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Confidence Interval ML 7.1

For a Proportion ()

Chapter 8

- A proportion is a mean of data whose only values are 0 or 1.

Confidence Interval for a Proportion ()

Chapter 8

Applying the CLT

- The distribution of a sample proportion p = x/n is symmetric if p = .50.
- The distribution of p approaches normal as nincreases, for any p.

Confidence Interval for a Proportion ()

Chapter 8

When Is It Safe to Assume Normality of p?

- Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both np 10 and n(1p) 10.

Sample size to assume normality:

Rule: It is safe to assume normality of p = x/n if we have at least 10 “successes” and 10 “failures” in the sample.

Table 8.9

Confidence Interval for a Proportion ()

Chapter 8

Confidence Interval for p

- The confidence interval for the unknown p (assuming a large sample) is based on the sample proportion p = x/n.

Estimating from Finite Population

Chapter 8

N = 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 ).

To estimate a population mean with a precision of +E (allowable error), you would need a sample of size n. Now,

Sample Size Determination ML 7-2

Chapter 8

Sample Size to Estimate m

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.

Sample Size Determination for a Mean

Chapter 8

How to Estimate s?

- 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

Chapter 8

Using MegaStat

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:

To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size n.

Sample Size Determination for a Mean

Chapter 8

- Since p is a number between 0 and 1, the allowable error E is also between 0 and 1.

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.

Sample Size Determination for a Mean

Chapter 8

How to Estimate p?

Sample Size Determination for a Mean

Chapter 8

Using MegaStat

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)?.

One-Sample Hypothesis Tests ML 7-3

Chapter 9

Learning Objectives

LO9-1: List the steps in testing hypotheses.

LO9-2: Explain the difference between H0and H1.

LO9-3:Define Type I error, Type II error, and power.

LO9-4: Formulate a null and alternative hypothesis for μ or π.

Logic of Hypothesis Testing

Chapter 9

Logic of Hypothesis Testing

Chapter 9

LO9-2: Explain the difference between H0 and H1.

- 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

Logic of Hypothesis Testing

Chapter 9

State the Hypothesis

- 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.

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

Chapter 9

LO9-3: Define Type I error, Type II error, and power.

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.

Logic of Hypothesis Testing

Chapter 9

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.

Logic of Hypothesis Testing

Chapter 9

Power of a Test

- A low b risk means high power.

- Larger samples lead to increased power.

Logic of Hypothesis Testing

Chapter 9

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.

Logic of Hypothesis Testing

Chapter 9

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.)

Statistical Hypothesis Testing

Chapter 9

LO9-4: Formulate a null and alternative hypothesis for μ or π.

- A statistical hypothesisis a statement about the value of a population parameter.
- A hypothesis testis 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.

Statistical Hypothesis Testing

Chapter 9

One-Tailed and Two-Tailed Tests

- The direction of the test is indicated by H1:

> indicates a right-tailed test

< indicates a left-tailed test

≠ indicates a two-tailed test

Statistical Hypothesis Testing

Chapter 9

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 ruleuses the known sampling distribution of the test statistic to establish the critical valuethat divides the sampling distribution into two regions.
- Reject H0 if the test statistic lies in the rejection region.

Statistical Hypothesis Testing

Chapter 9

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.

Statistical Hypothesis Testing

Chapter 9

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.

Statistical Hypothesis Testing

Chapter 9

Decision Rule for Left-Tailed Test

- Reject H0 if the test statistic < left-tail critical value.

Figure 9.2

Statistical Hypothesis Testing

Chapter 9

Decision Rule for Right-Tailed Test

- Reject H0 if the test statistic > right-tail critical value.

Statistical Hypothesis Testing

Chapter 9

Type I Error

also called a false positive

- 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.

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