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Happiness comes not from material wealth but less desire.

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Means and Proportions as Random Variables

Sampling distributionNormal curve approximation

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Definitions

A statistic is a numerical summary of a sample. Its value may differ for different samples.

A parameter is a numerical summary of a population, which is a (unknown) constant.

The sampling distribution of a statistic is the distribution of possible values of the statistic for repeated random samples of the same size taken from a population.

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Sample Proportion

In a random sample of size n, the sample proportion is the proportion of, say women, out of the sample. For example, if there are 8 women in a sample of 10 students, then the sample proportion for women is =8/10=0.8.

The population proportion is denoted as p.

Q: Identify each of the following as a statistic or a parameter: 1) p 2) .

p̂

p̂

p̂

p̂

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Sampling Distribution of

Mean:

Standard deviation:

Standard error, the estimated standard deviation:

ppE )ˆ(

p̂

n

pppStE

)ˆ1(ˆ)ˆ(

n

pppStD

)1()ˆ(

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Normal Curve Approximation for Sample Proportion

The sampling distribution of can be approximated by a normal distribution with the mean p and standard deviation

WHEN

both np > 5 and n(1-p) > 5.

Empirical Rule

p̂

n

pp )1(

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Example: Lottery

Suppose that probability is p=0.2 that a person purchasing an instant lottery ticket wins money, and this probability holds for every ticket purchased. Consider all random samples of 64 purchased tickets, and let be the sample proportion of winning tickets in a sample of 64 tickets.

p̂

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• What is the average (sample) proportion of winning tickets in 64 randomly selected tickets?

• What is the standard deviation of the sample pro-portion ?

• What is the probability that there are more than 32 winning tickets in 64 randomly selected tickets?

• Using Empirical Rule to fill the blanks:In about 95% of all random samples of 64 purchased tickets, the proportion of winning tickets will be between ____ and ____.

p̂

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Sample Mean X

The sample mean of a random sample of size n is the average in that sample.

Let and denote the population mean and standard deviation of the population sampled.

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Mean:

Standard deviation:

Standard error, the estimated standard deviation:

Sampling Distribution of X

)(XE

nXSD /)(

nsXSE /)(

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Normal Curve Approximation for Sample Mean

The sampling distribution of the sample mean, from a random sample of size n (where n is at least 30) can be approximated by a normal distribution with the mean and standard deviation .

Empirical Rule

x

n

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Example: Speed at 880

Vehicle speeds at highway 880 are believed to have mean 60 mph. The speeds for a randomly selected sample of n=36 vehicles at highway 880 were recorded and its standard deviation is s=6 mph.

What is the mean and standard error of (the sampling distribution of) sample mean?

Use the Empirical Rule to fill the blanks: For a random sample of 36 vehicles, there is about a 68% chance that the mean vehicle speed in the sample will be between ____ and ____.