Intro to Hypothesis Testing
Q1. Framework for
weight loss
A group of people have volunteered to try
out a diet for weight loss for 3 months.
How should the null and alternate hypotheses be
set up?
A.
H0: Diet increases weight;
Ha: Diet has no impact on weight
B.
H0: Diet reduces weight; Ha:
Diet has no impact on weight
C.
H0: Diet has no impact on
weight; Ha: Diet reduces weight
D.
H0: Diet has no impact on
weight; Ha: Diet increases weight
Correct Answer: H0: Diet has no impact on weight; Ha: Diet
reduces weight
- H0 (the null hypothesis) typically represents the
status quo or no effect, while Ha (the alternative hypothesis) represents
the effect or change you want to test.
- In this case, the null hypothesis (H0) assumes that
the diet has no impact on weight, while the alternative hypothesis (Ha)
suggests that the diet reduces weight.
- This is the most appropriate setup to test whether
the diet has an effect on weight loss, as we need evidence/proof to
conclude if this diet does in fact aid in weight loss
2. Q2. Marketing the
shampoo brand
Weekly sales of shampoo bottles has an average of 1800 . A
marketing company feels that this can be improved with right advertisement and
promotions.
What should be the null and alternate hypothesis, in
order to
validate their claim?
Let μ denote the average sales after
marketing.
i). H0:μ≤1800 and Ha:μ=1800
ii). H0:μ<1800 and Ha:μ≥1800
iii). H0:μ=1800 and Ha:μ=1800
iv). H0:μ=1800 and Ha:μ>1800
A.
i)
B. ii)
C.
iii)
D. iv)
- The default assumption should be that the marketing
has no effect, and the claim of the marketing company should manifest in
the alternate hypothesis Since the default assumption here is that the
marketing has no effect, we set up the null hypothesis as H0:μ=1800 .
- Now the claim is that the sales will improve. Thus,
the alternate hypothesis here should be Ha:μ>1800 .
- In this scenario, the burden of proof is on the
marketing company to show evidence suggesting that advertisement and
promotions can in fact increase average shampoo sales.
Q3. Should we build the gymnasium?
A software company is planning to build
a gymnasium for
its employees. But before they plan to put their idea into action, they would
like to know the interest of their employees.
They plan to survey a
sample of their employees to see if there is strong evidence that more than 45% of
the employees are interested, in which case they will consider building the
gymnasium.
The hypotheses the company is using are:
H0 : There is not enough evidence to
suggest that more than 45% employees are interested in gymnasium (i.e. at most
45% are interested).
Ha : There is enough evidence to suggest
that more than 45% employees are interested in gymnasium.
Which of the following would be a Type II error here?
A.
More than 45% are actually
interested, an it's not concluded that more than 45% are interested.
B.
More than 45% are actually
interested, and it's concluded that more than 45% percent are interested.
C.
At most 45% are actually
interested, and it's concluded that more than 45% are interested.
D. At most 45% are actually interested, and they concluded
that less than 45% are interested.
HINT-I
Type I error occurs when we
reject a true null hypothesis, and type II occurs when we fail to reject a
false null hypothesis. Based on this, we can check which statement is actually
making us fail to reject a false null hypothesis.
Type I error occurs when we reject a true null hypothesis, and type II occurs when we fail to reject a false null hypothesis.
- "More than 45% are actually interested, and
it's not concluded that more than 45% are interested." is an example
of type II error, since null hypothesis is false, but we reject it.
- Similarly, "At most 45% are actually
interested, and it's concluded that more than 45% are interested." is
a Type I error, since null hypothesis is true, but we reject it.
Q4. Soft Drinks
A soft drink manufacturing company claims that
the volume
of drink of their bottles is 15 oz. A
consumer group suspects the bottles are under‐filled and plans to conduct a test.
What is the Type I error in
this situation?
A.
The consumer group has
evidence that the volume of the bottles is not 15 oz.
B.
The consumer group does not
conclude that the soft drink bottles have less than 15 oz. when the mean
actually is less than 15 oz.
C.
The consumer group concludes
that the soft drink bottles have less than 15 oz. when the mean actually is 15
oz.
D. The consumer group has evidence that the claim is
correct.
Here based on the question, we
define our hypothesis as:
- Null hypothesis: Mean volume of soft drink in bottle is 15 oz,
i.e. μ=15
- Alternate hypothesis: Mean volume of soft drink in bottle is not
equal to 15oz, i.e. μ < 15
Based on this, for the given
situation, the Type 1 error is when the consumer group concludes that the soft
drink bottles have less than 15 oz, when the mean actually is 15 oz.
Q5. Other name?
A data scientist is working on a credit scoring model for a bank.
The goal is to determine if an applicant is creditworthy (has a low credit
risk) or not creditworthy (has a high credit risk) based on various financial
factors.
The bank has set a credit score threshold (significance level), and
applicants above this threshold are considered creditworthy, while those below
it are considered not creditworthy.
Consider the following scenarios:
What names can we give to these cases respectively?
A.
True Negative, Type - 2
error, Type - 1 error, True Positive
B.
True Positive, Type - 2
error, Type - 1 error, True Negative
C.
Type - 2 error, True
Positive, True Negative, Type - 1 error
D. True Negative, Type - 1 error, Type - 2 error, True
Positive
Correct Option: True Negative, Type - 2 error, Type - 1
error, True Positive
if pvalue < threshold:
Reject H0
else:
Fail to reject H0
It is called that out that
“applicants above this threshold are considered creditworthy, while those below
it are considered not creditworthy.”, i.e.
if pvalue < threshold:
Not Creditworthy
else:
Creditworthy
Hence, it is indirectly given
in our question that the hypothesis is to be setup as:
- H0: Applicant is creditworthy
- Ha: Applicant is not creditworthy
So, based on this,
- Case A: The model correctly identifies the applicant
as not creditworthy, which is a True Negative.
- Case D: The model correctly identifies the applicant
as creditworthy, which is a True Positive.
Q6. Hypothesis and Conclusion
As a data scientist you are working for an
e-commerce company, and you want to determine if the introduction of a new
algorithm has led to an increase in
the average order value of customer purchases.
The significance level (α) is set at 0.05.
What would be the appropriate hypotheses and
conclusion to this situation?
H0: New algorithm has no effect on average order value
Ha: New algorithm leads to a higher average order value.
Conclusion: Introduction of the new algorithm has led to a significant increase in the average order value.
H0: New algorithm has no effect on average order value
Ha: New algorithm leads to a higher average order value.
Conclusion: Introduction of the new algorithm has not led to a increase in the average order value
H0: New algorithm leads to a higher average order value
Ha: New algorithm has no effect on average order value.
Conclusion: Introduction of the new algorithm has led to a significant increase in the average order value.
H0: New algorithm leads to a higher average order value
Ha: New algorithm has no effect on average order value.
Conclusion = Introduction of the new algorithm has not led to a increase in the average order value
A.
a
B.
b
C.
c
D. d
From historical data, it is known that the
mean weight of airline passengers with carry-on baggage is 175lb, and the
standard deviation is 5lb.
Which of the following would be the
most appropriate way
to test the
claim that the mean
weight of airline passengers with carry-on baggage
is at
most 195lb, with
a 95% confidence level?
A.
Two tailed test
B.
Left tailed test
C.
Right tailed test
D. None of the above.
- H0 = µ ≥ 195 (Greater than or Equal to
195lb)
- H1 = µ < 195 (At most 195lb)
Hence, based on the alternate
hypothesis, we can see that our test would one directional, towards the left.
Q8. Appropriate test
I. Is there
a difference in memory retention for individuals at the age of 20 compared to
their memory at age 60?
II. Do people
who take daily vitamin live longer than the people who don’t take ?
Which of the following would be the appropriate test for
the above two statements?
A.
I .Two tailed test , II. One
tailed test.
B.
I. Two tailed test , II. Two
tailed test.
C.
I. One tailed test , II. One
tailed test.
D.
I. One tailed test , II. Two
tailed test.
- H0: (memory retention)age=20 = (memory retention)age=60
- H1: (memory retention)age=20 ≠ (memory retention)age=60
Hence, we would need to use a
Two Tailed Test here.
- H0: (Life Span)Daily Vitamin = (Life Span)No Daily Vitamin
- H1: (Life Span)Daily Vitamin > (Life Span)No Daily Vitamin
Hence, we would need to use a
One Tailed Test here.
Q1. Frame work for GRE verbal reasoning
The verbal reasoning section in the GRE
exam, has an average score
of 150 and
a standard
deviation of 8.5.
A coaching centre claims to improve these
numbers for their students. How should the null and alternate hypotheses
be set up?
A.
H0: Coaching improves score;
Ha: Coaching does not improve score
B.
H0: Coaching reduces score;
Ha: Coaching improves score
C.
H0: Coaching does not improve
score; Ha: Coaching reduces score
D. H0: Coaching does not improve score; Ha: Coaching
improves score
Correct Answer: H0: Coaching does not improve score; Ha:
Coaching improves score
By default, we would assume
that a student’s score would conform to given distribution, irrespective of
whether are enrolled in coaching or not.
- Hence, this becomes our Null Hypothesis.
- i.e. H0: Coaching does not improve score
However, there is a coaching
center that claims to improve the scores.
- This means that they have burden of proof.
- Hence, this becomes our Alternate hypothesis.
- i.e. Ha: Coaching improves score.
Q2. Judge the right way
In a court case, the null hypothesis is that
the defendant is innocent.
Identify the Type-2 error among
the following.
A.
The defendant is innocent,
and the judge pronounces him innocent
B.
The defendant is innocent,
but the judge pronounces him guilty
C.
The defendant is guilty, and
the judge pronounces him guilty
D. The defendant is guilty, but the judge pronounces him
innocent
The null and alternate hypothesis are:
Q3. Ride the bike
When you ride your bike, the null
hypothesis H0 is that the
bike is safe to drive.
Which of these is a Type-1 error?
A.
The bike is not safe, but you
think it is safe
B.
The bike is not safe, and you
think it is not safe
C.
The bike is safe, but you
think it is not safe
D. The bike is safe, and you think it is safe
Q4. Appropriate Conclusion
You perform a one-tailed hypothesis test
with a significance level of 0.01.
If the p-value is 0.015, what is
the appropriate conclusion?
A.
Reject the null hypothesis
B.
Fail to reject the null
hypothesis
C.
Inconclusive result
D. P value is not reliable enough to draw conclusions
Correct Option: Fail to reject the null hypothesis
Explanation:
- In a hypothesis test, the p-value represents the
probability of observing the data, or something more extreme, under the
assumption that the null hypothesis is true.
- When the p-value is less than or equal to the chosen
significance level (α), it provides evidence to reject the null hypothesis
in favor of the alternative hypothesis.
- However, when the p-value is greater than the
significance level, there is not enough evidence to reject the null
hypothesis.
- In this case, since the p-value is 0.015 and the
significance level is 0.01, the appropriate conclusion is to “fail to
reject the null hypothesis.”
