Data Analytics: Chapter 8: Sampling Distributions and Estimation

The conservative assumption for π when conducting polls or survey research is ________

.50

The quick rule of three says that if a sample of 20 restaurant customers were asked if their meal was satisfactory and none replied yes, then the upper limit on the confidence interval for π would be __________.

3/20 = .15

When constructing a confidence interval from a non-normal population with known variance, the common rule of thumb for sample size is that n is greater than or equal to ________.

30

A sample of 120 customers were asked if they were satisfied with their service. 75 responded yes. The z-score for a 95% confidence interval for π is:

1.96

For a normal population with u=25 and o=5, we would expect 95% of all x's (with a line) calculated from n=9 to fall between _____ and _____.

25+1.960(5/sqrt9)=28.27 25-1.960(5/sqrt9)=21.73

In order to improve the precision of a confidence interval estimate for u one can ______

1. decrease the confidence level 2. increase the sample size

The estimators x (with a line) and p from populations with normal distributions have the following properties:

1. efficient 2. unbiased

The standard deviation of x(with a line) is called the ______.

standard error of x(with a line)

Each t distribution is dependent on _______

the size of the sample

estimate

the value of the estimator in a particular sample

An internal estimate is typically preferred to a point estimate because ________

the interval estimate expresses out uncertainty in the point estimate

unbiased estimator

neither overstates nor understates the true parameter on average

The central limit theorem states that the sampling distribution of x (with a line) will approach a ______ distribution as the sample size n increases.

normal

bias

the difference between the expected value of the estimator and the true parameter

The following three MVUE are also consistent estimators for u, o2, and π.

x with a line, s2 and p

A 100 (1-α)% confidence interval of the population mean when the population standard deviation is not known is calculated by:

x(line top)±tα/2 s/√n

Both the z and t distributions have the following properties :

1. symmetric arounds 0 with asymptomatic tails 2. bell-shaped

Four factors play a role in statistical inference. Which factors are controllable and which are uncontrollable?

sample size = controllable desired confidence in the estimate = controllable sampling variation = uncontrollable population variation = uncontrollable

Standard error of an estimator is NOT affected by the ____

population size

A report states that the 90% confidence interval for the national average salary for a high school math teacher is $46,000 +/- $4000. This means that ____________.

1. We are 90% confident that the national average salary for high school math teachers is between $42,000 and $50,000 2. If a sample from a specific school district has an mean of $40,000 (x with a line over it), we can conclude that this district has a lower average salary than the nation

central limit theorem

allows us to approximate the shape of the sampling distribution of X (with a line over top) even when we don't know what the population looks like.

As a population proportion deviates from = 0.50, we need a ___________ sample size to satisfy a normal approximation.

larger

sampling distribution

the probability distribution of all possible values the statistic may assume when a random sample of size n is taken

True or false: Because random samples vary from sample to sample, an estimator from a random sample is a random variable.

true

If the population standard deviation of AAA batteries (with a population mean of 9 hours) is 0.5 hours, the margin of error for a 95% confidence interval for u (n=23 batteries) would be ____________. (carry to three decimals).

0.196

The sample size required for a 95% confidence level to estimate u with a margin of error =2 and with o=6 is n= ______.

35

The sampling distribution of a sample mean is ______ if the population from which the sample is drawn is normally distributed.

normal

The normal distribution approximation for x (with a line) is typically considered appropriate when the sample size n is greater than or equal to ___________

30

The three characteristics required to properly describe a sampling distribution are _____

1. mean 2. variance 3. shape

For a population proportion π, the sampling distribution of p is approximately normal if the sample size n is sufficiently _____.

large

For a sample size of 12 and 90% confidence interval, the t statistic used to estimate u would be ______

1.796

Match the t statistic to the correct degrees of freedom and confidence level 1.833 3.182 2.763

1.833 = df=9, 90% CL 3.182 = df=3, 95% CL 2.763 = df=28, 99% CL

The standard error of p is dependent on both _____

1. π 2. n

Given a confidence level of 95% and n=11, if o is known use z=______. If o is unknown use t=________. (Round z to two decimals and t to 3 decimals).

z= 1.96 t=2.228

standard error of the mean

the sampling error of the sample mean is described by its standard deviation

degrees of freedom

used to determine the value of the t statistic used in the confidence interval formula - tells us how many observations we used to calculate s, the sample standard deviation, less the number of intermediate estimates we used in out calculation.

For a 90% confidence interval the value for zα/2 is ______. (round to three decimal places.)

1.645

The expected value of x (with a line) is equal to the _______ mean

population

The sample statistic x (with a line) is an estimator of which population parameter?

u

AAA batteries are advertised to have a life of about 9 hours of use. With a certain level of confidence, you can advertise that the life is between 8-10 hours. In this example, 9 hours is the point estimate and what is the margin of error?

1 hour

Using the t distribution when the population is NOT normal can provide reliable results as long as _______.

1. the sample size is not too small 2. the population distribution is not badly skewed

Choose which parameters have the correct MVUE.

1. u; x with a line on top 2. o2; s2

Match the t statistic to the correct edges of freedom and confidence level 1.833 3.182 2.763

1.833 = df of 9; 90% CL 3.182 = df of 3, 95% CL 2.763 = df=28, 99% CL

In which application would it be safe to assume that σ is known?

Quality control studies

consistent estimator

converges toward the parameter being estimated as the sample size increases

Proportions are often easier to estimate than other parameters (such as mean) because proportions arise from _____ things.

counting

True or false: If an estimator is unbiased then there will be no sampling error.

false

An estimator is more efficient that other estimators if it is closer on average to the true value of the ___________.

parameter

The expected value of the sample proportion, p, is the population ________.

proportion

efficiency

refers to the variance of the estimator's sampling distribution - smaller variance means a more efficient estimator

The unbiased estimators of u and π are the ______ mean and proportion.

sample

As the ______ size increases, a consistent estimator's value will get closer an closer to the _____ value.

sample; parameter

Match the correct terms sampling error bias systematic random

sampling error = random bias = systematic

For a population proportion, π =.4 and n=25, the standard error of p= _____. (round to 3 decimals.)

sqrt(.4(.6)/25)=.0979=.098

As n increases, the variability of x with a line over it decreases implying the x with a line over it is a ____.

consistent estimator of u

central limit theorem for a mean

if a random sample of size n is drawn from a population with mean µ and standard deviation σ, the distribution of the sample mean X (with a line over top) approaches a normal distribution with mean µ and standard deviation σx⎯⎯=σ/square root of n as the sample size increases

Some of the desirable properties of a point estimator include all of the following except ____.

variability

sampling error

the difference between an estimate and the corresponding population parameter. Sampling error= sample mean (x with a line over top) - population mean

The confidence interval of for µ when σ is unknown will be wider than if σ were known because __________.

1. a t statistic will be greater than the z-score for the same level of confidence 2. we have more uncertainly about the population when estimating σ with s

Order the steps for using excel's data analysis tool to calculate the margin of error for a confidence interval for U when o is unknown.

1. enter a sample of data in a column 2. choose data anlysis' 3. choose descriptive statistics' 4. check confidence interval for mean

3 important facts about the sample mean

1. if the population is normal, the sample mean has a normal distribution centered at µ, with a standard error equal to σ/√n. 2. As sample size n increases, the distribution of sample means converges to the population mean µ 3. even if your population is not normal, by the Central Limit Theorem, if the sample size is large enough, the sample means will have approximately a normal distribution

The width of a confidence interval for π depends on:

1. sample proportion 2. sample size 3. confidence level

To find the t statistic for a 95% confidence level with n=23 using Excel, type the following function:

T.INV.2&(.05,22)

estimator

a statistic derived from a sample to infer the value of the population parameter. - random variable

An estimator is consistent if it approaches the population parameter of interest as the sample size. ____________.

increases

A confidence interval to estimate u will be more precise for a population with _____ variation than a population with ____ variation, keeping n and 1-a constant.

less; greater

The expected value of x (with a line) is equal to the _______ mean.

population

minimum variance estimator (MVUE)

shows two unbiased estimators

The variance of the sample means is _______ than the variance of the individual observations

smaller

point estimate

a sample mean calculated from a random sample

An ________ is a statistic used to infer a value of a population parameter.

estimator

Precision is directly linked with the width of a confidence interval. The smaller the interval, the _________ the precision.

greater

Rule of Three

if in n independent trials no events occur, the upper 95 percent confidence bound is approximately 3/n