Concepts of Sampling distributions of means and proportions
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| 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.
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| 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”.
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| 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.
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| 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 .
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| 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).
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| 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”.
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| 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.
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| 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.
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| 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.
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| 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
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| 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
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| 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.
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| 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.
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| 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.
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| 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”.
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| 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”.
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| 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.
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| 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.
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| 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.
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| 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.
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