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# Ch8_Sampling Dist.

### Concepts of Sampling distributions of means and proportions

Is a sample mean, “x-bar”, a random variables? Why? Yes, x-bar is a random variables because its value varies from sample to sample.
Do the sample means, “x-bars”, have an associated probability distribution? Yes. Just like any other random variables, the “x-bars” have probability distributions associated with them. That is, the sample means have a “shape”, “center” and “spread”.
What do we mean by the “sampling distribution” of a statistic? The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n.
What do we mean by the “sampling distribution of the sample mean (x-bar)”? The sampling distribution of the sample mean, “x-bar”, is the probability distribution of ALL possible values of the random variable, x-bar, computed from a sample of size “n” taken from a population with mean  and standard deviation .
Given a simple random sample of size n drawn from a large population with mean  and standard deviation , what do we know about the “mean” and “standard deviation” of the sampling distribution of x-bar? The “sampling distribution of x-bar” will have mean, µ(x-bar) = µ, and standard deviation, σ(x-bar) = σ/sqrt(n).
What is the standard deviation of the sampling distribution of “x-bar” called? The standard deviation of the sampling distribution of “x-bar” is called the “standard error of the mean”.
If a random variable X is normally distributed, what do we know about the distribution of the sample means, “x-bars” If a random variable X is normally distributed, the distribution of the sample means, “x-bars”, is automatically normally distributed.
What does the “Central Limit Theorem” tell us? According to the “Central Limit Theorem”, regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) becomes approximately normal as the sample size n increases.
How large does the sample size, “n, have to be before the distribution of the sample means is approximately normal? Regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) will be approximately normal if the sample size n ≥ 30.
Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean. A) SHAPE: The distribution of the “x-bar” is approximately normal because the sample size, n = 35, is greater than 30; B) CENTER: the mean = 11.4; C) SPREAD: the standard deviation = 3.2/sqrt(35) = 0.5409
Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. If a random sample of n = 35, what is the probability the mean oil change time is less than 11 minutes? Solution: P(x < 11) = P[Z < (11 – 11.4)/0.5409] = P(Z < -0.74) = 0.23
What is a reasonable “point estimate” for a POPULATION proportion, designated “p”? A SAMPLE proportion, designated “p-hat”, is a reasonable point estimate for a population proportion.
How is a sample proportion, “p-hat”, calculated? The sample proportion (“p-hat”) is given by p-hat = x/n, where “x” is the number of individuals in the sample with the specified characteristic and “n” is the sample size.
Is a sample proportion, “p-hat”, a random variables? Why? Yes, “p-hat” is a random variables because its value varies from sample to sample.
Do the sample proportions, “p-hats”, have an associated probability distribution? Yes. Just like any other random variables, the “p-hats” have probability distributions associated with them. That is, the sample proportions have a “shape”, “center” and “spread”.
What do we mean by the “sampling distribution of the sample proportions (p-hat)”? The sampling distribution of the sample proportions, “p-hat”, is the probability distribution of ALL possible values of the random variable, p-hat, computed from a sample (number of trials) “n”.
For sample proportions, is there a condition on the size of the sample with respect to the size of the population? Yes. If the sample size is denoted as “n” and the population size is denoted as “N”, the following condition is needed to satisfy the INDEPENDENCE of trials requirement: n ≤ 0.05N.
In terms of the SHAPE of the distribution of the sample proportions (“p-hats”), when can we assume it is approximately normal? The shape of the sampling distribution of “p-hat” is approximately normal provided np(1 – p) ≥ 10.
In terms of the CENTER of the distribution of the sample proportions (“p-hat”), what is the mean of the sample proportions? The mean of the sample proportions, µ(p-hat) = p, here “p” is the population proportion.
In terms of the SPREAD of the distribution of the sample proportions (“p-hat”), what is the standard deviation of the sample proportions? The standard deviation of the sample proportions, σ(p-hat) = sqrt[p(1 – p)/n], “n” is the sample size and “p” is the population proportion.
Created by: wgriffin410