Business Stats Test 3
X be normally distributed
in order to derive a confidence interval of mu, it is essential that....
confidence interval
provides a range of values that, with a certain level of confidence, contains the population parameter of interest
z table
provides cumulative probabilities for a given z
selection bias
refers to a systematic exclusion of certain groups from consideration for the sample
parameter p
represents the proportion of successes in the population, where success is defined by a particular outcome
statistics
sampling distributions describe the distribution of
variability of the estimator
since numerous samples of size n can be drawn from the underlying population, the ______________ is captured by its standard deviation, or its standard error
family of distributions identified by the df parameter
since the tdf distribution is a ____________________________ the t table is not as comprehensive as the z table; it only lists probabilities corresponding to a limited number of values
variance of X
smaller than the variance of the individual observation; this is an intuitive result, suggesting that averages have less variation than individual observations
sampling distribution
the probability distribution of the sample mean X; since X is a random variable, its sampling distribution is simply the probability distribution derived from all possible samples of a given size from the population
expected value of X
the same as the expected value of the individual observation; if we were to sample repeatedly from a given population, the average value of the sample means will equal the population mean from the underlying population
stratified sampling
the sample consists of elements from each group; preferred when the objective is to increase precision
consistency
another desirable property which is often considered a minimum requirement for an estimator
approximate percentages
appropriate for many real-world applications where the normal distribution is used only as an approximation; for normally distributed random variables, these percentages are exact
approximately normal whenever the sample size is sufficiently large (n > 30), generated by repeatedly taking samples of size n and computing the sample means, and the mean of the sampling distribution of the sample mean is equal to mu
the sampling distribution of the sample mean....
the standard deviation of the sampling distribution of the sample mean is NOT equal to funky o
the sampling distribution of the sample mean:
is never larger than the standard deviation of the population, decreases as the sample size increases, and measures the variability of the mean from sample to sample
the standard error of the mean
point estimate
the value of the point estimator derived from a given sample
narrower for 90% confidence than for 95% confidence
the width of a confidence interval estimate for a proportion will be
does not contain mu
this is the allowed probability that the estimation procedure will generate an interval that....
random variable X
this represents a certain characteristic of a population under study
desirable properties of a point estimator:
1) unbiased-ness 2) consistency 3) efficiency
important feature of the sampling distribution of the sample mean X
irrespective of the sample size "n", X is normally distributed if the population X from which the sample is drawn is normal; in other words.... if X is normal with expected value "mu" and standard deviation "funky o", then X is also normal with expected value "mu" and standard deviation "funky o/square root of n"
primary requisite for a "good" sample:
it be representative of the population we are trying to describe
normal distribution
it is required that X follows a ______________ in estimating the population mean
tdf
like the z distribution, this distribution is bell-shaped and symmetric around 0 with asymptotic tails
z table
lists z values along with the corresponding cumulative probabilities
symmetry
noncumulative probabilities can be evaluated using this
the population mean and the population variance
normal distribution is completely described by these two parameters
standard deviation of X
calculated as the positive square root of the variance
parameter of interest
describes a population that is qualitative rather than quantitative
population mean
describes the central location
population variance
describes the dispersion of the distribution
approximately normal if the sample size "n" is sufficiently large
for any population X with expected value mu and standard deviation "o", the sampling distribution X will be.....
approximately normal if the sample size n is sufficiently large
for any population proportion p, the sampling distribution of P is....
Yellow note:
given the symmetry of the normal distribution and the fact that the area under the entire curve is one, other probabilities can be easily computed; we can also use the table to compute z values for given cumulative probabilities
empirical rule
gives the approximate percentage of values that fall within 1, 2, or 3 standard deviations of the mean
t distribution
has slightly broader tails than the z distribution
examples of random variables that closely follow a normal distribution
heights and weights of newborn babies, scores on the SAT, and cumulative debt of college graduates
increase the sample size and decrease the confidence interval
suppose a 95% confidence interval for mu turns out to be (1,000, 2,100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width?
for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of population
the central limit theorem is important in statistics because
probability of error/ level of significance
the greek letter a denotes this
symmetric normal distribution
the mean, the median, and the mode are all equal for a normally distributed random variable
n greater than or equal to 30
the normal distribution approximation is justified when
np is greater than or equal to 5 and n(1-p) is greater than or equal to 5
the normal distribution approximation is justified when
quantitative
the population mean mu and population variance o2 describes __________ data
qualitative
the population proportion p is the essential descriptive measure when the data type is __________
cluster sampling
the sample consists of elements from the selected groups; preferred when the objective is to reduce costs
point estimator
the sample mean is a ___________ of the population mean and the sample proportion is a ____________ of the population proportion
binomial distriubtion
the sampling distribution of P is based on this and we can approximate it by a normal distribution for large samples, according to the central limit theorem
it has more area in the tails and less in the center than does the normal distribution, it is bellshaped and symmetrical, and as the number of degrees of freedom increases, the t distribution approaches the normal distribution
the student's t distribution
assumes the population is normally distributed, approaches the normal distribution as the sample size increases, and has more area in the tails than does the normal distribution
the t distribution....
t table
unlike the cumulative probabilities in the z table, the _______ provides the probabilities in the upper-tail of the distribution
degrees of freedom (df)
each t distribution is identified by this
confidence coefficient
(1-a)
expected value
The ___________________ of the sample means is equal to the population mean irrespective of the sample size
population parameter
is constant even though its value may be unknown
unbiased
an estimator is ___________ if, based on repeated sampling from the population, the average value of the estimator equals the population parameter
increases
an estimator is consistent if it approaches the population parameter of interest as the sample size ______________
smaller
an estimator is deemed efficient if its variability between samples is _________ than that of other unbiased estimators
point estimators
X and P are ___________ of their population counterparts mu and p; each of them provides a single value or point as an estimate of the unknown population parameter
unbiased estimators
X and P are the ____________________ of mu and p; this property is independent of the sample size
if all possible samples are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval and we have 99% confidence that we have selected a sample whose interval does include the population mean
a 99% confidence interval estimate can be interpreted to mean that
margin of error
a confidence interval is generally associated with this that accounts for the variability of the estimator and the desired confidence level of the interval
point estimator
a function of the random sample used to make inferences about the value of an unknown population parameter
normal curve/bell curve
a graph depicting the normal probability density function is often referred to as this
inferential statistics
a major portion of statistics is concerned with this where we examine the problem of estimating population parameters or testing hypotheses about such parameters
estimate
a particular value of the estimator is called this
simple random sample
a sample of "n" observations which has the same probability of being selected from the population as any other sample of "n" observations
standard normal distribution
a special case of the normal distribution with a mean equal to zero and a standard deviation (or variance) equal to one
sample statistic
we use a calculated _____________ to make inferences about the unknown population parameter
sample proportion P
we use this as the point estimator of the population proportion p
estimator
when a stat is used to estimate a parameter, it is referred to as this
1) the sample size n or df=n-1 2) alpha
when determining the value of tadf, we need two pieces of info:
bias
when the info from a sample is not typical of info in the population in a systematic way, this has occurred
asymptotic tails
when the tails get closer and closer to the horizontal axis but never touch it