Hypothesis Testing: Population Mean and Proportion

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Hypothesis Testing Population Mean and
Proportion

• Chapter 9

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Chapter 9 - Learning Objectives

• Conduct a HYPOTHESIS TEST for a SINGLE population
  MEAN or PROPORTION.
• Describe the logic of developing the null and
  alternative hypotheses.
• Describe what is meant by Type I and Type II
  errors.
• Test a hypothesis about population mean, when
  variance of the population is known
• Test a hypothesis about population mean, when
  variance of the population is unknown
• Test a hypothesis about population proportion
• Explain the relationship between confidence
  intervals and hypothesis tests.

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Introduction

• Hypothesis
• A belief/presumption/claim/assertion about
  something which can be rejected or accepted.
• A statement that could be accepted or refuted
  subject to the presence or lack of enough
  statistical evidence (data).

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Introduction

• Hypothesis Testing
• Is a general procedure of making inference about
  population parameter.
• The main purpose of hypothesis testing is to
  determine whether there is enough statistical
  evidence in favor of a certain belief about a
  parameter.
• Enables us to determine whether the belief about
  the population parameter is supported by the data
  (evidence).

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Introduction

• Examples
• Developing a drug requires some minimum level of
  efficacy.
• In a random sample of patients (half of whom are
  treated with a particular drug and the other half
  are given a placebo), is there enough evidence
  that support the belief that the drug is
  effective in curing a certain disease?

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Introduction

• Examples
• Entrepreneurs who develop new products want to
  know what proportion of potential customers could
  buy their products. Consider that a purchase of
  the product by 10 of the consumers is the
  minimum threshold.
• In a random sample of potential customers, is
  there enough statistical evidence that support
  the view that more than 10 of the potential
  customers will purchase a new products?

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Introduction

• Answering these kind of day to day real world
  problems involve testing the available
  wisdom/belief (hypothesis).
• If sufficient statistical evidence is not
  available, implementation of a new idea, product
  and policy is difficult.

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Introduction

• Two Types of Hypothesis
• 1.The Null hypothesis (H0)
• The belief or statement about something
• It is what is typical of the population. It is a
  term that represents the business as usual
  idea, where nothing out of the ordinary occurs.
• 2. The Alternative Hypothesis (H1)
• The alternative to the belief/statement
• It is what is being considered as the challenge
  to the existing idea. The view of about the
  population characteristic, that if it true, would
  trigger some new action, some change in
  procedures that had previously defined business
  as usual.

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The Null Hypothesis Note

• Non Directional, two-tail test
• The alternative is formulated in such a way that
  the response is non-directional (could be greater
  or lower).
• Directional (One tailed test)
• The alternative to the null hypothesis is
  specified in such a way that it is greater than
  what is stated in the null ( right-tailed test)
• The alternative to the null is specified in such
  a way that it is lower than what is sated in the
  null ( left-tailed test)

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Introduction

• Example
• A person is accused of committing a crime.
• Plaintiffs/Prosecutors bring the case to a court.
  
• Jury is selected and evidence is presented.
• Based on the evidence, not the accusation, jurors
  either convict or acquit the person.
• When the trial begins
• What would the jurors are expected to presume
  about the defendant?
• What would the plaintiff/prosecutors want the
  jurors to believe?
• What would the plaintiff/prosecutors need to
  provide to the jury?

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Example-1

• Two Types of Hypothesis
• H0 The defendant is innocent
• (Business as usual)
• 2. H1 The defendant is guilty as charged

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Example-2

• Two Types of Hypothesis
• H0 The new drug is not effective
• (Business as usual)
• 2. H1 The new drug is effective
• (challenge to the business as usual view)

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Introduction

• Jury decision
• 1. Convict the Defendant
• Find the defendant GUILTY. I.e., there is enough
  evidence to support the claim.
• Rejecting the null hypothesis H0
• 2. Acquit the defendant
• Find the defendant NOT GUILTY as charged. I.e.,
  No sufficient evidence is presented.
• Fail to reject the null hypothesis H0

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Two Potential Errors While Testing Hypothesis
State of Reality
H0 True
H0 False
H0 isTrue
Test Result Says
H0 isFalse
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Introduction

• Two Types of Errors while hypothesis testing
• Type-I error (a.)
• Rejecting a true null hypothesis. The
  probability of rejecting a true null hypothesis
  is also called significance level.
• Wrongly convicting an innocent person
• Type-II error (ß)
• Failing to reject a False null hypothesis. ß
  represents, the probability of failing to reject
  a false null hypothesis
• Acquitting a guilty defendant.
• Note that the two are inversely related.
  Minimizing one will lead to the maximization of
  the other error.

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9.2 Hypothesis Testing in practice

• Example
• Three highway patrol officers were assigned to
  check whether the average speed of cars traveling
  on certain stretch of a highway matches the
  posted limit of 55 mph.
• One officer claims that on the average the speed
  at which cars travel on the stretch is greater
  than the posted limit of 55 mph.

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9.2 Hypothesis Testing in practice

• Example
• Three highway patrol officers were assigned to
  check whether the average speed of cars traveling
  on certain stretch of a highway matches the
  posted limit of 55 mph.
• The other officer claims the contrary The speed
  at which cars travel on the stretch is less than
  the posted limit of 55 mph.

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9.2 Hypothesis Testing in practice

• Example
• Three highway patrol officers were assigned to
  check whether the average speed of cars traveling
  on certain stretch of a highway matches the
  posted limit of 55 mph.
• The third officer claims that on the average the
  speed at which cars travel on the stretch is
  different from the posted limit of 55 mph.

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9.2 Hypothesis Testing in practice

• Example
• To test these claims a random sample of 200 cars
  traveling over the same stretch of highway was
  taken. Results from the sample indicate the
  average speed of cars travelling on the stretch
  of the highway is 56mph. It is known that the
  standard deviation for the entire population of
  cars traveling over the stretch is just 5 mph.
• At 5 level of significance, test if each of the
  officers claims can be supported by enough
  statistical evidence.

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9.2 Hypothesis Testing in practice

• Developing a Hypothesis
• We want to know whether the mean speed of cars
  traveling on the highway is greater than 55 as
  stated by the second officer.
• As he trusts his officers, the sheriff wants to
  maintain each of the officers claim. However, he
  needs to know if there is sufficient information
  (evidence) to support either of the claims.

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9.2 Hypothesis Testing in practice

• Step-1 Determine the null and the alternative
  hypothesizes.
• In the null hypothesis, always we should specify
  a single value of the population parameter (in
  this case, the average speed of all cars
  traveling on the stretch, m).

H0 m 55
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9.2 Hypothesis Testing in practice

• Step-1 Determine the null and the alternative
  hypothesizes.
• In the alternative hypothesis, we state the
  belief we want to maintain (in this case,
  officers claim) provided that there is
  sufficient information that enables us to reject
  the null.

H0 m 55
H1 m gt 55
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9.2 Hypothesis Testing in practice

• H0 µ55
• H1 µgt55
• N200 Sample mean56 Std. Deviation5 a0.05

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Approaches to Testing

• There are two methods (approaches) to determine
  (test) whether or not there is enough information
  in the sample to support the hypothesis we want
  to maintain i.e., alternative hypothesis (H1).
• The (Critical Value or Rejection region method).
• The p-value method.

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The Rejection Region Method
This is a method in which we establish range of
values (critical value) such that if the test
statistic falls into the region, we reject the
null hypothesis in favor of the alternative
hypothesis.
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The Rejection Region Method for a Right - Tail
Test
Reject H0 here

• If the sample mean lies in the rejection
  region, then there is
• sufficient (enough) evidence to support the
  alternative
• claim(H1). That is, to reject the null(H0)

Critical value of the sample mean
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Step-2Compute the Test Statistic

• If information about population SD is available,
  and the distribution of the population is
  considered normal, then the test statistic is
  estimated as follows

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The standardized test statistic

• Using the information from the sample, we
  calculate the standardized test statistic as
  follows

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Step-3 Establish the Critical Value

• Establish the critical value (rejection region),
  the value based on which we want to say yes,
  there is sufficient information to support the
  hypothesis that we want to keep

H0 m 55
H1 m gt 55
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Step-3 Establish the Critical Value

• To establish the critical value, we must first
  determine the acceptable margin of error (the
  error that we are will to take in making the
  decision).
• In our case, it was 5

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Step-3 Establish the Critical Value

• For the 5(one sided) margin of error, the
  critical value is 1.645

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Step-4 Compare the Test statistic with
Established Critical value

• Compare the test statistic with established
  critical value, and make the appropriate decision

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Step-4 The Rejection Region Method- Compare the
Test statistic with Established Critical value

• Reject H0 if

Decision Rule Reject the null hypothesis in
favor of the alternative if the calculated value
of Z is greater than the critical value (Z at the
chosen level of significance)
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• Using the information from the sample, we
  computed the test statistic to be

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• Conclusion

• The calculated value of Z (2.83) is greater than
  the critical value(1.645). Thus we reject the
  null hypothesis.

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Step-5 Make a Conclusion

• If the computed sample mean falls in the
  rejection region, then we reject the null
  hypothesis. If we do so, it means that there is
  sufficient evidence that supports the idea we
  want to maintain (the one stated in the
  alternative hypothesis).
• Thus we can say that the officers claim that on
  average people are driving above the speed limit
  is valid.

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Repeat the procedure for each of the other
officers claim as well

• In each case formulate the null and the
  alternative decide on the level of significance
  (margin of error) at which you want to test the
  hypothesis
• Compute the test statistic and the critical
  values
• Make business decision (reject or fail to reject
  the null hypothesis and indicate the implication

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The P value method

• As an alternative to testing a hypothesis using
  the rejection region method, we can also use the
  p-value method.
• The p-value provides information about the amount
  of statistical evidence that supports the
  alternative hypothesis.

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The p-value Method
What is P-Value? It is the probability that
we reject the null hypothesis when it is
true. In other words, it is a method in which
we test the probability of observing a test
statistic at least as extreme as the one
computed, given that the null hypothesis is true.
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The p-value Method
Decision Rule when using the p-value
method Reject the null hypothesis (H0) only
if the P-Value is LESS THAN the level of
significance (a).
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10.2 Hypothesis Testing in practice

• Example
• In a a random sample of 200 cars traveling over
  the same stretch of high way, the average speed
  of cars is found to be 56mph. It is known that
  the standard deviation for the entire population
  of cars traveling over the stretch is just 5 mph,
  and the posted speed limit is 55mph.
• At 5 level of significance, using the p-value
  method test if cars are traveling faster than the
  speed limit.

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The p-value Method

• To use the p-value method
• Formulate the null and alternative hypothesis
• Compute the test statistic (Z calculated)
• Depending up on H1, find the P(ZgtZcal)
• P-Value P(Zgt2.83) 0.500-p(0ltzlt2.83)
• 0.500-0.4977
• 0.0023
• 4. Compare the p-value with the level of
  significance
• 5. Reject the null hypothesis only if the
  p-value is less than significance level
• 6. Make the appropriate inference

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The p-value Method
Decision P-value(0.0023) lt 0.05 (significance
level). So we reject the null hypothesis. What
does 0.0023 mean?
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Hands-On-Problem

• In the past, patrons of a cinema complex have
  spent an average of 2.50 for popcorn and other
  snacks, with a standard deviation of 0.90.
• It is known that the amount of expenditure is
  normally distributed. After an intensive public
  campaign about the health effects of popcorn, the
  mean expenditure of a sample of 18 patrons is
  found to be 2.10.
• At 5 significance level, test if this recent
  experience suggest a decline in spending?

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Hands-On-Problem

• Formulate the null and the alternative
  hypothesizes for the problem.
• Establish and the critical values in terms of the
  standardized (Z) values and test the hypothesis
  using the rejection region method.
• Compute the P-Value and test hypothesis using the
  p-value method.
• Interpret your results.

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Hands-On-Problem

• Using the rejection Region Method
• Formulate the Null and Alternative Hypothesis
• H0 µ2.50
• H1 µ lt2.50
• Z cal (test statistic)
• Z table Z0.05-1.645
• Decision
• Z cal gt Z table Thus reject H0.
• Conclusion
• There is enough evidence that supports the view
  that the public campaign has worked effectively
  (The campaign has resulted in reduced consumption
  of Pop Corn)

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Hands-On-Problem

• Using the p-Value method
• H0 µ2.50
• H1 µ lt2.50
• Z cal (test statistic)
• P(Zlt-1.89) P(Zgt1.89) 0.5000-0.4706 0.0294
• Level of significance is 0.05
• Decision
• P value0.0294 lt 0.05 . Thus
  reject H0.
• Conclusion
• There is enough evidence that supports the view
  that the public campaign has worked effectively
  (Has resulted in reduced consumption of Pop Corn)

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Hypothesis test-types

• The test methods we established can be classified
  in to two
• Two-Tail test
• We use two tail test when the alternative
  hypothesis asserts just a difference.
• Example The claim of officer C
• One-Tail test
• We use a one tail (left or right) test when the
  alternative asserts that the variable of interest
  is higher (right) or lower (left) than the value
  specified in the null hypothesis.
• 2.1. Right tail test (case of officer A)
• 2.2. Left tail test (case of officer B)

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Testing Proportions

• We apply the same procedure we used the previous
  examples.
• However, it is important to note the following
  when testing the proportions.
• 1. We apply only the Z distribution when testing
  sample proportion
• To apply the Z test npgt5 and n(1-p)gt5
• 3. Both rejection region and p-value method can
  be used to test hypothesis about proportions

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Interpreting the p-value

• If the p-value is less than 1, there is evidence
  that supports the alternative hypothesis. (at the
  significance level of 1 or more) Overwhelming

• If the p-value is between 1 and 5, there is a
  evidence that supports the alternative hypothesis
  (at 5 or more significance level) Strong
• If the p-value is between 5 and 10 there is a
  evidence that supports the alternative
  hypothesis. (at 10 ore more) Weak
• If the p-value exceeds 10, there is no evidence
  that supports the alternative hypothesis, unless
  we choose a margin of error that exceeds 10

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Testing Proportions

• Example
• An auto repair shop owner claims that no less
  than 70 of his customers are satisfied with his
  work. However, a survey of 150 customers reveals
  that only 66 are satisfied with the work
  performed. At 5 level of significance, can we
  conclude that less than 70 of the customers are
  satisfied?

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Practice-Problem

• The no smoking regulations in office buildings
  require workers who smoke to take a break and
  leave the building in order to satisfy their
  habits.
• A study indicates that such workers average 32
  minutes per day taking smoking breaks.
• To help reduce the average, break rooms with
  powerful exhausts were installed in the
  buildings. To check whether these rooms served
  their purpose, a random sample of 110 smokers
  produced a mean break time of 29.92 minutes. The
  sample standard deviation is 6 minutes.
• Is it possible to say that there has been a
  decrease in the amount of time smokers take for
  break? Test the hypothesis at 5 level of
  significance.

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Testing a Hypothesis about the mean of the
population-- Variance of the population is unknown

• We follow similar procedure that we have used in
  testing a hypothesis about the mean when variance
  of the population is known, except that we use a
  t-distribution instead of a Z-distribution

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Steps to solve such a Problem

• Formulate the null and the alternative
  hypothesizes.
• Establish the test statistic (calculated t) using
  the information from the sample and the critical
  value (table value of t) at the n-1 degrees of
  freedom and selected level of significance
• If using the rejection region method, compare the
  calculated t with the critical value of t, and
  reject the null hypothesis if the calculated
  value is greater than the table value.
• 4. If using the P-Value method, compute the
  p-value using the test statistic and reject the
  null hypothesis if the p-value is less than the
  significance level selected.
• 5. Make the appropriate inference depending up
  on the results from the test.