice, let's start with some
practice problems for Chapter 10.
A few years ago, Pepsi invited consumers to take
the Pepsi challenge. Consumers were asked to
decide which of two sodas, Coke or Pepsi. They
preferred an applied taste test. Pepsi was
interested in determining what factors played a
role in people's taste preferences. One of the
factors studied was the gender of the consumer.
Below are the results of the analysis comparing
the taste preferences of men and women with the
proportions depicting preference in or for Pepsi.
For meals, size
of 109. So that's your N1. And proportion.
for males is around 4.2. For females,
N2 equals 52, and proportion of females, 25%. The
difference between proportions of men and women or
males and females is 0.172, around 0.172. And this
statistic is given by 2.118, so approximately 2
.12. Now, based on this result, based on this
information, question number one,
To determine if a difference exists in the test
preferences of men and women, give the correct
alternative hypothesis that lives through a test.
A. B. Why B? Because the test defines between the
new form A and the new form B. Because if we say
that H1 is equal to U1 minus M equals F,
So the correct answer is B? B. So that's
incorrect. C. Why? Why C is the correct answer?
Because
Y is not equal because we have difference. So
since we have difference here, it should be not
equal to. And since we are talking about
proportions, so you have to ignore A and B. So A
and B should be ignored first. Then you either
choose C or D. C is the correct answer. So C is
the correct answer. That's for number one. Part
two. Now suppose Pepsi wanted to test to determine
if males preferred Pepsi more than females. Using
the test statistic given, compute the appropriate
p-value for the test. Let's assume that pi 1 is
the population proportion for males who preferred
Pepsi, and pi 2 for females who prefer Pepsi. Now
he asks about suppose the company wanted to test
to determine if males prefer Pepsi more than
females. Using again the statistic given which is
2.12 for example, compute appropriately value. Now
let's state first H0 and H8.
H1 pi 1
minus pi 2 is greater than zero.
Because it says that males prefer Pepsi more than
females. Bi-1 for males, Bi-2 for females. So I
should have Bi-1 greater than Bi-2 or Bi-1 minus
Bi-2 is positive. So it's upper case. Now, in this
case, my B value, its probability is B.
It's around this value.
1 minus b of z is smaller than 2.12. So 1 minus,
now by using the table or the z table we have.
Since we are talking about 2, 1, 12, 2, 1, 2, I'm
sorry, 2, 1, 2, 2, 1, 2, so the answer is 983. So
1 minus 893, so the answer is 017. So my b value.
equals 0 and 7. So A is the correct answer. Now if
the problem is two-tailed test, it should be
multiplied by 2. So the answer, the correct should
be B. So you have A and B. If it is one-third,
your correct answer is A. If it is two-thirds, I
mean if we are testing to determine if a
difference exists, then you have to multiply this
one by two. So that's your B value. Any question?
Number three. Suppose Babs wanted to test to
determine if meals If males prefer Pepsi less than
females, using the statistic given, compute the
product B value. Now, H1 in this case, B1 is
smaller than Z by 2, by 1 smaller than 1. Now your
B value, Z is smaller than, because here it is
smaller than my statistic 2.12.
We don't write negative sign. Because the value of
the statistic is 2.12. But here we are going to
test lower tail test. So my B value is B of Z
smaller than. So smaller comes from the
alternator. This is the sign under the alternator.
And you have to take the value of the Z statistic
as it is.
So B of Z is smaller than minus 3. so they need if
you got a correct answer D is the correct if B is
the correct answer you will get nine nine nine six
six that's incorrect answer any question the
correct is D number
four suppose
that Now for example, forget the information we
have so far for B value. Suppose that the two
-tailed B value was really 0734. Now suppose my B
value for two-tailed is 0734. That's for two
-tailed. This is my B value.
This is my B value. It's 0, 7, 3, 4. Now we have
four answers. Part A, B, C, and D. Which one is
the correct?
A says at 5% level, there is sufficient evidence
to conclude the proportion of males Preferring
Pepsi differs from the proportion of females
preferring Pepsi. Which one is the correct answer?
B value is 0.734.
B it says at alpha equals 10 percent. There is
sufficient evidence. to indicate the proportion of
males preferring Pepsi differs from the proportion
of females preferring Pepsi.
C. At 5%, there is sufficient evidence to indicate
the proportion of males preferring Pepsi equals
the proportion of females preferring Pepsi. D. At
8% level, there is insufficient evidence to
include to indicate the proportion of males
preferring babies differs from the proportion of
females preferring babies.
Again, suppose that here it's two-tailed test. It
says two-tailed test. Two-tailed means Y1 does not
equal Y2. So in this case, we are testing Y1
equals Y2. against by one is not by two and your B
value is zero seven three four. So which one is
the correct answer? B? D. Let's look at D. Let's
look at D.
Since B value is smaller than alpha, Since it
means we reject Insufficient means we don't reject
So D is incorrect D
is incorrect because here there is insufficient
Since
we if we reject it means that we have sufficient
evidence so support The alternative so D is
incorrect Now what's about C at five percent Five,
so this value is greater than five, so we don't
reject. So that's incorrect.
B. At five, at 10% now, there is sufficient
evidence. Sufficient means we reject. We reject.
Since this B value, 0.7, is smaller than alpha. 7%
is smaller than 10%. So we reject. That means you
have to read carefully. There is sufficient
evidence to include, to indicate the proportion of
males preferring Pepsi differs from the proportion
of females. That's correct. So B is the correct
state. Now look at A. A, at 5% there is sufficient
evidence? No, because this value is greater than
alpha, so we don't reject. For this one. Here we
reject because at 10% we reject. So B is the
correct answer. Make sense? Yeah, exactly, for
10%. If this value is 5%, then B is incorrect.
Again, if we change this one to be 5%, still this
statement is false.
It should be smaller than alpha in order to reject
the null hypothesis. So, B is the correct
statement.
Always insufficient means you don't reject null
hypothesis. Now for D, we reject null hypothesis
at 8%. Since this value 0.7 is smaller than alpha,
so we reject. So this is incorrect. Now for C, be
careful. At 5%, if this, if we change this one
little bit, there is insufficient evidence. What
do you think? About C. If we change part C as at 5
% there is insufficient evidence to indicate the
proportion of males preferring Pepsi equals.
You cannot say equal because this one maybe yes
maybe no you don't know the exact answer. So if we
don't reject the null hypothesis then you don't
have sufficient evidence in order to support each
one. So, don't reject the zero as we mentioned
before. Don't reject the zero does not imply
if zero is true. It means the evidence, the data
you have is not sufficient to support the
alternative evidence. So, don't say equal to. So
say don't reject rather than saying accept. So V
is the correct answer.
Six, seven, and eight. Construct 90% confidence
interval, construct 95, construct 99. It's
similar, just the critical value will be changed.
Now my question is, which is the widest continence
interval in this case? 99. The last one is the
widest because here 99 is the largest continence
limit. So that means the width of the interval is
the largest in this case.
For 5, 6 and 7. The question is construct either
90%, 95% or 99% for the same question. Simple
calculation will give the confidence interval for
each one. My question was, which one is the widest
confidence interval? Based on the C level, 99%
gives the widest confidence interval comparing to
90% and 95%. The exact answers for 5, 6 and 7, 0.5
to 30 percent. For 95 percent, 0.2 to 32 percent.
For 99, negative 0.3 to 0.37. So this is the
widest. Because here we start from 5 to 30. Here
we start from lower than 5, 2%, up to upper, for
greater than 30, 32. Here we start from negative 3
% up to 37. So this is the widest confidence
interval. Number six. Number six. number six five
six and seven are the same except we just share
the confidence level z so here we have one nine
six instead of one six four and two point five
seven it's our seven six next read the table e
Table A. Corporation randomly selects 150
salespeople and finds that 66% who have never
taken self-improvement course would like such a
course. So currently, or in recent,
It says that out of 150 sales people, find that 66
% would
like to take such course. The firm did a similar
study 10 years ago. So in the past, they had the
same study in which 60% of a random sample of 160
salespeople wanted a self-improvement course. So
in the past,
into 160, and proportion is 60%. The groups are
assumed to be independent random samples. Let Pi 1
and Pi 2 represent the true proportion of workers
who would like to attend a self-improvement course
in the recent study and the past study
respectively. So suppose Pi 1 and Pi 2. Pi 1 for
recent study and Pi 2 for the past study. So
that's the question. Now, question number one.
If the firm wanted to test whether this proportion
has changed from the previous study, which
represents the relevant hypothesis?
Again, the firm wanted to test whether this
proportion has changed. From the previous study,
which represents the relevant hypothesis in this
case?
Which is the correct? A is
the correct answer. Why A is the correct answer?
Since we are talking about proportions, so it
should have pi. It changed, it means does not
equal 2. So A is the correct answer. Now B is
incorrect because why B is incorrect? Exactly
because under H0 we have pi 1 minus pi 2 does not
equal 0. Always equal sign appears only under the
null hypothesis. So it's the opposite here. Now C
and D talking about Upper tier or lower tier, but
here we're talking about two-tiered test, so A is
the correct answer.
This sign null hypothesis states incorrectly,
because under H0 should have equal sign, and for
alternate it should be not equal to.
Number two.
If the firm wanted to test whether a greater
proportion of workers would currently like to
attend a self-improvement course than in the past,
currently, the proportion is greater than in the
past. Which represents the relevant hypothesis? C
is the correct answer. Because it says a greater
proportion of workers work currently. So by one,
greater than by two. So C is the correct answer.
It says that the firm wanted to test proportion of
workers currently study
or recent study by one represents the proportion
of workers who would like to attend the course so
that's by one greater than
In the past. So it means by one is greater than by
two. It means by one minus by two is positive. So
the alternative is by one minus two by two is
positive. So this one is the correct answer.
Exactly. If if here we have what in the past
should be it should be the correct answer.
That's to three. Any question for going to number
three? Any question for number two? Three. What is
the unbiased point estimate for the difference
between the two population proportions?
B1 minus B2 which is straight forward calculation
gives A the correct answer. Because the point
estimate in this case is B1 minus B2. B1 is 66
percent, B2 is 60 percent, so the answer is 6
percent. So B1 minus B2 which is 6 percent. I
think three is straightforward. Number four, what
is or are the critical values which, when
performing a z-test on whether population
proportions are different at 5%. Here, yes, we are
talking about two-tailed test, and alpha is 5%. So
my critical values, they are two critical values,
actually.
What is or are the critical values when testing
whether population proportions are different at 10
%? The same instead here we have 10 instead of 5%.
So A is the correct answer. So just use the table.
Now for the previous one, we have 0 to 5, 0 to 5.
The other one, alpha is 10%. So 0, 5 to the right,
the same as to the left. So plus or minus 164.
So 4 and 5 by using the z table.
So exactly, since alpha here is 1, 0, 2, 5, so the
area becomes smaller than, so it should be z
greater than. So 1.106, the other one 1.645,
number 6. What is or are? The critical value in
testing whether the current population is higher
than. Higher means above. Above 10. Above 10, 5%.
So which? B.
B is the correct. Z alpha.
So, B is the correct answer.
7.
7 and 8 we should have to calculate number 1. 7
was the estimated standard error of the difference
between the two sample proportions. We should have
a standard error.
Square root, B dash 1 minus B dash multiplied by 1
over N1 plus 1 over N2. And we have to find B dash
here. Let's see how can we find B dash.
B dash
equal x1 plus x2. Now what's the value of X1?
Exactly. Since B1 is X1 over N1. So that means X1
is N1 times B1. So N1 is 150 times 60%. So that's
99. And similarly, X2 N2, which is 160, times 60%
gives 96. So your B dash is x1 plus x2 divided by
N1 plus N2, which is 150 plus 310. So complete B
dash versus the bold estimate of overall
proportion So 9 and 9 plus 9 is 6.
That's just B-.
6 to 9.
6 to 9. So this is not your answer. It's just B-.
Now take this value and the square root of 6 to 9.
times 1.629 multiplied by 1 over N1 which is 150
plus 160. That's your standard error. B dash is
not standard error. B dash is the bold estimate of
overall proportion. Now simple calculation will
give C. So C is the correct answer.
What's the standard error of the difference
between the two proportions given by this
equation? Here first we have to compute P' by
using x1 plus x2 over n1 plus n2. In this example,
the x's are not given, but we have the
proportions. And we know that B1 equals X1 over
N1. So X1 equals N1 times B1. So I got 99.
Similarly, X2 and 2 times B2 is 96. So B dash is
629. So plug this value here, you will get 055.
What's the value that is satisfactory to use in
evaluating the alternative hypothesis? That there
is a difference in the two population proportions.
So we have to compute Z score, Z stat, which is V1
minus V2, which is 6%, minus 0, divided by this
amount, 0.55. Now, 0.6 over 0.5 around 1. Six over
five, so the answer is one. So that's my Z
statistic.
That's number eight.
So the answer is C is the correct answer. So
straightforward calculations for C and D gives C
correct answer for both seven and eight.
So C is correct for each one. Now 9.
In 9, the company tests to determine at 5% level
of significance whether the population proportion
has changed. from the previous study. As it
changed, it means we are talking about two-tiered
tests. Which of the following is most correct? So
here we are talking about two-tiered tests and
keep in mind your Z statistic is 1.093. And again,
we are talking about two-tiered tests. So my
rejection regions are
Negative 196, critical values I mean. So the
critical regions are 1.96 and above or smaller
than minus 1.96. Now, my z statistic is 1.903. Now
this value falls in the non-rejection region. So
we don't reject the non-hypothesis.
Ignore A and C, so the answer is either B or D.
Now let's read B. Don't reject the null and
conclude that the proportion of employees who are
interested in self-improvement course has not
changed.
That's correct. Because we don't reject the null
hypothesis. It means there is no significant
difference. So it has not changed. Now, D, don't
reject the null hypothesis and conclude the
proportion of Obliques who are interested in a
certain point has increased, which is incorrect.
So B is the correct answer. So again, since my Z
statistic falls in the non-rejection region, we
don't reject the null hypothesis. So either B or D
is the correct answer. But here we are talking
about none or don't reject the null hypothesis.
That means we don't have sufficient evidence
support that there is significant change between
the two proportions. So there is no difference. So
it has not changed. It's the correct one. So you
have to choose B. So B is the most correct answer.
Now, 10, 11, and 12. Talking about constructing
confidence interval 99, 95, and 90%. It's similar.
And as we mentioned before, 99% will give the
widest confidence interval. And the answers for
these are 14, 11, 14, is negative 0.8 to 20%. For
11, 0.5, negative 0.5 to 17. For 90%, negative 0.3
to 0.15. So this is the widest confidence
interval, which was for 99%. So similar as the
previous one we had discussed. So for 99, always
we get The widest confidence interval. Any
question? That's all. Next time shall start
chapter 12, Chi-square test of independence.