27 September, 2011 STAT 101 -- Part VI 1

Highlight the last lecture

27 September, 2011 STAT 101 -- Part VI 2

Highlight the last lecture (cont’d)

Assumption: Population is normal

n is large • Resampling

• nonparametric

yesyes

No No

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VII. Confidence Intervals

Point and interval estimations ofMeans of normal distribution and non-normal distributionProportion parameter of binomial distribution

Determining sample sizeFor the meanFor the proportion

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Confidence interval estimation

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Confidence Intervals for the population mean

Assumptions: Population is normally distributed Standard deviation of population is given

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Example: protein intake Find the 95% confidence interval for the average daily

protein intake of men aged 20-25. Population standard deviation is 58.6 grams. The random sample with size 267 men aged 20-25 is

observed. The margin of error is Before collected the data from 267 men, we can say that

there are 95% chance of the random interval will include

It is noted that sample mean is still a random variable before collecting any data and we are still talking about probability.

After collecting the daily protein intake of these 267 men and calculating the sample mean of 72.1 grams.

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After obtained the numerical result from sampling, we can not say that the population mean falls between 65.071g and 79.129g with 95% chance.

The correct way to present the result is: The 95% confidence interval for the average

daily protein intake for men aged 20-25 is (65.071g, 79.129g)

Having determined a numerical result from one specific sample, it is no longer sensible to speak about the probability of its covering the fixed quantity

If many repeated samples with same sample size were taken from the same population and the confidence intervals were constructed, the proportion of intervals containing would be approximately 0.95.

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Excel output of the example (Protein Intake):

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Interpretation of confidence intervals

Values below true mean Values above true mean

True population mean

http://www.socr.ucla.edu/Applets.dir/ConfidenceInterval.html

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Confidence intervals for the mean with unknown population variance

The only assumption is the population distribution is normal. The population standard deviation is unknown. It is reasonable to estimate the population standard deviation from

the sample standard deviation.

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Why is it t-distribution?

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Insurance example (cont’d)

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Excel output of the example (insurance):

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Large sample size cases No assumptions of normal population distribution or the

population variance. If the sample size is sufficiently large, the Central-Limit

Theorem may be applied to guarantee that

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Flow Chart for determining the distributions

Is population distribution normal?

Is population standard deviation given?

Normal tables t-distribution tables

Normal tables

Large sample

size (>120)

yes

yes

no

no

Is sample size sufficiently large (n

>=30), such that CLT applied?

Use other methods

yes no

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Factors affecting the length of a confidence interval The shorter the length of confidence interval, the better the

estimation Consider the confidence interval for population mean

The length of confidence interval is then The length depends on S, n and

n n increases, length decreases

increases (confidence level decrease), length decreases

S S increases, length increases

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Determining sample size for the mean

The required sample size can be found to reach a desired margin of error with a specified level of confidence.

The margin of error is also called sampling error

The margin of error can be interpreted as

the amount of imprecision in the estimate of the population parameter

the amount added and subtracted to the point estimate to form the confidence interval

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Requirements of determining sample size

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Numerical example A consumer group wants to estimate the mean electric bill for

the amount of July for single-family homes in a large city. Based on studies conducted in other cities, the standard

deviation is assumed to be $25. The group wants to estimate the mean bill for July to within ±

$5 with 99% confidence. What sample size is needed?

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Estimation for the binomial distribution

Recall the common structure of the binomial distribution: A sample of n independent trials Each trial can have only two possible

outcomes which are denoted as `success’ and `failure’

The probability of a success at each trial is assumed to be constant p

The parameters of the binomial distribution are n and p

Now, assume that p is unknown and we want to use the sample proportion to estimate p

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Point estimation: sample proportion

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Sampling distribution of

Sampling distribution of sample proportion

Population

1st sample of n

3rd sample of n

kth sample of n

2nd sample of n

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Sampling distribution of sample proportion

In previous section, we discussed that normal approximation to the binomial distribution

In fact, the normal approximation can be justified on the basis of the Central-Limit Theorem since sample proportion is just a sample mean

The textbook uses the rule of CLT: By the CLT, we get

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Example During June and July of 2001, the European Union Executive

Commission conducted a study of 6,543 European adults. Of those surveyed, 56% said that the euro single currency would promote economic growth and 73% knew the correct date of the changeover (January 1, 2002).

Construct a 95% confidence interval estimate for the proportion of European adults who believe that the euro would promote economic growth.

Interpret the interval constructed.

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Excel output of the example (Euro)

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Requirements of determining sample size for the proportion

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Numerical example: A study of 658 CEOs conducted by the Conference Board reported

that 250 stated that their company’s greatest concern was sustained and steady top-line growth (“CEOs’ Greatest Concerns,” USA Today Snapshots, May 8, 2006, P1D).

To conduct a follow-up study to estimate the population proportion of CEOs whose greatest concern was sustained and steady top-line growth to within ±0.01 with 95% confidence, how many CEOs would you survey?

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Useful and interesting websites

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http://www.socr.ucla.edu/Applets.dir/ConfidenceInterval.html Confidence Intervals

simulations

http://en.wikipedia.org/wiki/Confidence_intervalConfidence Intervals information

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10 15 20 25 30 35 40

G2

G1

Assignment 2: Box-and-Whisker Plot

Stem-and-Leaf Display

Stem unit: 1

Statistics 12 0Sample Size 94 13Mean 37.08511 14Median 40 15Std. Deviation 4.760183 16Minimum 12 17Maximum 40 18

1920212223 0242526 0 027 028 0 02930 031 0 0 032 0 033 0 034 0 0 0 035 0 0 0 0 036 0 0 0 037 0 0 0 0 0 0 038 0 0 0 039 0 0 0

40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0