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BIOSTAT 6 - Estimation• There are two types of inference: estimation and

hypothesis testing; estimation is introduced first.

• The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

• E.g., the sample mean ( ) is employed to estimate the population mean ( ).

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Estimation

• The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

• There are two types of estimators:

• Point Estimator

• Interval Estimator

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Estimation

• For example, suppose we want to estimate the mean weekly income for business students at the university. For n=25 students,

• is calculated to be 500 $/week.

• point estimate interval estimate

• An alternative statement is:• The mean income is between 480 and 520 $/week.

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Confidence Interval Estimator for • If you know the value for σ

• What is Z/2 ? Traditionally this is the Z-score required in order to get /2 of the area in the “RIGHT HAND TAIL” of the normal distribution.

• However, this text will use Z(1-/2) for this value.

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Notation Issue You will Just Have to Live With.

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Normal Z-Scores Used

• These Z-Scores will cover most problems you will need for confidence intervals as well as hypothesis testing later.

• Keep in mind the notation differences in this text.

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Example: Confidence Interval

• A nurse records the time in minutes it takes a patient to recover after an operation before they are placed in a private room:

• Its is known that the standard deviation of recovery time is 75 minutes. We want to estimate the mean time with 95% confidence in order to help determine the number of beds needed in recovery.

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Example:

• “We want to estimate the mean recovery time with 95% confidence”

• The parameter to be estimated is the population mean:

• The 95% confidence interval is

• Mechanically, all we need to do is fill in the numbers for each symbol in this formula.

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Example:

• For this problem, we will need to calculate the sample mean [later on we will also need to calculate the sample standard deviation to use for an unknown σ]

• The lower and upper confidence limits are 340.76 and 399.56

• Meaning: The odds are pretty high (95%) that this interval actually contains the true mean.

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Confidence Intervals

• What affects the width of these confidence intervals.

• Effect of “n”• Effect of “σ”• Effect of Confident Level (Z-Score)• Effect of

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Sample Size to Achieve a Given Confidence Interval

• We can control the width of the confidence interval by determining the sample size necessary to produce a desired confidence interval.

• Suppose we want to estimate the mean demand “to within 5 units”;• i.e. we want to the interval estimate to be:

• From the formula for the confidence interval

• This implies that

• Solving for n results in

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Sample Size to Achieve a Given Confidence Interval

• The general formula for the sample size needed to estimate a population mean within + W with (1-)100% confidence is

• This author will use “d” instead of “W”• Note that you need to know σ in order to

determine a sample size (n). Where do you get a value for σ??

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Sample Size Example

• A lumber company wishes to estimate the mean diameter of trees in order to determine the amount of money they will pay the land owner. They need to estimate this to within 1 inch at a confidence level of 99%. The tree diameters are normally distributed with a standard deviation of 6 inches.

• How many trees need to be sampled?

• How would you take this ramdom sample????

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Confidence Interval for Unknown σ

• When the population standard deviation is unknown and the population is normal, we will use the sample standard deviation, s, instead of the unknown value for σ. The confidence interval now becomes

• Note that “t” replaces “Z” and “s” replaces “σ” in the confidence interval formula

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The “t” Statistic• The t-distribution looks very much like the

standard normal distribution but is spread out wider (like someone sat on the normal distribution and bulged it out). The t statistic has one parameter called the degrees of freedom (d.f., , or some other notation). In this case the d.f. for a confidence interval is d.f. = n – 1

• See Table E in back of text. Note that the same notation variation applies here as it did for the Z statistics earlier.

• t/2 = t statistic to get /2 area to right (standard notation used by most people)

• t1-/2 = t statistic to get 1- /2 to left (this text)

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Confidence Interval Unknown Standard Deviation

• A nurse records the time in minutes it takes a patient to recover after an operation before they are placed in a private room: A sample of 25 times were recorded and the sample mean calculated to be 370.12 and sample standard deviation calculated to be 80.8. [same data as previous example]

• We want to estimate the mean time with 95% confidence in order to help determine the number of beds needed in recovery.

• Students calculate this in class!!!!!!!!!!!!!!!!!

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HW: Estimation

• 6.2.1, 6.2.3, 6.3.1, 6.3.5, 6.5.1, 6.5.3, 6.7.1, 6.7.3, 6.8.1, 6.8.3

• Chap 6: Review questions and exercises

• 3, 6, 9