Lecture Estimation

7/29/2019 Lecture Estimation

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Binomial distributions for sample counts

Binomial distributions are models for some categorical variables, typically

representing the number of successes in a series of n trials.

The observations must meet these requirements:

The total number of observations n is fixed in advance.

Each observation falls into just 1 of 2 categories: success and failure.

The outcomes of all n observations are statistically independent.

All n observations have the same probability of success, p.

We record the next 50 births at a local hospital. Each newborn is either a

boy or a girl; each baby is either born on a Sunday or not.


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Applications for binomial distributions

Binomial distributions describe the possible number of times that

a particular event will occur in a sequence of observations.

They are used when we want to know about the occurrence of an

event, not its magnitude.

In a clinical trial, a patients condition may improve or not. We study the

number of patients who improved, not how much better they feel.

Is a person ambitious or not? The binomial distribution describes the

number of ambitious persons, not how ambitious they are. In quality control we assess the number of defective items in a lot of

goods, irrespective of the type of defect.


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Reminder: Sampling variability

Each time we take a random sample from a population, we are likely to get adifferent set of individuals and calculate a different statistic. This is called sampling

variability.

If we take a lot of random samples of the same size from a given population, the

variation from sample to samplethe sampling distributionwill follow a

predictable pattern.


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Binomial mean and standard deviation

The center and spread of the binomial

distribution for a count Xare defined by the mean

m and standard deviation s:

)1( pnpnpqnp

Effect of changing p when n is fixed.

a) n = 10, p = 0.25

b) n = 10, p = 0.5

c) n = 10, p = 0.75

For small samples, binomial distributions

are skewed when p is different from 0.5. 00.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

Number of successes

P(X=x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

Number of successes

P(X=x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

Number of successes

P(X=x)

a)

b)

c)


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Sample proportions

The proportion of successes can be more informative than the count. Instatistical sampling the sample proportion of successes, , is used to estimate the

proportion p of successes in a population.

For any SRS of size n, the sample proportion of successes is:

n

X

np

samplein thesuccessesofcount

In an SRS of 50 students in an undergrad class, 10 are Hispanic:

= (10)/(50) = 0.2 (proportion of Hispanics in sample)

The 30 subjects in an SRS are asked to taste an unmarked brand of coffee and rate it

would buy or would not buy. Eighteen subjects rated the coffee would buy.

= (18)/(30) = 0.6 (proportion of would buy)

p

p

p


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Sampling distribution of the sample proportionThe sampling distribution of is never exactly normal. But as the sample size

increases, the sampling distribution of becomes approximately normal.

The normal approximation is most accurate for any fixed n when p is close to 0.5, and

least accurate when p is near 0 or near 1.

pp


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Estimation

Estimation A process whereby we select

a random sample from a population and use

a sample statistic to estimate a population

parameter.


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Point and Interval Estimation

Point Estimate A sample statistic used toestimate the exact value of a population

parameter

Confidence interval (interval estimate) Arange of values defined by the confidence levelwithin which the population parameter is

estimated to fall.

Confidence Level The likelihood, expressedas a percentage or a probability, that a specified

interval will contain the population parameter.


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A population distribution variation in the larger

group that we want to know about.

A distribution of sample observations variation in the sample that we can observe.

A sampling distribution a normal distribution

whose mean and standard deviation are unbiased

estimates of the parameters and allows one to infer

the parameters from the statistics.

Inferential Statistics involves

Three Distributions:


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What does this Theorem tell us: Even if a population distribution is skewed, we know that the

sampling distribution of the mean is normally distributed

As the sample size gets larger the mean of the sampling

distribution becomes equal to the population mean As the sample size gets larger the standard error of the mean

decreases in size (which means that the variability in the sample

estimates from sample to sample decreases as n increases).

It is important to remember that researchers do not

typically conduct repeated samples of the same

population. Instead, they use the knowledge of theoretical

sampling distributions to construct confidence intervals

around estimates.

The Central Limit Theorem

Revisited


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A range of reasonable guesses at a population value,for example, a mean.

Confidence level = chance that range of guessescaptures the population value.

Most common confidence level is 95%


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General Format of a Confidence Interval

estimate +/- margin of error


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Accuracy of a mean

A sample of n=36 college women hasmean pulse = 75.3.

The SD of these pulse rates = 8 . How well does this sample mean estimate

the population mean ?


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Standard Error of Mean

SEM = SD of sample / square root of n

SEM = 8 / square root ( 36) = 8 / 6 = 1.33

Margin of error of mean = 2 x SEM Margin of Error = 2.66 , about 2.7


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Interpretation

95% confidence that the sample mean iswithin 2.7 (pulse beats) of the population

mean.

A 95% confidence interval for thepopulation mean

sample mean +/- margin of error 75.3 +/-2.7 ; 72.6 to 78.0


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C.I. for mean pulse of men

n=49

sample mean=70.3, SD = 8

SEM = 8 / square root(49) = 1.1 margin of error=2 x 1.1 = 2.2 Interval is 70.3 +/- 2.2 68.1 to 72.5


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Do men and women differ in

mean pulse? C.I. for women is 72.6 to 78.0 C.I. for men is 68.1 to 72.5 No overlap between intervals We say that population means differ


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Confidence Levels:

Confidence Level The likelihood, expressed as a

percentage or a probability, that a specified interval

will contain the population parameter.

95% confidence level there is a .95 probability that

a specified interval DOES contain the population

mean. In other words, there are 5 chances out of 100

(or 1 chance out of 20) that the interval DOES NOT

contains the population mean.

99% confidence level there is 1 chance out of 100that the interval DOES NOTcontain the population

mean.


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Constructing a

Confidence Interval (CI)

The sample mean is the point estimate of the

population mean.

The sample standard deviation is the pointestimate of the population standard deviation.

The standard error of the mean makes it

possible to state the probability that an

interval around the point estimate contains

the actual population mean.


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Standard error of the mean the standard

deviation of a sampling distribution

n

x

x

Standard Error

The Standard Error


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n

x

x

Since the standard error is generally not known, we

usually work with the estimated standard error:

n

ss xx

Estimating standard errors


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)( xSEZXCI

Determining a

Confidence Interval (CI)

)(

n

sZXCI x

Given a large enough sample, any confidence interval for the

population mean may be constructed:

Where z is chosen from a standard normal distribution table to

obtain a desired degree of confidence.


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Confidence Level Increasing our confidence levelfrom 95% to 99% means we are less willing to draw

the wrong conclusion we take a 1% risk (ratherthan a 5%) that the specified interval does not contain

the true population mean.If we reduce our risk of being wrong, then we need a

wider range of values . . . So theinterval becomeslessprecise.

)(n

sZX x

Confidence Interval Width


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Confidence Interval Z Values


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Sample Size Larger samples result in smallerstandard errors, and therefore, in sampling

distributions that are more clustered around the

population mean. A more closely clustered sampling

distribution indicates that our confidence intervals

will be narrower and more precise.


)(n

sZX x


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Standard Deviation Smaller sample standarddeviations result in smaller, more precise confidence

intervals.

(Unlike sample size and confidence level, the

researcher plays no role in determining the standard

deviation of a sample.)


)(n

sZX x


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Finding confidence interval of the mean years of education of

voters. (Table 9.4, Hamilton)

Mean = 12.97 years

Standard deviation = 2.42 years

Number of cases n= 155

Calculation of 95 percent confidence interval.

)(n

sZX x

)

155

42.2(96.197.12

38.097.12

So the interval is 12.59 13.35


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Interpretation

Informal: Based on our analysis of thisparticular sample, we are about 95% confident

that the mean education among all voters in

this town lies between 12.59 and 13.35 years.

Formal: If we took a large number of random

samples, each with 155 cases, and calculated

confidence intervals in this manner for each

sample, about 95% of those confidence

intervals should include the true population

mean .


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Estimating the standard error of a proportion

basedon the Central Limit Theorem, a sampling distribution of

proportions is approximately normal, with a mean, ,

equal to the population proportion, , and with a standard

error of proportions equal to:

n

1

Since the standard error of proportions is generally not

known, we usually work with the estimated standarderror:

n

s

1

Confidence Intervals for Proportions


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Determining a Confidence Interval

for a Proportion

n

ZSEZ

1)(

Large sample confidence intervals for proportions

are found as

Where z is chosen from a table of the standard normal

distribution to give the desired degree of confidence.


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Finding an approximate 95% confidence interval for the

proportion favoring school closings.

Sample statistics:

Proportion favoring school closed = 0.431

Number of cases n = 153

Confidence interval for population proportion

n

ZSEZ

1)(

153

431.01431.096.1431.0

078.0431.0 So the interval is 0.353 0.509


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Interpretation

Informal: Based on our analysis of this one sample weare about 95% confident that the proportion in favor

of closing schools, among all voters in this town, lies

between 0.353 and 0.509.

Formal: If we took a large number of randomsamples, each with 153 cases, and calculated

confidence intervals in this manner for each sample,

about 95% of those confidence intervals should

include the true population proportion .

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