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Week 7 October 13-17
Three Mini-Lectures QMM 510Fall 2014
8-2
• A proportion is a mean of data whose only values are 0 or 1.
Ch
apter 8
Confidence Interval ML 7.1For a Proportion ()
Number of "successes"
1's 56
1 0 0 0 0 1 1 0 0 00 0 1 0 1 0 0 1 0 01 1 1 1 0 1 1 1 1 10 1 0 1 1 0 0 1 0 11 1 1 1 1 1 1 0 0 00 1 1 1 1 0 1 1 1 01 0 1 1 1 1 1 1 1 00 1 1 1 0 0 0 1 0 00 0 0 0 0 1 0 1 0 00 1 1 0 1 1 1 1 1 1
Sample of 100 Binary Outcomes
8-3
• The distribution of a sample proportion p = x/n is symmetric if p = .50.
• The distribution of p approaches normal as n increases, for any .p
Applying the CLT
Ch
apter 8
Confidence Interval for a Proportion ()
8-4
• Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both np 10 and n(1p) 10.
When Is It Safe to Assume Normality of p?
Sample size to assume normality:
Table 8.9
Ch
apter 8
Confidence Interval for a Proportion ()
Rule: It is safe to assume normality of p = x/n if we have at least 10 “successes” and 10 “failures” in the sample.
8-5
Confidence Interval for p
• The confidence interval for the unknown p (assuming a large sample) is based on the sample proportion p = x/n.
Ch
apter 8
Confidence Interval for a Proportion ()
8-6
Example: Auditing
Ch
apter 8
Confidence Interval for a Proportion ()
8-7
Ch
apter 8
Estimating from Finite PopulationN = population size; n = sample size
• The FPCF narrows the confidence interval somewhat.• When the sample is small relative to the population, the FPCF has little
effect. If n/N < .05, it is reasonable to omit it (FPCF 1 ).
8-8
• To estimate a population mean with a precision of + E (allowable error), you would need a sample of size n. Now,
Sample Size to Estimate m
Ch
apter 8
Sample Size Determination ML 7-2
8-9
• Method 1: Take a Preliminary SampleTake a small preliminary sample and use the sample s in place of s in the sample size formula.
• Method 2: Assume Uniform PopulationEstimate rough upper and lower limits a and b and set s = [(b a)/12]½.
How to Estimate s?
Ch
apter 8
• Method 3: Assume Normal PopulationEstimate rough upper and lower limits a and b and set s = (b a)/4. This assumes normality with most of the data within m ± 2s so the range is 4s.
• Method 4: Poisson ArrivalsIn the special case when m is a Poisson arrival rate, then s = .m
Sample Size Determination for a Mean
8-10
Using MegaStat
Ch
apter 8
Sample Size Determination for a Mean
For example, how large a sample is needed to estimate the population mean age of college professors with 95 percent confidence and precision of ± 2 years, assuming a range of 25 to 70 years (i.e., 2 years allowable error)? To estimate σ, we assume a uniform distribution of ages from 25 to 70:
2(70 25)13
12
2 2 2 2
2 2
(1.96) (13)163
2
zn
E
8-11
• To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size n.
• Since p is a number between 0 and 1, the allowable error E is also between 0 and 1.
Ch
apter 8
Sample Size Determination for a Mean
8-12
• Method 1: Assume that p = .50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary.
• Method 2: Take a Preliminary SampleTake a small preliminary sample and use the sample p in place of p in the sample size formula.
• Method 3: Use a Prior Sample or Historical DataHow often are such samples available? p might be different enough to make it a questionable assumption.
How to Estimate p?
Ch
apter 8
Sample Size Determination for a Mean
8-13
Using MegaStat
Ch
apter 8
Sample Size Determination for a Mean
2 2
2 2
(1 ) (1.96) (.50)(1 .50)2401
(.02)
zn
E
For example, how large a sample is needed to estimate the population proportion with 95 percent confidence and precision of ± .02 (i.e., 2% allowable error)?.
9-14
One-Sample Hypothesis Tests ML 7-3C
hap
ter 9
Learning Objectives
LO9-1: List the steps in testing hypotheses.
LO9-2: Explain the difference between H0 and H1.
LO9-3: Define Type I error, Type II error, and power.
LO9-4: Formulate a null and alternative hypothesis for μ or π.
9-15
Ch
apter 9
Logic of Hypothesis Testing
9-16
• Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about some fact about a population.
• One statement or the other must be true, but they cannot both be true.
• H0: Null hypothesisH1: Alternative hypothesis
• These two statements are hypotheses because the truth is unknown.
State the Hypothesis
Ch
apter 9
LO9-2: Explain the difference between H0 and H1.
Logic of Hypothesis Testing
9-17
• Efforts will be made to reject the null hypothesis.• If H0 is rejected, we tentatively conclude H1 to be the case. • H0 is sometimes called the maintained hypothesis.• H1 is called the action alternative because action may be required if we
reject H0 in favor of H1.
State the Hypothesis
Ch
apter 9
Can Hypotheses Be Proved?
• We cannot accept a null hypothesis; we can only fail to reject it.
Role of Evidence
• The null hypothesis is assumed true and a contradiction is sought.
Logic of Hypothesis Testing
9-18
Types of Error• Type I error: Rejecting the null hypothesis when it is true. This occurs
with probability a (level of significance). Also called a false positive.• Type II error: Failure to reject the null hypothesis when it is false. This
occurs with probability b. Also called a false negative.
Ch
apter 9
LO9-3: Define Type I error, Type II error, and power.
Logic of Hypothesis Testing
9-19
Probability of Type I and Type II Errors
• If we choose a = .05, we expect to commit a Type I error about 5 times in 100.
• b cannot be chosen in advance because it depends on a and the sample size.
• A small b is desirable, other things being equal.
Ch
apter 9
Logic of Hypothesis Testing
9-20
Power of a Test
Ch
apter 9
• A low b risk means high power.• Larger samples lead to increased power.
Logic of Hypothesis Testing
9-21
Relationship between a and b
• Both a small a and a small b are desirable.• For a given type of test and fixed sample size, there is a trade-off
between a and b.• The larger critical value needed to reduce a risk makes it harder to
reject H0, thereby increasing b risk.• Both a and b can be reduced simultaneously only by increasing the
sample size.
Ch
apter 9
Logic of Hypothesis Testing
9-22
Consequences of Type I and Type II Errors
• The consequences of these two errors are quite different, and the costs are borne by different parties.
• Example: Type I error is convicting an innocent defendant, so the costs are borne by the defendant. Type II error is failing to convict a guilty defendant, so the costs are borne by society if the guilty person returns to the streets.
• Firms are increasingly wary of Type II error (failing to recall a product as soon as sample evidence begins to indicate potential problems.)
Ch
apter 9
Logic of Hypothesis Testing
9-23
• A statistical hypothesis is a statement about the value of a population parameter.• A hypothesis test is a decision between two competing mutually exclusive and
collectively exhaustive hypotheses about the value of the parameter.• When testing a mean we can choose between three tests.
Ch
apter 9
Statistical Hypothesis Testing
LO9-4: Formulate a null and alternative hypothesis for μ or π.
9-24
• The direction of the test is indicated by H1:
Ch
apter 9
> indicates a right-tailed test< indicates a left-tailed test≠ indicates a two-tailed test
Statistical Hypothesis Testing
One-Tailed and Two-Tailed Tests
9-25
Decision Rule
• A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error.
• The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions.
• Reject H0 if the test statistic lies in the rejection region.
Ch
apter 9
Statistical Hypothesis Testing
9-26
Decision Rule for Two-Tailed Test
• Reject H0 if the test statistic < left-tail critical value or if the test statistic > right-tail critical value.
Ch
apter 9
Statistical Hypothesis Testing
9-27
When to use a One- or Two-Sided Test
• A two-sided hypothesis test (i.e., µ ≠ µ0) is used when direction (< or >) is of no interest to the decision maker.
• A one-sided hypothesis test is used when - the consequences of rejecting H0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher.
• Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal.
Ch
apter 9
Statistical Hypothesis Testing
9-28
Decision Rule for Left-Tailed Test• Reject H0 if the test statistic < left-tail critical value.
Figure 9.2
Ch
apter 9
Statistical Hypothesis Testing
9-29
Decision Rule for Right-Tailed Test• Reject H0 if the test statistic > right-tail critical value.
Ch
apter 9
Statistical Hypothesis Testing
9-30
Type I Error
• A reasonably small level of significance a is desirable, other things being equal.
• Chosen in advance, common choices for a are .10, .05, .025, .01, and .005 (i.e., 10%, 5%, 2.5%, 1%, and .5%).
• The a risk is the area under the tail(s) of the sampling distribution.• In a two-sided test, the a risk is split with a/2 in each tail since there
are two ways to reject H0.
Ch
apter 9
Statistical Hypothesis Testing
also called a false positive