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Chapter 5

Chapter 5

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Department
Statistics
Course Code
STAB22H3
Professor
Moras

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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.
www.notesolution.com
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
www.notesolution.com

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Description
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 = Xn is used when a random variable had 2 possible outcomes. Ex: p- hat = 8402000 = 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. www.notesolution.com
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