Introduction to Hypothesis Testing

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Introduction to Hypothesis Testing
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1 Introduction

• The purpose of hypothesis testing is to determine
  whether there is enough statistical evidence in
  favor of a certain belief about a parameter.
• Examples
• Is there statistical evidence in a random sample
  of potential customers, that support the
  hypothesis that more than 10 of the potential
  customers will purchase a new products?
• Is a new drug effective in curing a certain
  disease? A sample of patients is randomly
  selected. Half of them are given the drug while
  the other half are given a placebo. The
  improvement in the patients conditions is then
  measured and compared.

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What is a hypothesis?
A tentative statement about a population
parameter that might be true or wrong
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Types of hypotheses

• Null hypothesis
• Research/alternative hypothesis

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2 Concepts of Hypothesis Testing

• The critical concepts of hypothesis testing.
• Example
• An operation manager wants to determine if the
  mean demand during lead time is greater than 350.
• If so, the ordering policy should be changed.
• The two hypotheses about a population mean
• H0 The null hypothesis m 350
• H1 The alternative hypothesis m gt 350

This is what you want to prove
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2 Concepts of Hypothesis Testing

• Assume the null hypothesis is true (m 350).

m 350

• Sample from the demand population, and build a
  test statistic related to the parameter
  hypothesized (the sample mean).
• Decide, is the value obtained big enough to
  reject the null hypothesis?

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2 Concepts of Hypothesis Testing

• Assume the null hypothesis is true (m 350).

m 350

• In this case the mean m is not likely to be
  greater than 350. Do not reject the null
  hypothesis.

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Types of Errors

• Two types of errors may occur when deciding
  whether to reject H0 based on the statistic
  value.
• Type I error Reject H0 while the truth is, it is
  true.
• Type II error Do not reject H0 while the truth
  is, it is false.
• Example continued
• Type I error Reject H0 (m 350) in favor of H1
  (m gt 350) while the truth is, the real value of m
  is 350.
• Type II error Do not reject H0 (m 350) while
  the truth is, the real value of m is greater than
  350.

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Controlling the probability of conducting a type
I error

• Recall
• H0 m 350 and H1 m gt 350.
• H0 is rejected if is sufficiently large
• Thus, a type I error is made if
  when m 350.
• By properly selecting the critical value we can
  limit the probability of conducting a type I
  error to an acceptable level.

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What is a critical value?
A value needed to determine weather to reject or
not to reject the null hypothesis.
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3 Testing the Population Mean When the
Population Standard Deviation is Known

• The manager of a department store is
  thinking about establishing a new billing system
  for the stores credit customer. After a through
  financial analysis, she determines that the new
  system will be cost effective only if the mean
  monthly account is more than 170. A random
  sample of 400 monthly account is drawn, for which
  the sample mean is 178. The manager knows that
  the accounts are approximately normally
  distributed with standard deviation 65. Can the
  manager conclude from this available information
  that the new system will be cost-effective?

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3 Testing the Population Mean When the
Population Standard Deviation is Known

• Example 1
• A new billing system for a department store will
  be cost- effective only if the mean monthly
  account is more than 170.
• A sample of 400 accounts has a mean of 178.
• If accounts are approximately normally
  distributed with s 65, can we conclude that
  the new system will be cost effective?

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Testing the Population Mean (s is Known)

• Example 1 Solution
• The population of interest is the credit accounts
  at the store.
• We want to know whether the mean account for all
  customers is greater than 170.

H1 m gt 170

• The null hypothesis must specify a single value
  of the parameter m,

H0 m 170
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Approaches to Testing

• There are two approaches to test whether the
  sample mean supports the alternative hypothesis
  (H1)
• The rejection region method is mandatory for
  manual testing (but can be used when testing is
  supported by a statistical software)
• The p-value method which is mostly used when a
  statistical software is available.

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The Rejection Region Method
The rejection region is a range of values such
that if the test statistic falls within that
range, the null hypothesis is rejected in favour
of the alternative hypothesis.
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Steps in rejection region method

• Construct appropriate hypotheses
• Determine a test statistics to be used
• Determine the critical value
• Compare the test statistic with the critical
  value. Reject the null hypothesis if the former
  is greater than the latter.
• Make an appropriate conclusion.

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The Rejection Region Method for a Right - Tail
Test

• Example 1 solution continued
• Recall H0 m 170 H1 m gt 170
  therefore,
• It seems reasonable to reject the null
  hypothesis and believe that m gt 170 if the
  sample mean is sufficiently large.

Reject H0 here
Critical value of the sample mean
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The Rejection Region Method for a Right - Tail
Test

• Example 1 solution continued
• Define a critical value for that is
  just large enough to reject the null
  hypothesis.

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Determining the Critical Value for the Rejection
Region

• Allow the probability of committing a Type I
  error be a (also called the significance level).
• Find the value of the sample mean that is just
  large enough so that the actual probability of
  committing a Type I error does not exceed a.
  Watch

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Determining the Critical Value for a Right
Tail Test
Example 1 solution continued
P(commit a Type I error) P(reject H0 given
that H0 is true)
is allowed to be a.

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Determining the Critical Value for a Right
Tail Test
Example 1 solution continued
a
0.05
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Determining the Critical value for a Right -
Tail Test
Conclusion Since the sample mean (178) is greater
than the critical value of 175.34, there is
sufficient evidence to infer that the mean
monthly balance is greater than 170 at the 5
significance level.
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The standardized test statistic

• Instead of using the statistic , we can use
  the standardized value z.
• Then, the rejection region becomes

One tail test
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The standardized test statistic

• Example 1 - continued
• We redo this example using the standardized test
  statistic.
• Recall H0 m 170
• H1 m gt 170
• Test statistic
• Rejection region z gt z.05 1.645.

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The standardized test statistic

• Example 1 - continued

Conclusion Since Z 2.46 gt 1.645, reject the
null hypothesis in favor of the alternative
hypothesis.
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P-value Method

• The p-value provides information about the amount
  of statistical evidence that supports the
  alternative hypothesis.

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P-value Method
The probability of observing a test statistic at
least as extreme as 178, given that m 170 is
The p-value
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Interpreting the p-value

• Because the probability that the sample mean
  will assume a value of more than 178 when m 170
  is so small (.0069), there are reasons to believe
  that m gt 170.

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Interpreting the p-value
We can conclude that the smaller the p-value the
more statistical evidence exists to support the
alternative hypothesis.
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Interpreting the p-value

• Describing the p-value
• If the p-value is less than 1, there is
  overwhelming evidence that supports the
  alternative hypothesis.

• If the p-value is between 1 and 5, there is a
  strong evidence that supports the alternative
  hypothesis.
• If the p-value is between 5 and 10 there is a
  weak evidence that supports the alternative
  hypothesis.
• If the p-value exceeds 10, there is no evidence
  that supports the alternative hypothesis.

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The p-value and the Rejection Region Methods

• The p-value can be used when making decisions
  based on rejection region methods as follows
• Define the hypotheses to test, and the required
  significance level a.
• Perform the sampling procedure, calculate the
  test statistic and the p-value associated with
  it.
• Compare the p-value to a. Reject the null
  hypothesis only if p-value lta otherwise, do not
  reject the null hypothesis.

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Conclusions of a Test of Hypothesis

• If we reject the null hypothesis, we conclude
  that there is enough evidence to infer that the
  alternative hypothesis is true.
• If we do not reject the null hypothesis, we
  conclude that there is not enough statistical
  evidence to infer that the alternative hypothesis
  is true.

The alternative hypothesis is the more
important one. It represents what we are
investigating.
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A Left - Tail Test

• The SSA Envelop Example.
• The chief financial officer in FedEx believes
  that including a stamped self-addressed (SSA)
  envelop in the monthly invoice sent to customers
  will decrease the amount of time it take for
  customers to pay their monthly bills.
• Currently, customers return their payments in 24
  days on the average, with a standard deviation of
  6 days.

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A Left - Tail Test

• The SSA envelop example continued
• It was calculated that an improvement of two days
  on the average will cover the costs of the
  envelops (checks can be deposited earlier).
• A random sample of 220 customers was selected and
  SSA envelops were included with their invoice
  packs.
• The times customers payments were received were
  recorded (SSA.xls)
• Can the CFO conclude that the plan will be
  profitable at 10 significance level? ?

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A Left - Tail Test

• The SSA envelop example Solution
• The parameter tested is the population mean
  payment period (m).
• The hypotheses areH0 m 22H1 m lt 22 (The
  CFO wants to know whether the plan will be
  profitable)

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A Left - Tail Test

• The SSA envelop example Solution continued
• The rejection region It makes sense to believe
  that m lt 22 if the sample mean is sufficiently
  smaller than 22.
• Reject the null hypothesis if

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A Left -Tail Test
Left-tail test

• The SSA envelop example Solution continued
• The standardized one tail left hand test is

Define the rejection region
Since -.91 gt 1.28 do not reject the null
hypothesis. The p value P(Zlt-.91)
.1814Since .1814 gt .10, do not reject the null
hypothesis
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A Two - Tail Test

• Example 2
• ATT has been challenged by competitors who
  argued that their rates resulted in lower bills.
• A statistics practitioner determines that the
  mean and standard deviation of monthly
  long-distance bills for all ATT residential
  customers are 17.09 and 3.87 respectively.

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A Two - Tail Test

• Example 2 - continued
• A random sample of 100 customers is selected and
  customers bills recalculated using a leading
  competitors rates (see Xm11-02).
• Assuming the standard deviation is the same
  (3.87), can we infer that there is a difference
  between ATTs bills and the competitors bills
  (on the average)?

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A Two - Tail Test

• Solution
• Is the mean different from 17.09?

H0 m 17.09

• Define the rejection region

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A Two Tail Test
Solution - continued
17.09
We want this erroneous rejection of H0 to be a
rare event, say 5 chance.
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A Two Tail Test
Solution - continued
17.09
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A Two Tail Test
Two-tail test
There is insufficient evidence to infer that
there is a difference between the bills of ATT
and the competitor.
Also, by the p value approach The p-value P(Zlt
-1.19)P(Z gt1.19) 2(.1173) .2346 gt .05
a/2 0.025
a/2 0.025
-1.19
1.19