ResEcon 10

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. What value will be used as the point estimate for the mean endowment of all private colleges in the United States?

$180.975

the probability is 0.95 that a randomly chosen sample of size n will result in a sample mean that falls between

(u - 1.96o/sqrt of n, u+1.96o/sqrt of n)

the probability is 0.95 that the unknown population mean falls within

(xbar - 1.96o/sqrt of n, xbar + 1.96o/sqrt of n)

the (1-a)% confidence interval (estimator) for the population mean, constructed from a random sample of size n and known standard deviation o

(xbar - z1-a/2 * o/sqrt of n, xbar + z1-a/2 * o/sqrt of n) where z1-a/2 = the critical value from the standard normal table (z table) for the (1-a)% confidence level

denote z to be 1 if a randomly surveyed individual believes in global warming

, and 0 if that individual does not believe in global warming

At a computer manufacturing company, the actual size of computer chips is normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A random sample of 12 computer chips is taken. What is the standard error for the sample mean?

0.029 0.1/sqrt of 12

2 approaches to measuring the reliability of an estimator

1. sampling error 2. bias

keep in mind that:

1. we typically do not know the population parameter 2. the estimator is a random variable

keep in mind that:

1. we typically do not know the population parameter. if you did we would not need an estimator for it. 2. the estimator is a random variable

the length of this interval is

2*1.96o/sqrt of n

the intervals of the same length constructed around the sample means will contains the population mean

95% of the time

the sample means will fall within the interval constructed around the population mean

95% of the time

the expected value of the sample mean (proportion) is the population mean

E(Xbar) = ux E(p) = pi

A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.True or False: The confidence interval is valid only if the lengths cut are normally distributed.

F

A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results.True or False: If the population had consisted of 10,000 stores, the confidence interval estimate of the mean would have been wider in range. (T or F)

F

The Central Limit Theorem (CLT) says that a histogram of the sample means will have a bell shape, even if the population is skewed and the sample is small. (T or F)

F (The CLT says that a histogram of the sample means will have a bell shape, even if the population is skewed only when the sample size is large enough. The rule of thumb is a sample size of 30 or larger.)

A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, and 31. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3. (T or F)

False

Central Limit Theorem (CLT) 2

If a population is with mean u and standard deviation o, then the sampling distribution of the sample mean x bar approaches a normal distribution with mean u and standard deviation o/square root of n as the sample size gets larger

Central Limit Theorem (CLT) 1

If the population is normally distributed with mean u and standard deviation o, then the sample distribution of the sample mean x bar is also normally distributed with mean u and standard deviation o/square root of n for any sample size n

Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100). Give a definition of what it means to be "95% confident" in an inference.

In repeated sampling, 95% of the intervals constructed would contain the population mean

Why is the Central Limit Theorem so important to the study of sampling distributions?

It allows us to disregard the shape of the population when n is large

A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results.True or False: We do not know for sure whether the true population mean is between 5.623 and 6.377 lawnmowers sold

T

The Central Limit Theorem is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known. (T or F)

T

The actual voltages of power packs labeled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, 11.77. A confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal. (T or F)

T

A sample is taken and a confidence interval is constructed for the mean of the distribution. At the center of the interval is always which value?

The sample mean.

A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, and 31. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3. True or False: 95% of all confidence intervals constructed similarly to this one will contain the mean of the population.

True

estimator

a statistic derived from the sample distribution to make inferences about a population parameter

confidence interval

an interval around a sample statistic such that, if all the possible samples of a given size were taken and an interval was constructed around each of them , then a percentage of these intervals would include the true value of the population parameter

interval estimator

an interval constructed from the random sample (before it is drawn) for which statements can be made about the likeliness the unknown population mean falls within that interval

the sample mean is a

continuous random variable regardless of whether the random variable is continuous or not

sampling error of the sample mean

difference between the estimator and the corresponding parameter - it measures how far away (or how wrong) your estimator for the population mean is from the population mean - it is a random variable, whose variability tells us how reliable our estimator is

sampling error

difference between the estimator and the corresponding population parameter

different random samples result in

different sample means

the sample means and proportions are

estimators - functions of the random sample

statistic

function of a random sample, and is therefore a random variable (eg. sample mean, sample standard deviation, median, relative frequency, etc.)

CLT for a proportion

if a population is with proportion pi the sampling distribution oft he sample proportion p approaches a normal distribution with mean pi and standard deviation op = square root of pi(1-pi)/n as the sample size n gets larger

unbiased estimator

if the average of all possible values of the estimator equals the parameter value

distribution of x normal --> sampling distribution is normal with mean u(pi) and standard deviation

if x is not normal --> sampling distribution is not normal

we therefore need a system by which we can make statements about how confident we are

in our sample mean as an estimator of the unknown population mean

the mean (expected value) of the sample mean

is equal to the population mean. E(x) = ux = u the sample mean is an unbiased estimator for the population mean

a point estimate (mean proportion, etc)

is not perfect

because the estimator is a random variable,

it follows a distribution

z is called a binary variable

it only has two possible outcomes

for any random sample of size n, contruct an interval of the sample length around the resulting sample mean x bar.

margin of error 2* o/sqrt of n

Theoretically, the standard error of the mean

measures the variability of the sample mean from sample to sample, is never larger than the standard deviation of the population, decreases as the sample size increases

the sample statistics (mean, variance, etc.) become

more centered around their population counterparts.

with larger sample sizes, the distributions of the samples become

more representative of the population distribution

in the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a ---- interval.

narrower

as long as the sample size is large enough, the shape of the sampling distribution of the mean becomes

normal

the estimator is defined without

observing the sample

the distribution of some random samples may do a bad job representing the distribution

of the population distribution from which it is drawn

the fraction of individuals surveyed that believe in global warming is

p = 1/n* sum of zi and x - sum of zi

the means calculated from these random samples will be

poor estimates of the population mean

because the sample mean is a random variable it has a

probability distribution, called the sampling distribution

the sample mean and sample proportion are

random variables

Central Limit Theorem (CLT):

regardless of how x is distributed, the sampling distribution of the sample mean is approximately normal for sufficiently large samples (n>=30, or npi.+10, (n(1-pi)>=10), with mean and variance given above.

how large is large?

rule of thumb: n>=30

how large is large?

rule of thumb: npi>=10 and n(1-pi)>=10

large samples will, most of the time, produce

smaller sampling errors than smaller samples.

consider a random sample of a binary variable taken from a population with proportion pi. then the standard error for the sample proportion p = x/n is

square root of pi(1-pi)/n

standard error of the sample mean

standard deviation of the sample mean - it decreases as the sample size increases - equal to the standard deviation of the sample divided by the square root of the sample size

the standard deviation of the sample mean (proportion) is called the

standard error, and is given by ox = ox/sqrt of n, op = sqrt of pi(1-pi)/n

Sampling distributions describe the distribution of

statistics

the sample mean is an unbiased estimator for the population mean

the average of all possible values of the sample mean is equal to the population mean.

bias

the expected or average difference between the estimator and the corresponding parameter

population proportion

the fraction of a population with a certain characteristic or attribute

sample proportion p

the fraction, x, of a sample of size n with a certain characteristic or attribute p = x/n

for any fixed interval length and any random sample

the interval constructed around the population mean contains sample mean if and only if the interval constructed around the sample mean contains the population mean

rule of thumb:

the larger the sample size, the closer to the normal shape the distribution of the sample mean becomes

the confidence interval

the most frequently used interval estimator

standard deviation of the sample mean

the population standard deviation divided by the square root of the sample size

sampling distribution of an estimator

the probability distribution of all the possible values the stastistic may take from a random sample size of n

samples may not be representative of the population they are drawn from, but

the representativeness of the samples tend to improve as their sizes increase

The width of a confidence interval for μ is not affected by:

the sample mean

estimate

the value of the estimator when evaluated with a particular sample (it is simply a number (constant). it does not have variation, because once you have the sample, thats it)

the sample statistics tend to be more centered around

their population counterparts as the sample size increases

the sample statistics (mean, variance, etc) vary with the random sample:

they are functions of the random sample and are therefore themselves random variables.

the sample mean is an

unbiased estimator for the population mean

natural response when using an estimator for inferring about the population parameter:

well, how do you measure the quality/reliability of an estimator?

natural question when using an estimator for inferring about the population parameter:

what is the quality of my estimator? how reliable is my estimator?