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Parameter Estimation and Standard Error in Statistics, Study notes of Psychology

University of South Florida (USF)Psychology

An introduction to parameter estimation using statistics, focusing on the concepts of bias, standard error, and sampling distribution. It covers the difference between biased and unbiased estimators, the role of standard error in measuring the precision of an estimator, and the relationship between standard error, population sd, and sample size.

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

koofers-user-h48 🇺🇸

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Slide 1

Estimating Parameters

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Slide 2 Parameter Estimation

We use statistics to estimate parameters,

e.g., effectiveness of pilot training, psychotherapy.

We want to know how good

our estimates are.

Most common ways to examine goodness of a

statistic as an estimator are biasand standard error.

We will define both, but first:

µ

→X

σ

→

SD

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Slide 3 Sampling Distribution

• A sampling distribution is a distribution of a

statistic over many samples.

• To get a sampling distribution,

– 1. Take a sample of size N (a given number like

5, 10, or 1000) from a population

– 2. Compute the statistic (e.g., the mean) and

record it.

– 3. Repeat 1 and 2 a lot (infinitely).

– 4. Plot the resulting sampling distribution, a

distribution of a statistic over repeated samples.

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Discover Study notes of Psychology University of South Florida (USF)

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Slide 1

Estimating Parameters

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Slide 2 (^) Parameter Estimation

e.g., effectiveness of pilot training, psychotherapy.We use statistics to estimate parameters, We want to know how goodour estimates are. Most common ways to examine goodness of astatistic as an estimator are bias and standard error. We will define both, but first:

X → μ SD → σ

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Slide 3 (^) Sampling Distribution

A sampling distribution is a distribution of astatistic over many samples.
To get a sampling distribution,– 1. Take a sample of size N (a given number like
- 1. Compute the statistic (e.g., the mean) and5, 10, or 1000) from a population
- 1. Repeat 1 and 2 a lot (infinitely).record it.
- 1. Plot the resultingdistribution of a statistic over repeated samples sampling distribution, a.

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Slide 4 (^) Sampling Distribution

Class exercise• Find some people’s height, graph it.
Take subsamples of different sizes NFind the mean. and compute mean height. Graph theresults.
What happens as N gets larger?

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Slide 5 (^) Bias

If the mean of the sampling distributionequals the parameter, the statistic is said
If the mean of the sampling distributionto be^ unbiased. does not equal the parameter, thestatistic is biased.
The mean is an unbiased estimator.The average value of X is μ.

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Slide 6 (^) Bias

The sample standard deviation andvariance are biased estimators of their population values. Fortunately, theestimators can be made unbiased with a simple correction. Use in the denominator. All stat packages N-1 instead of N (SPSS) do this. ˆ ( 1 ) 2 2 σ = ∑ (^) NX −− X ˆ ( 1 ) 2 (hat means sample estimate of parameter)^ σ=^ ∑^ NX^ −− X

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Slide 10 (^) Standard Error

This means that the standard error getslarge when the population SD is large and when our sample size is small.
We can make our estimates as preciseas we want (small standard error) by increasing the size of the sample, thatis, by using more participants in our research.

σ (^) X =^ σ N

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Slide 11 (^) Review

What do these terms mean?• Sampling distribution
Bias
Standard error

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Slide 12 (^) Central Limit Theorem

As N increases, the samplingdistribution of means becomes Normal.

0.0 Saturation of Sugar 1.

Tea Preferences as a function of sweetness

0.0 0.5 1.

Population and Sampling Distributions Population preference Sampling DistributionN=

Sampling DistributionN=

Notice:1. Location of means. 2.3. Size of sampling variances.Shape of distributions.

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Slide 13 (^) Descriptive vs. Inferential Statistics• The mean and standard deviation can be used in 2 ways. One way is to describe thedistribution of data (our mean is Xbar).

The other way is to infer something about apopulation (is the population mean 25+?).
Because the sampling distribution of themean is normally distributed, we can use the normal to show how close the parameter islikely to be to the sample mean and to make decisions about treatments.

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Slide 14 (^) Statistical Tables

There are several well-studied statisticaltables that are used for conducting
One of these isstatistical tests. z , the unit normal. This table shows areas or percentages thatcorrespond to various distances from the mean when measured in SD units.We use z for large sample tests.

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Slide 15 (^) Statistical Tables

Another commonly used table isvalues of t are basically the same as t. The z , butsample size gets small. t spreads out more and more as the
t SD with small samples. The values of takes into account the error in and z and t are virtually identical if N>100.

X

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Slide 19 (^) Estimated Mean With Confidence Interval (2)• Let’s say we want to create a 95% confidence interval so 95/100 times, CIwill contain the population mean.

95%CI =
Sample mean plus/minus the value ofthe t distribution that contains 95 percent of the distribution times thestandard error of the mean. The 95 percent value is called a critical value.

X ± t. 05 S X

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Slide 20 (^) Estimated Mean With Confidence Interval (3)

-3 -2 Standard Errors of the Mean-1 0 1 2 3

Frequency

Confidence Interval

-3 -2 Standard Errors of the Mean-1 0 1 2 3

Confidence Interval

approx 75 percent interval approx 95 percent interval

X = μˆ

Find the value of t

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Slide 21 (^) Example Confidence Interval

Want to estimate height of students at USF.Sampled N=100 students. Found mean =
in and SD = 6 in.Best guess for population mean is 68 inches
plus or minus some.95%CI =
• 95%CI=68±(1.98)[6/sqrt(100)]68 ±1.98(.6) = 68 ±1.
Interval is 66.81 to 69.19. Such an intervalwill contain the mean 95% of the time.

X ± t. 05 S X

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Slide 22 (^) Example Confidence Interval

(^63636363 66666666) Height 69696969 72727272 75757575

er boundnce Interval Confidence IntervalUpper bound Population freq dist

Note. Sample mean is close topopulation mean. Confidence interval is computed about the sample mean. The confidence interval contains the populationmean! Yay!

Parameter Estimation and Standard Error in Statistics, Study notes of Psychology

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