INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 5: Sampling Distributions

(Part 1)

Dr. Nahid Sultana

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Chapter 5: Sampling Distributions

5.1 The Sampling Distribution of a Sample Mean 5.2 Sampling Distributions for Counts and Proportions

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5.2 Sampling Distributions for Counts and Proportions

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Binomial Distributions for Sample Counts

Binomial Distributions in Statistical Sampling

Binomial Mean and Standard Deviation

Sample Proportions

Normal Approximation for Counts and Proportions

Binomial Formula

Study of sampling distributions with the simplest case of a random variable, where there are only two possible outcomes.

Example: A sample survey asks 2000 college students whether they think that parents put too much pressure on their children. We will view the responses of these sampled students as an SRS from a population. Here we have only two possible outcomes for a random variable.

Count and Sample proportion

Example: Here n = 2000 college students and X=840 is the number of students who think that parents put too much pressure on their children.

What is the sample proportion of students surveyed who think that parents put too much pressure on their children? 0.42 840/2000 p ==

Let n = the sample size and X = represent the r.v. that gives the count for the outcome of interest. When a random variable has two possible outcomes, we can use the sample proportion, , as a summary.

X/n p =

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Binomial distribution for sample counts

The distribution of a count X depends on how the data are produced. Here is a simple but common situation.

The Binomial Setting

1. There are a fixed number n of observations. 2. The n observations are all independent. 3. Each observation falls into one of just two categories, which for

convenience we call “success” and “failure.” 4. The probability of a success, call it p, is the same for each

observation.

A binomial setting arises when we perform several independent trials (also called observations) of the same chance process and record the number of times that a particular outcome occurs. There are four conditions for a binomial setting:

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The count X of successes in a binomial setting has the binomial distribution with parameters n and p, denoted by B(n, p). where n is the number of trials of the chance process and p is the probability of a success on any one trial. The possible values of X are the whole numbers from 0 to n.

Note: Not all counts have binomial distributions;

Binomial Distribution

Example (Binomial Setting): Tossing a coin n times. 1. n is fixed; 2. each toss gives either heads or tails; 3. knowing the outcome of one toss does not change the probability

of an outcome on any other toss i.e. independent; 4. If we define heads as a success, then p=1/2 is the probability of

a head and is same for any toss. The number of heads in n tosses is a binomial random variable X. The probability distribution of X is called a binomial distribution.

Binomial distribution for sample counts (cont…)

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Binomial distribution for sample counts (cont…)

Example (Binomial Distribution Toss a fair coin 15 times. Give the distribution of X, the number of heads that you observe.

We have 15 independent trials, each with probability of success (heads) equal to 0.5.

So X has the B(15, 0.5) distribution.

Note: The binomial distributions are an important class of discrete probability distributions.

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Binomial Distributions in Statistical Sampling

The binomial distributions are important in statistics when we want to make inferences about the proportion p of successes in a population. Sampling Distribution of a Count Choose an SRS of size n from a population with proportion p of successes. When the population is much larger than the sample, the count X of successes in the sample has approximately the binomial distribution B(n, p).

Note: Usually we use the binomial sampling distribution for counts when the population is at least 20 times as large as the sample.

The accuracy of this approximation improves as the size of the population increases relative to the size of the sample.

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Binomial Mean and Standard Deviation

If a count X has the binomial distribution based on n observations with probability p of success, what is its mean and standard deviation? Here are the facts:

If a count X has the binomial distribution with number of trials n and probability of success p, the mean and standard deviation of X are:

Mean and Standard Deviation of a Binomial Random Variable

)1( pnp

np

X

X

−=

=

σ

μ

Note: These formulas work ONLY for binomial distributions. They can’t be used for other distributions!

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Normal Approximation for Binomial Distributions

As n gets larger, something interesting happens to the shape of a binomial distribution.

Suppose that X has the binomial distribution with n trials and success probability p. When n is large, the distribution of X is approximately Normal with mean and standard deviation

As a rule of thumb, we will use the Normal approximation when n is so large that np ≥ 10 and n(1 – p) ≥ 10.

Normal Approximation for Binomial Distributions

µX = np

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Example:

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A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time-consuming.” Suppose that exactly 60% of all adult U.S. residents would say “Agree” if asked the same question. Let X = the number in the sample who agree. Estimate the probability that 1520 or more of the sample agree.

49.24)40.0)(60.0(2500)1(

1500)60.0(2500

==−=

===

pnp

np

σ

μ

Check the conditions for using a Normal approximation. Since np = 2500(0.60) = 1500 and n(1 – p) = 2500(0.40) = 1000 are both at least 10, we may use the Normal approximation.

3) Calculate P(X ≥ 1520) using a Normal approximation.

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Sampling Distribution of a Sample Proportion

As n increases, the sampling distribution becomes approximately Normal.

Sampling Distribution of a Sample Proportion

. ison distributi sampling theof The pmean

:Then successes. of proportion sample thebe ˆLet successes. of proportion with size of population a from size of SRSan Choose

ppNn

npp

p

)1( ison distributi sampling theofdeviation standard The ˆ

−=σ

For large n, ˆ p has approximately the N(p, p(1− p) /n distribution.

ˆ p =count of successes in sample

size of sample=

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Xp

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Binomial Formula

The number of ways of arranging k successes among n

observations is given by the binomial coefficient

for k = 0, 1, 2, …, n.

Note: n! = n(n – 1)(n – 2)•…•(3)(2)(1) ; and 0! = 1.

nk

=

n!k!(n − k)!

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Binomial Probability

If X has the binomial distribution with n trials and probability p

of success on each trial, the possible values of X are 0, 1, 2,

…, n. If k is any one of these values,

P(X = k) =nk

pk (1− p)n −k