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Confidence Intervals

Chapter Contents

Confidence Interval for a Mean (μ) with Known σConfidence Interval for a Mean (μ) with Unknown σConfidence Interval for a Proportion (π)Estimating from Finite PopulationsSample Size Determination for a MeanSample Size Determination for a Proportion

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Chapter Learning Objectives (LO’s)Construct a 90, 95, or 99 percent confidence interval for μ.Know when to use Student’s t instead of z to estimate μ.Construct a 90, 95, or 99 percent confidence interval for π.Construct confidence intervals for finite populations.Calculate sample size to estimate a mean or proportion.Construct a confidence interval for a variance (optional).

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Confidence Intervals

What is a Confidence Interval?

Chapter 8

Confidence Interval for a Mean () with known ()

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What is a Confidence Interval?• The confidence interval for with known is:

Chapter 8

Confidence Interval for a Mean () with known ()

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• A higher confidence level leads to a wider confidence interval. Choosing a Confidence Level

• Greater confidence implies loss of precision (i.e. greater margin of error).

• 95% confidence is most often used.

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Confidence Intervals for Example 8.2

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• A confidence interval either does or does not contain .• The confidence level quantifies the risk.• Out of 100 confidence intervals, approximately 95% may contain ,

while approximately 5% might not contain when constructing 95% confidence intervals.

Interpretation

Chapter 8

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 distributionIs 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.

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

Student’s t Distribution

Chapter 8Confidence Interval for a Mean () with Unknown ()

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Student’s t Distribution

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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.

Figure 8.11

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Comparison of Normal and Student’s t

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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 , 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.

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• 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.

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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.

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

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Example GMAT Scores Again

Figure 8.13

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Example GMAT Scores Again• Construct a 90% confidence interval for the mean GMAT score of

all MBA applicants.

x = 510 s = 73.77

• Since 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 T-table.

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• 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 valuesas well as to construct confidence Intervals.

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

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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).

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Using T-Table

• 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.

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

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Confidence Interval for a Proportion ()

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

Applying the CLT

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

When is it Safe to Assume Normality of p?

Sample size to assume normality:

Table 8.9

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Confidence Interval for

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

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Example Auditing

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N = population size; n = sample size

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

Sample Size to Estimate

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Sample Size determination for a Mean

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• Method 1: Take a Preliminary SampleTake a small preliminary sample and use the sample s in place of in the sample size formula.

• Method 2: Assume Uniform PopulationEstimate rough upper and lower limits a and b and set = [(b-a)/12]½.

How to Estimate ?

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• Method 3: Assume Normal PopulationEstimate rough upper and lower limits a and b and set = (b-a)/4. This assumes normality with most of the data with ± 2 so the range is 4.

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

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

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

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• Method 1: Assume that = .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 in the sample size formula.

• Method 3: Use a Prior Sample or Historical DataHow often are such samples available? Unfortunately, might be different enough to make it a questionable assumption.

How to Estimate ?

Chapter 8