STATS CH 7 & 8
True of False: The standard deviation of the sampling distribution of the sample mean is equal to σ
FALSE
We use a calculated sample _____ to make inferences about an unknown population _____.
Statistic, Parameter
True or False: The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large (n>=30)
TRUE
True or False: The students t distribution assumes the population is normally distributed
TRUE
Suppose you are constructing a confidence interval for the population mean. For a given confidence level and standard deviation, the width of the interval is wider for a
smaller sample size
In _________ the population is divided up into mutually exclusive and collectively exhaustive groups called strata. The sample consists of elements from each stratum.
stratified random sampling
As a general guideline, the normal distribution approximation can be used to describe the sampling distribution of the sample mean when
n is > or = 30
A population consists of all items of interest in a statistical
...
_________ is the value of the point estimator derived from a given sample.
A point estimate
Refers to a tendency of a sample statistic to systematically over or under estimate a population parameter
Bias
________ provides a range of values for the population parameter
Interval estimator
________ provides a single value for the population parameter
Point estimator
True or False: The width of a confidence interval estimate for a population proportion will be narrower for 90% confidence than for 95% confidence (larger sample, smaller interval)
True
_________ of x̅ is smaller than the variance of the individual observation. This is an intuitive result, suggesting that averages have less variation than individual observations.
Variance
In a statistical problem, a population consists of
all items of interest
The central limit theorem states that for any distribution, as n gets larger, the sampling distribution of the sample proportion
approaches a normal distribution
The standard error of the sample mean is not affected by the
confidence level
An estimator is _________ if it approaches the population parameter of interest as the sample size increases.
consistent
A population parameter is a ____ even though its value may be unknown
constant
An estimator is deemed _______ if its variability between samples is smaller than that of other unbiased estimators.
efficient
A population consists of all items of interest in a statistical problem, whereas a _______ is a subset of the population
sample
The probability distribution of the sample mean is commonly referred to as the
sampling distribution of x bar
An estimator is unbiased if the average value of the estimator is equal to
the parameter being estimated
measures the variability of the mean from sample to sample
the standard error of the mean
An estimator is _________ if, based on repeated sampling from the population, the average value of the estimator equals the population parameter.
unbiased
A 99% confidence interval estimate can be interpreted to mean that
we have 99% confidence that we have selected a sample whose interval does not include the population mean.
Suppose you are constructing a confidence interval for the population mean. For a given sample size and standard deviation, the width of the confidence interval is______ for a greater confidence level.
wider
An estimator is consistent if it approaches the population parameter of interest as the sample size
increases
The branch of statistics that uses sample statistics to estimate a population parameter or test a hypothesis about such a parameter is BEST referred to as ______
inferential statistics
The further a population proportion deviates from p= 0.50 the _____ the sample size required in order to satisfy a normal approximation.
larger
Suppose you are constructing a confidence interval for the population mean. For a given confidence level and sample size, the width of the interval is wider for a
larger standard deviation
In general, the variability between sample means is _____ the variability between observations.
less than
The allowed probability that an interval estimate of a population mean will not contain μ is referred to as the
level of significance
A confidence interval is constructed by using the point estimate as a base, to which we ass and subtract the
margin of error
accounts for the variability of the estimator and the desired confidence level of the interval
margin of error
If the shape of the population from which a simple random sample is drawn is normal, then the shape of the sampling distribution of X-(X bar) is
normal
In order to construct a confidence interval for μ, the sampling distribution of the estimator x̅ must follow or approximately follow a ______ distribution.
normal
In order to derive a confidence interval for μ, the estimator x̅ must have a
normal sampling distribution
A particular value of an estimator
parameter
_________ is a function of the random sample used to make inferences about the value of an unknown population parameter.
point estimator
the parameter p represents the
population proportion
The mean SAT score for Division 1 student-athletes is 947 with a standard deviation of 205. If you select a random sample of 60 of these students, what is the probability the mean is below 900?
.0375 HOW z= x̅ - μ/ σ / Sqrt(n) 900-947/205/Sqrt(60) = -1.78 (Use table to find answer) = .0375
38% of business students are accounting majors. A sample of 100 students is selected.What is the probability that our sample proportion will be less than .36
.34 HOW p=.38 n=100 PBAR=.36 z= PBAR-p/ Sqrt(p(1-p)/n) .36-.38/Sqrt(.38(1-.38)/100) =.3446
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.
.625 (SDev/sq root of n) 10/(sqrt)256= .0625
A 99% confidence interval estimate can be interpreted to mean that
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
When those responding to a survey or poll differ systematically from the non-respondents. This is an example of a
Nonresponse bias
Is a value, usually unknown (and which therefore has to be estimated), used to represent a certain population characteristic Is used to describe a population
Prameter
True or False: The students t distribution has more area in the tails than does in the normal distribution
TRUE
True or False: The students t distribution is bell shaped and symmetrical
TRUE
True or Fasle: The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means
TRUE
Desirable properties for an estimator include
unbiased-ness, efficiency, and consistency
Is a sample of n observations which has the same probability of being selected from the population as any other sample of n observations
simple random sample
The science that deals with the collection, classification, analysis, and interpretation of numerical facts or data. Is used to describe parameters
statistics
A statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.
the central limit theorem
A confidence interval narrows if what is accomplished
the sample size increases the chosen confidence interval decreases
True or False: A statistic is a random variable whose value depends on the chosen random sample
true
True or False: Many different samples could be drawn from a population in order to analyze that population.
true
True or False: The central limit theorem is important in statistics because for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population
True
a random variable is also known as
a statistic
True or False: The student's t distribution is used to construct confidence intervals for the population mean when the population standard deviation is known
FALSE (UNKNOWN)
True or False: The mean of the sampling distribution of the sample mean is equal to μ
TRUE
True or False: The students t distribution approaches the normal distribution as the sample size increases
TRUE
True or False: The students t distribution has more area in the tails and less in the center than does the normal distribution
TRUE
TRUE or FALSE: Since x̅ is a random variable, its sampling distribution is simply the probability distribution derived from all possible samples of a given size from the population
True
True or False : The standard error of the mean is never larger than the standard deviation of the population
True
True or False: Cluster sampling works best when most of the variation in a population is within groups and not between groups
True
True or False: the standard error of the mean decreases as the sample size increases
True
True or False:The sample size required to approximate the normal distribution depends on how much the population varies from normality
True
Is x̅ a point estimator?
YES
When the sample is not a representative of the population we are trying to describe. This is a characteristic of ______ sample.
a bad sample
this interval provides a range of values that, with a certain level of confidence, contains the population parameter of interest
confidence interval
If an unbiased estimator is ______, its standard error must be lower than that of other unbiased estimators
efficient
When a statistic is used to estimate a parameter, it is referred to as an ______
estimator
when a sample is used to make inferences about a population parameter, it is referred to as an
estimator
A 95% confidence interval for the population mean implies that
for repeated samples 95% of the sample means will fall within the interval
The estimator x̅ follows a normal distribution when the underlying population follows a _____ distribution.
normal
A ________ is a descriptive measure for a qualitative variable
proportion
Sampling distributions describe the distribution of
statistics
Stratified sampling is preferred to cluster sampling when the objective is
to increase precision
True or False: A parameter is a constant although its value may be unknown
true
A particular value of the estimator is called an
estimate
True or False: The students t distribution, as the number of degrees of freedom increases, the t distribution approaches the normal distribution
TRUE
_________ of x̅ is the same as the expected value of the individual observation. In other words, if we were to sample repeatedly from a given population, in the average value of the sample means will equal the population mean from the underlying population
Expected Value
If the population from which the sample is drawn is normally distributed, then the sampling distribution of the sample mean is
always normally distributed
If a simple statistic consistently over or under estimates a population parameter then there is
bias
90% of restaurant customers order coffee with desert. If we take a sample of 144 customers who ordered desert, what is the probability that greater than .92 ordered coffee?
.2119 How PBAr= .92 n=144 p=.90 z= .92-.90 ---------- = .80 Sqrt(.9(1-.9)/144) Then use z table and .80 to get .7881 After that do this --> 1-.7881 = .2119
A manufacturer of batteries for portable hand tools wishes to investigate the length of time a battery will last between charges at a fixed load. A sample of 45 batteries had a mean length of time of 4.3 hours with a standard deviation of 0.25 hours. Construct a 95 percent confidence interval for the population mean.
4.23 to 4.37 We are 95% confident that the mean of a sample of batteries is between 4.23 and 4.37 How n= 45 x̅ = 4.3 s= .25 4.3 +- 1.96(.25/Sqrt(45)) = 4.23 to 4.37
Police Chief Aaron Ard of River City reports 500 traffic citations were issued last month. A sample of 35 of these citations showed the mean amount of the fine was $54, with a standard deviation of $4.50. Construct a 95 percent confidence interval for the mean amount of a citation in River City.
52.51 to 55.49 We are 95% confident that the mean amount of citation in River City is between 52.51 and 55.49 How n= 35 x̅ = 54 s=4.5 54+- 1.96(4.5/Sqrt(35)) =52.51 to 55.49
A range of values used to estimate an unknown population parameter
A confidence interval
When examining the possible outcome of an election, what type of confidence interval is most suitable for estimating the current support for a candidate?
Confidence interval for the population proportion
Suppose a 95% confidence interval for 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. What will result in a reduced interval width?
an increased sample size and decrease in the confidence level.