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Applied Statistics in Business & Economics, 4th edition David P. Doane and Lori E. Seward

Prepared by Lloyd R. Jaisingh

Chapter Contents

8.1 Sampling Variation

8.2 Estimators and Sampling Errors

8.3 Sample Mean and the Central Limit Theorem

8.4 Confidence Interval for a Mean (μ) with Known σ

8.5 Confidence Interval for a Mean (μ) with Unknown σ

8.6 Confidence Interval for a Proportion (π)

8.7 Estimating from Finite Populations

8.8 Sample Size Determination for a Mean

8.9 Sample Size Determination for a Proportion

8.10 Confidence Interval for a Population Variance, 2 (Optional)

Chapter 8

Chapter Learning Objectives (LO’s)

LO8-1: Define sampling error, parameter, and estimator.

LO8-2: Explain the desirable properties of estimators.

LO8-3:State the Central Limit Theorem for a mean.

LO8-4:Explain how sample size affects the standard error.

LO8-5:Construct a 90, 95, or 99 percent confidence interval for μ.

Chapter 8

Chapter Learning Objectives (LO’s)

LO8-6:Know when to use Student’s t instead of z to estimate μ.

LO8-7:Construct a 90, 95, or 99 percent confidence interval for π.

LO8-8:Construct confidence intervals for finite populations.

LO8-9: Calculate sample size to estimate a mean or proportion.

LO8-10: Construct a confidence interval for a variance (optional).

Chapter 8

8.1 Sampling Variation

Chapter 8

- Sample statistic– a random variable whose value depends on which population items are included in the random sample.
- Depending on the sample size, the sample statistic could either represent the population well or differ greatly from the population.
- This sampling variationcan easily be illustrated.

8-5

8.1 Sampling Variation

Chapter 8

- Consider eight random samples of size n = 5 from a large population of GMAT scores for MBA applicants.

- The sample means tend to be close to the population mean (m = 520.78).

8-6

8.1 Sampling Variation

Chapter 8

- The dot plots show that the sample means have much less variation than the individual sample items.

8.2 Estimators and Sampling Distributions

LO8-1

Chapter 8

- Estimator– a statistic derived from a sample to infer the value of a population parameter.
- Estimate – the value of the estimator in a particular sample.
- Population parameters are usually represented by Greek letters and the corresponding statistic by Roman letters.

LO8-1: Define sampling error, parameter and estimator.

Some Terminology

8-8

8.2 Estimators and Sampling Distributions

LO8-1

Chapter 8

Examples of Estimators

Sampling Distributions

- The sampling distributionof an estimator is the probability distribution of all possible values the statistic may assume when a random sample of size n is taken.
- Note: An estimator is a random variable since samples vary.

8.2 Estimators and Sampling Distributions

LO8-1

Chapter 8

- Bias is the difference between the expected value of the estimator and the true parameter. Example for the mean,

- Sampling erroris the difference between an estimate and the
- corresponding population parameter. For example, if we use the sample
- mean as an estimate for the population mean, then the

Bias

- An estimator is unbiased if its expected value is the parameter being estimated. The sample mean is an unbiased estimator of the population mean since

- On average, an unbiased estimator neither overstates nor understates the true parameter.

8-10

8.2 Estimators and Sampling Distributions

LO8-1

Chapter 8

8-11

8.2 Estimators and Sampling Distributions

LO8-2

Chapter 8

- Efficiency refers to the variance of the estimator’s sampling distribution.
- A more efficient estimator has smaller variance.

LO8-2: Explain the desirable properties of estimators.

Note: Also, a desirable property for an estimator is for it to be unbiased.

Efficiency

Figure 8.6

8-12

8.2 Estimators and Sampling Distributions

LO8-2

Chapter 8

A consistent estimator converges toward the parameter being estimated

as the sample size increases.

LO8-2: Explain the desirable properties of estimators.

Consistency

Figure 8.6

8-13

8.3 Sample Mean and the Central Limit Theorem

LO8-3

Chapter 8

LO8-3: State the Central Limit Theorem for a mean.

The Central Limit Theorem is a powerful result that allows us to

approximate the shape of the sampling distribution of the sample

mean even when we don’t know what the population looks like.

8-14

8.3 Sample Mean and the Central Limit Theorem

LO8-3

Chapter 8

- If the population is exactly normal, then the sample mean follows a normal distribution.

- As the sample size n increases, the distribution of sample means narrows in on the population mean µ.

8-15

8.3 Sample Mean and the Central Limit Theorem

LO8-3

Chapter 8

- If the sample is large enough, the sample means will have approximately a normal distribution even if your population is notnormal.

8.3 Sample Mean and the Central Limit Theorem

LO8-3

Chapter 8

Illustrations of Central Limit Theorem

Using the uniform

and a right skewed distribution.

Note:

8-17

8.3 Sample Mean and the Central Limit Theorem

LO8-3

Chapter 8

Applying The Central Limit Theorem

The Central Limit Theorem permits us to define an interval within which the sample means are expected to fall. As long as the sample size n is large enough, we can use the normal distribution regardless of the population shape (or any n if the population is normal to begin with).

8-18

8.3 Sample Mean and the Central Limit Theorem

LO8-4

Chapter 8

Even if the population standard deviation σ is large, the sample means will fall within a narrow interval as long as n is large. The key is the standard error of the mean:.. The standard error decreases as n increases.

LO8-4: Explain how sample size affects the standard error.

Sample Size and Standard Error

For example, when n = 4 the standard error is halved. To halve it again requires n = 16, and to halve it again requires n = 64. To halve the

standard error, you must quadruple the sample size (the law of diminishing returns).

8-19

8.3 Sample Mean and the Central Limit Theorem

Chapter 8

Illustration: All Possible Samples from a Uniform Population

- Consider a discrete uniform population consisting of the integers {0, 1, 2, 3}.
- The population parameters are: m = 1.5, s = 1.118.

8-20

8.3 Sample Mean and the Central Limit Theorem

Chapter 8

Illustration: All Possible Samples from a Uniform Population

- The population is uniform, yet the distribution of all possible sample means of size 2 has a peaked triangular shape.

8-21

8.4 Confidence Interval for a Mean () with known ()

LO8-5

Chapter 8

LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.

What is a Confidence Interval?

8-22

8.4 Confidence Interval for a Mean () with known ()

LO8-5

Chapter 8

What is a Confidence Interval?

- The confidence interval for m with known s is:

8-23

8.4 Confidence Interval for a Mean () with known ()

LO8-5

Chapter 8

Choosing a Confidence Level

- A higher confidence level leads to a wider confidence interval.

- Greater confidence implies loss of precision (i.e. greater margin of error).
- 95% confidence is most often used.

Confidence Intervals for Example 8.2

8-24

8.4 Confidence Interval for a Mean () with known ()

LO8-5

Chapter 8

Interpretation

- A confidence interval either does or does not contain m.
- The confidence level quantifies the risk.
- Out of 100 confidence intervals, approximately 95% may contain m, while approximately 5% might not contain when constructing 95% confidence intervals.

When Can We Assume Normality?

- If is known and the population is normal, then we can safely use the
- formula to compute the confidence interval.
- If is known and we do not know whether the population is normal, a common
- rule of thumb is that n 30 is sufficient to use the formula as long as the distribution
- Is approximately symmetric with no outliers.
- Larger n may be needed to assume normality if you are sampling from a strongly
- skewed population or one with outliers.

8-25

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

- Use the Student’s t distributioninstead of the normal distribution when the population is normal but the standard deviation s is unknown and the sample size is small.

LO8-6: Know when to use Student’s t instead of z to estimate .

Student’s t Distribution

8-26

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

LO8-6: Know when to use Student’s t instead of z to estimate .

Student’s t Distribution

8-27

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Student’s t Distribution

- t distributions are symmetric and shaped like the standard normal distribution.
- The t distribution is dependent on the size of the sample.

Comparison of Normal and Student’s t

Figure 8.11

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Degrees of Freedom

- Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic.
- Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation. The d.f for the t distribution in this case, is given by d.f. = n -1.

- As n increases, the t distribution approaches the shape of the normal distribution.
- For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Comparison of z and t

- For very small samples, t-values differ substantially from the normal.
- As degrees of freedom increase, the t-values approach the normal z-values.
- For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 = 30.

So for a 90 percent confidence interval, we would use t = 1.697, which is only slightly larger than z = 1.645.

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Example GMAT Scores Again

Figure 8.13

x = 510 s = 73.77

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Example GMAT Scores Again

- Construct a 90% confidence interval for the mean GMAT score of all MBA applicants.

- Since s is unknown, use the Student’s t for the confidence interval with d.f. = 20 – 1 = 19.
- First find t/2 = t.05 = 1.729 from Appendix D.

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

- For a 90% confidence interval, use Appendix D to find t0.05 = 1.729 with d.f. = 19.

Note: One can use Excel,

Minitab, etc. to

obtain these values

as well as to

construct confidence

Intervals.

We are 90 percent confident that the true mean GMAT score might be within the interval [481.48, 538.52]

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Confidence Interval Width

- Confidence interval width reflects - the sample size, - the confidence level and - the standard deviation.
- To obtain a narrower interval and more precision- increase the sample size or - lower the confidence level (e.g., from 90% to 80% confidence).

8.5 Confidence Interval for a Mean () with Unknown ()

LO8-6

Chapter 8

Using Appendix D

- Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10.
- If the table does not give the exact degrees of freedom, use the t-value for the next lower degrees of freedom.
- This is a conservative procedure since it causes the interval to be slightly wider.
- A conservative statistician may use the t distribution for
- confidence intervals when σ is unknown because
- using z would underestimate the margin of error.

LO8-7

8.6 Confidence Interval for a Proportion ()

Chapter 8

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

LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.

8-36

LO8-7

8.6 Confidence Interval for a Proportion ()

Chapter 8

Applying the CLT

- The distribution of a sample proportion p = x/n is symmetric if p = .50 and regardless of p, approaches symmetry as n increases.

8-37

LO8-7

8.6 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 np10 and n(1- p) 10.

Sample size to assume normality:

Table 8.9

8-38

LO8-7

8.6 Confidence Interval for a Proportion ()

Chapter 8

Confidence Interval for p

- Since p is unknown, the confidence interval for p = x/n (assuming a large sample) is

8-39

LO8-7

8.6 Confidence Interval for a Proportion ()

Chapter 8

Example Auditing

8-40

LO8-8

8.7 Estimating from Finite Population

Chapter 8

LO8-8: Construct Confidence Intervals for Finite Populations.

N = population size; n = sample size

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

LO8-9

8.8 Sample Size determination for a Mean

Chapter 8

LO8-9: Calculate sample size to estimate a mean or proportion.

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.

- Method 4: Poisson ArrivalsIn the special case when m is a Poisson arrival rate, then s = m .

LO8-9

8.8 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 with m ± 2s so the range is 4s.

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

8.9 Sample Size determination for a Proportion

LO8-9

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? Unfortunately, p might be different enough to make it a questionable assumption.

8.9 Sample Size determination for a Proportion

LO8-9

Chapter 8

How to Estimate p?

LO8-10

8.10 Confidence Interval for a Population Variance (2)

- If the population is normal, then the sample variance s2 follows the chi-square distribution (c2) with degrees of freedom d.f.= n – 1.
- Lower (c2L) and upper (c2U) tail percentiles for the chi-square distribution can be found using Appendix E.

LO8-10: Construct a confidence interval for a variance (optional).

Chi-Square Distribution

8-46

LO8-10

8.10 Confidence Interval for a Population Variance (2)

- Using the sample variance s2, the confidence interval is

LO8-10: Construct a confidence interval for a variance (optional).

Confidence Interval

- To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds.

8-47

LO8-10

8.10 Confidence Interval for a Population Variance (2)

You can use Appendix E to find critical chi-square values.

8-48

LO8-10

8.10 Confidence Interval for a Population Variance (2)

Caution: Assumption of Normality

- The methods described for confidence interval estimation of the variance and standard deviation depend on the population having a normal distribution.
- If the population does not have a normal distribution, then the confidence interval should not be considered accurate.

8-49