Hypothesis Testing

1
Hypothesis Testing

• In statistics a hypothesis is a claim about a
  property of a population
• A common guideline for statistical reasoning is
• Analyze a sample in an attempt to distinguish
  between results that can easily occur and results
  that are highly unlikely
• Statistical hypothesis testing is a
  decision-making process for evaluating claims
  about a population.

2
Objectives

• Understand the definitions used in hypothesis
  testing.
• State the null and alternative hypotheses.
• Find critical values for the z test.
• State the five steps used in hypothesis testing.
• Test means for large samples using the z test.
• Test means for small samples using the t test.

3
Objectives (contd.)

• Test proportions using the z test.
• Test variances or standard deviations using the
  chi square test.
• Test hypotheses using confidence intervals.
• Explain the relationship between type I and type
  II errors and the power of a test.

4
Introduction

• In hypothesis testing, the researcher must
• define the population under study,
• state the particular hypotheses that will be
  investigated,
• give the significance level,
• select a sample from the population,
• collect the data,
• perform the calculations required for the
  statistical test,
• reach a conclusion.

5
Methods to Test Hypotheses

• The three methods used to test hypotheses are
• 1. The traditional method.
• 2. The P-value method.
• 3. The confidence interval method.

6
Statement of a Hypothesis

• A statistical hypothesis is a conjecture about a
  population parameter which may or may not be
  true.
• There are two types of statistical hypotheses for
  each situation the null hypothesis and the
  alternative hypothesis.

7
The Null Hypothesis H0

• The null hypothesis, symbolized by H0, is a
  statistical hypothesis that states that there is
  no difference between a parameter and a specific
  value, or that there is no difference between two
  parameters.
• The null hypothesis must contain the condition of
  equality
• written with the symbols , ?, ?
• stated in three possible forms

8
The Alternative Hypothesis H1

• The alternative hypothesis, symbolized by H1, is
  a statistical hypothesis that states the
  existence of a difference between a parameter and
  a specific value, or states that there is a
  difference between two parameters.
• The alternative hypothesis is a statement that
  must be true if the null hypothesis is false
• The alternative hypothesis may be written in
  three possible forms
• The alternative hypothesis determines whether the
  statistical test is a one-tailed test or
  two-tailed test

9
The Hypotheses Form a Logical Pair

• The null hypothesis must contain the condition of
  equality
• written with the symbols , ?, ?
• The alternative hypothesis is a statement that
  must be true if the null hypothesis is false
• written with the symbols ?, lt, gt

10
Design of the Study

• After stating the hypotheses, the researchers
  next step is to design the study.
• The researcher selects the correct statistical
  test, chooses an appropriate level of
  significance, and formulates a plan for
  conducting the study.

11
Statistical Test

• A statistical test uses the data obtained from a
  sample to make a decision about whether or not
  the null hypothesis should be rejected.
• The numerical value obtained from a statistical
  test is called the test value.

12
Levels Of Significance

• The probability of rejecting the null hypothesis
  when it is true is called the significance level
• denoted with the Greek letter ?
• ? choices are typically ?.05, ? .01 or ?
  .10
• ? is just the area in the tails of the
  distribution
• P(type I error) ?
• The probability of a type II error is ?

13
Critical Region

• The set of all values of the test statistic that
  would cause us to reject the null hypothesis
• The critical or rejection region is the range of
  values that indicates a significant difference
  between the sample data and the null hypothesis
  parameter
• The remaining region is the non-critical region
  which indicates a difference due to chance- we
  fail to reject the null hypothesis

14
Controlling Type I and Type II Errors

• Mathematically it can be shown that ?, ? and
  sample size n are related
• For a fixed ?, an increase in sample size will
  cause a decrease in ?
• For a fixed n, a decrease in ? causes an
  increase in ?
• To decrease both ? and ? increase sample size
• Thus ? and ? are related in that decreasing one
  increases the other.

15
Setting Significance Levels

• Consider a package of M Ms
• contains 1498 candies
• package weight labeled as 1361 g ? .9085g/candy
• Consider a package of Bufferin
• contains 30 tablets
• 325 mg/ tablet
• What are the consequences if the MMs dont have
  a mean population weight of .9085g?
• What are the consequences if the Bufferin tablets
  have too much aspirin?

16
Setting Significance Levels

• What are the consequences if the MMs dont have
  a mean population weight of .9085g?
• not critical to test the claim that ? .9085g
  
• we choose n 100 , ? .05
• What are the consequences if the Bufferin tablets
  have too much aspirin?
• more critical, choose n500 and ? .01

17
Conclusions in Hypothesis Testing

• The initial conclusion will always be one of the
  following
• 1. Fail to reject the null hypothesis H0
• 2. Reject the null hypothesis
• Wording is very important
• Notice that we are never proving the null
  hypothesis

18
Tailed Tests

• Tails in a distribution are the extreme regions
  bounded by critical values
• Two-tailed test used when H1 ?
• A one-tailed test is either right-tailed or
  left-tailed, depending on the direction of the
  inequality of H1

19
Hypothesis-Testing (Traditional Method)

• Step 1 State the hypothesis, and identify the
  claim.
• Step 2 Find the critical value from the
  appropriate table.
• Step 3 Compute the test value.
• Step 4 Make the decision to reject or not
  reject the null hypothesis.
• Step 5 Summarize the results with appropriate
  wording

20
Testing Claims About the Mean of a Population

• Assumptions for the z -test
• 1. Sample size is large (n ? 30)
• When applying the Central Limit Theorem use ?,
  or use the sample standard deviation , s, as an
  estimate of ?
• 2. If the sample size is small then the parent
  population must be normally distributed, and ?
  must be known

21
The z Test Formula

• The z test is a statistical test for the mean of
  a population

22
The z Test When ? is Unknown

• The central limit theorem states that when the
  population standard deviation ? is unknown, the
  sample standard deviation s can be used in the
  formula as long as the sample size is 30 or more.

23
The P-value

• The P-value (or probability value) is the
  probability of getting a sample statistic that
  is at least as extreme as the one found from the
  sample data, assuming that the null hypothesis is
  true.
• When using a P-value
• Reject the null hypothesis H0 if the P-value ? ?
• Fail to reject the null hypothesis if the
  P-value gt ?
• Or simply report the P-value

24
The P-value (contd.)

• The P-value is the actual area under the standard
  normal distribution curve (or other curve
  depending on what statistical test is being used)
  representing the probability of a particular
  sample statistic or a more extreme sample
  statistic occurring if the null hypothesis is
  true.

25
Hypothesis-Testing (P-Value Method)

• Step 1 State the hypothesis, and identify the
  claim.
• Step 2 Compute the test value.
• Step 3 Find the P value from the appropriate
  table.
• Step 4 Make the decision to reject or not
  reject the null hypothesis.
• Step 5 Summarize the results with appropriate
  wording

26
Statistical vs. Practical Significance

• The researcher should distinguish between
  statistical significance and practical
  significance.
• When the null hypothesis is rejected at a
  specific significance level, it can be concluded
  that the difference is probably not due to chance
  and thus is statistically significant. However,
  the results may or may not have any practical
  significance.

27
Interpretations of P-Values

• Interpretation
• Highly statistically significant Very strong
  evidence against the null hypothesis
• Statistically significant Adequate evidence
  against the null hypothesis
• Insufficient evidence against the null hypothesis

• P -Value
• Less than 0.01
• 0.01 to 0.05
• Greater than 0.05

28
Confidence Intervals Hypothesis Testing

• There is a relationship between confidence
  intervals and hypothesis testing.
• When the null hypothesis is rejected in a
  hypothesis testing situation, the confidence
  interval for the mean using the same level of
  significance will not contain the hypothesized
  mean.
• Likewise, when the null hypothesis is not
  rejected, the confidence interval computed will
  contain the hypothesized mean.

29
The t Test

• The t test is a statistical test of the mean of a
  population and is used when the population is
  normally or approximately normally distributed, ?
  is unknown, and the sample size is less than 30.
• The formula for the t test is
• The degrees of freedom are d.f. n1.

30
z Test for a Proportion

• A hypothesis test involving a population
  proportion can be considered as a binomial
  experiment when
• there are only two outcomes and
• the probability of a success does not change
  from trial to trial.

31
Binomial Experiment

• A probability experiment with
• Each trial has only two outcomes
  we consider the outcomes as
  success (yes) or failure (no)
• There are a fixed number of trials
• The outcomes of each trial are independent
• The probability of success remains the same for
  each trial

32
Examples of Proportions Viewed as Binomial
Experiments

• The percentage of late-night viewers who watch
  The Late Show with David Letterman is equal to
  18
• Based on a sample survey, fewer than 1/4 of all
  college graduates smoke.
• If a fatal car crash occurs, there is a .44
  probability that it involves a driver who has
  been drinking

33
Assumptions Used When Testing a Claim About a
Population Proportion

1. The conditions for a binomial experiment are
met 2. The conditions np? 5 and nq ? 5 are
both satisfied so the binomial
distribution of sample proportions can be
approximated by a normal distribution with
? np
34
Formula for the z Test for Proportions
35
Assumptions for Chi-Square Test for Single
Variance

• 1. The sample must be randomly selected from the
  population.
• 2. The population must be normally distributed
  for the variable under study.
• 3. The observations must be independent of each
  other.

36
Robustness of Inferences

• Tests for claims about standard deviations are
  not as robust as other test claims
  the inferences can be
  very misleading if the population does not have a
  normal distribution
• In this section, the condition of a normally
  distributed population is a much stricter
  requirement
• Compare this to the student t distribution
  where we required the population to be
  approximately normal

37
Robust Inferences

• For the student t distribution where we
  required the population to be approximately
  normal
• We say that inferences for the mean are fairly
  robust non-extreme departures from normality
  still lead to reasonable conclusions

38
Point Estimates for Variance and Standard
Deviation

• The sample variance s2 is the best point estimate
  of the population variance
• However, s is not the best point estimate of the
  standard deviation of the population UNLESS THE
  SAMPLE SIZE IS LARGE

39
The 3 Properties of the Chi-Square Distribution

• Not symmetric
• Non-negative
• Shape depends on degrees of freedom
• d.f. n - 1
• as n increases the shape approaches a normal
  distribution

40
Chi-Square Test for Single Variance

• The formula is

41
Summary

• A statistical hypothesis is a conjecture about a
  population.
• There are two types of statistical hypotheses
  the null hypothesis states that there is no
  difference, and the alternative hypothesis
  specifies a difference.

42
Summary (contd.)

• The z test is used when the population standard
  deviation is known and the variable is normally
  distributed or when ? is not known and the sample
  size is greater than or equal to 30.
• When the population standard deviation is not
  known and the variable is normally distributed,
  the sample standard deviation is used, but a t
  test should be conducted if the sample size is
  less than 30.

43
Summary (contd.)

• Researchers compute a test value from the sample
  data in order to decide whether the null
  hypothesis should or should not be rejected.
• Statistical tests can be one-tailed or
  two-tailed, depending on the hypotheses.

44
Summary (contd.)

• The null hypothesis is rejected when the
  difference between the population parameter and
  the sample statistic is said to be significant.
• The difference is significant when the test value
  falls in the critical region of the distribution.
  
• The critical region is determined by ?, the level
  of significance of the test.

45
Summary (contd.)

• The significance level of a test is the
  probability of committing a type I error.
• A type I error occurs when the null hypothesis is
  rejected when it is true.
• The type II error can occur when the null
  hypothesis is not rejected when it is false.
• One can test a single variance by using a
  chi-square test.

46
Conclusions

• Researchers are interested in answering many
  types of questions. For example
• Will a new drug lower blood pressure?
• Will seat belts reduce the severity of injuries
  caused by accidents?
• These types of questions can be addressed through
  statistical hypothesis testing, which is a
  decision-making process for evaluating claims
  about a population.