sampling distribution of the sample mean

X -bar=µ (sampling distribution = population distribution) true or false?

false

X-bar=x-bar true or false?

false

sampling distribution?

find the distribution of all possible values of the sample statistic (from all possible samples of size n) -find its mean, variability (standard error) and shape -use this info to make inferences about the population parameter

If the random variable X has a normal distribution, then the random variable X-bar has a shape which is a _______ distribution

normal

sample to study..? statistic to get..?

sample to study population statistic to get parameter

If X has a normal distribution, then X-bar has a normal distribution no matter what. true or false?

true

case 2: X does not have a normal distribution, but still has mean u and SD o. the shape of the distribution of x is: the shape of the sampling distribution of x bar is:

-anything but normal -approx normal if n large enough

case 1: X has a normal distribution with mean u and SD o. the shape of the distribution of X is: the shape of the sampling distribution of X bar is:

-normal -also normal with same mean and smaller SE

what is a sampling distribution?

set of all values of a statistic for all possible samples of size n from the population

How large does n generally have to be in order for the Central Limit Theorem to take effect? (Assume X does not have a normal distribution.) A. n can be any value B. n > 30 C. np ≥ 10 D. np at least 10 and n(1-p) at least 10

b.

If X has a normal distribution, which of the following is true about the shape (distribution) of the random variable X-bar? A. X-bar is approximately normal for any n B. X-bar is exactly normal for any n C. X-bar is exactly normal if n > 30 D. X-bar is approximately normal if n > 30

b.

What is the symbol we use to represent the mean of the random variable X-bar? A.Mu with subscript X B.Mu with subscript X-bar C.Capital X-bar

b.

How large does n have to be for the CLT to take effect, assuming X did not have a normal distribution? A. n at least 30 B. np at least 10 and n(1-p) at least 10 C. n can be any number D. None of these

a.

Let X-bar be the sample mean from X's distribution. Suppose X does not have a normal distribution. The distribution (shape) of X-bar is: A. Exactly normal B. Approximately normal for any sample size by the Central Limit Theorem C. Approximately normal for large enough samples by the Central Limit Theorem D. None of the above

c.

Suppose Bob wants to estimate the current GPA of all freshmen students at OSU. He takes a random sample of 100 freshmen at OSU and finds their current mean GPA is 3.15. Describe the population, parameter, sample and statistic

population = all freshman students at OSU parameter = the average current GPA of all freshman students at OSU sample = 100 freshman students at OSU statistic = the average current GPA (3.15) of the 100 freshman students at OSU sampled

population? parameter? sample? statistic?

population = entire group of interest parameter = one # the describes the population (ex: avg price in US) sample = subset of population statistic = one number that describes the sample (ex: avg gas price of 100 stations)

The Central Limit Theorem tells us about the _____________________ of the sampling distribution of X-bar when X does not have a normal distribution:

shape

If the values in the population are skewed, the sampling distribution of X-bar can be approximated by the normal distribution if the sample contains at least 30 observations. true or false?

true

Let X-bar be the sample mean from X's distribution. The standard error of X-bar is: A. The same as the standard deviation of X B. Greater than the standard deviation of X C. Less than the standard deviation of X

C.

The population mean: A. Is always larger than or equal to the sample mean B. Is always smaller than or equal to the sample mean C. Can be smaller than, or larger than, or equal to the sample mean D. None of the above

C.

The mean of X-bar equals the mean of X when? A. Only if X has a normal distribution B. Only if n is large C. Both a and b are needed D. This statement is always true no matter what.

D.

If X has a normal distribution, could you find a probability for X-bar if n was only 10? A. Yes, because X has a normal distribution already. B. No, because you always need n to be at least 30.

a.

The Central Limit Theorem states that, if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean X-bar: A. Is approximately normal if n ≥ 30 B. Is approximately normal if n < 30 C. Is approximately normal if the underlying population is normal D. Has the same variance as the population

a.

The Central Limit theorem tells us important results that pertain to: A. The shape (type) of the distribution of X-bar B. All of the above C. The standard error of the distribution of X-bar D. The mean of the distribution of X-bar

a.

A company has developed a new battery, but the average lifetime of all of the batteries it makes is unknown. In order to estimate this average, a sample of 500 batteries is tested and the average lifetime of this sample is found to be 225 hours. The 225 hours is the value of a:

statistic

what affects the standard error and how?

1. as n increases, SE decreases (more info, more data, better precision, closer to Mx) 2. as ox increases, more diversity/variability in the population (SE also increases)

As n increases, which of the following statements is true? A. The population standard deviation decreases. B. The standard error of the sample mean decreases. C. The population standard deviation increases. D. The standard error of the sample mean increases.

B.

The Central Limit Theorem is important in statistics because: A. For n large, it says the distribution of the sample mean is approximately normal, regardless of the shape of the population. B. For any sample size, it says the sampling distribution of the sample mean is approximately normal. C. For n large, it says the distribution of the sample mean is exactly normal, regardless of the shape of the population. D. For any sample size, it says the sampling distribution of the sample mean is exactly normal

a.

what two ways can we do statistical inference?

a. estimate a population parameter using info from a sample b. test a population parameter using info from a sample

If X does not have a normal distribution, which of the following is true about the shape (distribution) of the random variable X-bar? A. X-bar is exactly normal if n > 30 B. X-bar is approximately normal for any n C. X-bar is exactly normal for any n D. X-bar is approximately normal if n > 30

d.

statistic interference?

goal: estimate or test a population parameter using info from the sample steps: 1. take a sample 2. get a sample statistic -measure and take into account the variability from sample to sample -use this info to draw conclusions about the population based on your sample

CLT

if X has ANY distribution (not normal) then the shape of the sample distribution of x bar is approx normal, as long as n>30 --> CLT only applies to shapes, not mean or SE -averages average out to the middle, they get closer to Mx as n --> infinity (appears to look normal as n increases)

mean of all sample means = mean of the population a. write this sentence in symbols b. explain why this equality makes sense

a. u(xbar) = ux b. makes sense bc our population of all possible values of X and our sampling distribution is the set of all possible samples which includes all possible values of X, n times

If X has ___ distribution EXCEPT a _____ distribution, then the shape of the sampling distribution of X-bar is approximately _____ if n is large enough.

any, normal, normal

Which of the following describes the relationship between parameters and statistics? A. We use parameters to estimate or test statistics B. We use statistics to estimate or test parameters C. Statistics and parameters are the same thing

b.

Which of the following is true about the mean of the random variable X-bar as n increases? A. The mean of X-bar increases B. The mean of X-bar stays the same C. The mean of X-bar decreases D. Not enough information to tell

b.

Which of the following is true about the standard error of the random variable X-bar as n increases? A. The standard error of X-bar stays the same. B. The standard error of X-bar decreases. C. The standard error of X-bar increases. D. Not enough information to tell.

b.

Suppose X has a normal distribution and X-bar represents the average of a sample of size n. Which of the following is true? A. X-bar has an approximate normal distribution for any value of n. B. X-bar has an approximate normal distribution if n is large enough. C. X-bar has an exact normal distribution for any value of n. D. X-bar has an exact normal distribution if the sample size is large enough.

c.

The set of all possible sample means from all possible samples of size n from the population is known as the: A. Population probability distribution of X B. Not enough information to tell C. Sampling distribution of X-bar D. Sample mean of the population of X

c.

What aspects of X-bar does the CLT apply to? Shape Mean Standard Error

shape