Chapter 5

Sampling distributions of statistics provide the link between probability and data.

Sampling distribution tells us about the results we are likely to see.

The Distribution of a statistics

A statistic from a random sample or randomized experiment is a random variable; the probability

distribution of the statistic is it sampling distribution.

Ex 5.1 Heights of young women

We call N(64.5, 2.5) the population distribution.

Population Distribution

The population distribution of a variable is the distribution of its values for all members of the

population. The population distribution is al the probability distribution of the variable when we

choose one individual at random from the population.

5.1 Sampling distributions for counts and proportions

Simplest case of a random variable has only 2 possible outcomes.

N=sample size

X= # of counts that represent outcome of interest

Sample proportion, p-hat = X/n is used when a random variable had 2 possible outcomes.

Ex: p- hat = 840/2000 = 0.42

The Binomial distributions for sample counts

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 success, call it p, is the same for each observation.

Ex: tossing a coin

N= # of tosses

Heads = success, so p is the probability of heads

X= # of heads that show up

Binomial Distributions

The distribution of the count X of successes in the binomial setting is called the binomial

distribution with parameters n and p.

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N= # of observations

P = probability of success on any one observation

X = are the possible values of X that are whole numbers from 0 to n.

X is B(n,p)

Binomial distributions in statistical sampling

Binomial distributions are important in statistics when making inferences about the proportion p

of “success” in a population

Sampling Distributions of a count

A population contains proportions p of successes.

If the population is much larger than the sample, the count X of successes in an SRS of size n has

approximately the binomial distribution B(n,p).

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

the size of the sample.

As a rule of thumb, we will use the binomial sampling distribution for counts when the

population is at least 20 times as large as the sample.

Finding binomial probabilities:

We use table c to find the probabilities P(X = k) of individual outcomes for a binomial random

variable X.

When using the table always stop to ask whether you must count successes or failures,

since the probability on table c does not exceed 0.5, if the success probability is greater than 0.5

than we must count for failures instead.

Ex 5.9 She makes 75% of her free throws.

P(probability of free throws) = 0.75,

since free throws are greater than 0.5 we instead count for P(probabilities of misses) = 0.25

so binomial of misses B(12,0.25)

P(probability of missing 5 or more) = P(X = 5) = P(X = 5) + P(X = 6) + … + P(X = 12) = 0.1576

Therefore, 5 or more out of 12 free throws will be missed by about 16% of the time.

Binomial mean and standard deviation

If a count X has the B(n,p) distribution

µX = np

X = square root of np(1-p)

Sample proportions

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