SB 8.1-8.2

A theorem that allows us to use the normal probability distribution to approximate the sampling distribution of the sample mean whenever the sample size is large is known as

the Central Limit Theorem.

You discover that if μ =10, then the probability of observing x(bar) ≤ 8.8 is 0.0034. Your observed value of x(bar) = 8.8. What do you conclude?

μ < 10

The standard deviation of the sampling distribution of x(bar) is equal to

σ√n.

The standard deviation of p̂ equals

√(p(1−p))/(n)

A random sample of size 100 is taken from a population whose population proportion is 0.40. The expected value of the sample proportion is

0.40

A population has a mean of 50 and a standard deviation of 10. A random sample of 256 is selected. The standard deviation of x(bar) is equal to

0.625

The probability distribution describing the set of all possible values of s2 is called

the sampling distribution of s2.

A list of all possible values of the sample mean and the probabilities of these values being observed is known as the ______.

sampling distribution of x(bar)

The formula √(p(1-p))/n calculates the ________

standard deviation of p̂

The shape of the sampling distribution of p̂ becomes more normal as _________

the sample size increases.

Suppose we find that if a population proportion is 0.72, the chance of seeing a sample proportion that is less than or equal to 0.70 is 0.1251. If we observe p̂ = 0.70,

we have insufficient evidence to doubt that p = 0.72.

The sampling distribution of ____ represents a list of all possible values of the sample mean and the probabilities of these values being observed.

x(bar)

A population has a mean of 100 and a standard deviation of 10. A random sample of 25 is selected. The expected value of x(bar) is equal to

100

If we assume that a population has proportion p = 0.2 and we choose a random sample of size 400, what is the chance the sample population p̂ is greater than or equal to 0.25?

.0062

Suppose we choose a sample of size 100 from a population of monthly cable bills having standard deviation $20. If we assume the population mean bill is $65, what is the probability mean of our sample is greater than $70?

.0062 Reason: Since x(bar) has expected value 65 and standard deviation 20√100 = 2, this is P{Z > 2.5} = 0.0062.

Suppose you choose a sample of size 100 from a population having 25% successes. The sample proportion of successes will have standard deviation

.0433 Reason: The standard deviation is √(p(1-p))/n = √(.25(.75))/(100) = 0.0433.

Consider a population having mean μ =100. If μμ̂ is an unbiased point estimate of μ, then the mean of μ̂ is

100.

A population has a mean of 100 and a standard deviation of 12. A random sample of 36 is selected. The standard deviation of x(bar) is equal to

2

The central limit theorem states that the distribution of the sample mean will be approximately normal if the sample size is sufficiently large; as a general guideline n≥ ______

30

Consider a population having mean μ = 100 and variance σ2 = 36. Given that s2 is an unbiased estimate of σ2, the mean value of s2 must be

36

A population has a mean of 50 and a standard deviation of 10. A random sample of 144 is selected. The expected value of xx is equal to

50

The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution of the sample mean becomes

closer to a normal distribution.

In general, the variability between values of the sample mean is _____ the variability between all individual observations.

less than

The sampling distribution of the sample mean is ______.

likely to look more similar to a normal curve than the population distribution does.

As the sample size increases, the shape of the sampling distribution of p̂ becomes

more normal.

In many situations, the distribution of the population of all possible sample means looks like a

normal curve.

For any population proportion p, the sampling distribution of the sample proportion is approximately normally distributed if

np ≥5 and n(1 - p) ≥5.

As a general guideline, the normal distribution approximation can be used to describe the sampling distribution of the sample mean when

n≥30.

The sampling distribution of the sample median tells us about the set of all possible values

of the sample median.

The expected value of p̂ is the

proportion of successes in the population.