Stats DAT 565 week 3

True or false: A 95% confidence interval implies that the probability that μ falls in that interval is 0.95.

False

The common confidence level used in polls is

95%

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

The desired confidence level is often called _____.

1-α

Match the t statistic to the correct degrees of freedom and confidence level.

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

When constructing a confidence interval from a non-normal population with known variance, the common rule of thumb for sample size is that n ≥

30

A confidence interval

is a range of values used to estimate an unknown population parameter.

The standard error for p, the sample proportion, equals

π(1−π)n‾‾‾‾‾‾√

Order the steps for using Excel's data analysis tool to calculate the margin of error for a confidence interval for μ when σ 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

The standard error decreases

as n increases.

When constructing a confidence interval from a normal population the sample size

doesn't matter.

The two components of a confidence interval are the point ± the of error.

estimate margin

The estimated calculated sample size based on a margin of error is

rounded up to the next whole number.

The standard deviation of x is calculated as

σn

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

larger

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

normal

When assuming a uniform population the best technique for estimating sigma to determine a sample size is

set σ =(b−a)212where b is the upper bound and a is the lower bound.

The three characteristics required to properly describe a sampling distribution are

variance mean shape

By increasing the sample size, n, and keeping the confidence level the same, the width of the confidence interval

will decrease

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

x± tα/2s√n.

The sample size required for a 95% confidence level to estimate μ with a margin of error = 2 and with σ = 6 is n =

35

Because x is an unbiased estimator for μ

E(x) = μ

The sampling distribution of a sample mean refers to

The probability distribution of all x values calculated from all possible samples of size n.

A sample of 120 customers were asked if they were satisfied with their service. 75 responded Yes. The sample proportion of customers satisfied with their service is

.625, 0.625, 75/120, or 62.5%

The proportion calculated from a sample of size 50 is approximately normally distributed if x is between 10 and ___

40

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

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

True or false: It is rare that the population variance is known.

True

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

.15

For a normal population with μ = 25 and σ = 5, we would expect 95% of all x's calculated from n =9 to fall between _____ and _____. (Round to two decimals.)

21.73 and 28.27

The normal distribution approximation for x is typically considered appropriate when the sample size n ≥

30

The formula for a confidence interval for μ when the population standard deviation is unknown replaces the z score with a statistic and replaces σ with . (Please enter one word per blank.)

Blank 1: t Blank 2: s

If observations appear to be unusual or outliers one should

Investigate to determine if the data are in error. investigate to determine if the outliers are causing skewness. investigate to determine if the observations came from a different population.

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

Large

The CLT states that the sampling distribution of x will approach ________ as n increases.

N(μ, σn)

True or false: As sample size increases the t distribution approaches a standard normal distribution.

True

True or false: For a given sample size n and population standard deviation σ, a lower confidence level implies a narrower confidence interval.

True

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

We are 90% confident that the national average salary for high school math teachers is between $42,000 and $50,000 If a sample from a specific school district has an x = $40,000, we can conclude that this district has a lower average salary than the nation.

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

We have more uncertainty about the population when estimating σ with s. A t statistic will be greater than the z score for the same level of confidence.

A confidence interval to estimate μ will be more precise for a population with ______ variation than a population with _______ variation, keeping n and 1-α constant.

less; greater

The degrees of freedom for the student t distribution is defined as

n - 1

x will have a normal distribution if X is normally distributed. However the sample size

n can be any size

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

normal

Proportions frequently occur in surveys and are most often expressed as

percents

The expected value of x is equal to the __The expected value of the sample proportion, p, is the population mean.

population

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

proportion

The probability distribution of the sample mean is commonly referred to as the distribution of the sample mean.

sampling

When assuming a normal population the best technique for estimating sigma to determine a sample size is

set σ = b−a6 where b is the upper bound and a is the lower bound.

A t distribution has

slightly broader tails than the z distribution. a symmetrical shape.

The standard deviation of x is called the

standard error of x

Both the z and t distributions have the following properties:

symmetric around 0 with asymptotic tails bell-shaped

The CLT applies to populations with distributions

that are not normal or are unknown.

An underlying assumption for using the t distribution is

that the population should have a normal distribution.

An interval estimate is typically preferred to a point estimate because

the interval estimate expresses our uncertainty in the point estimate.

A sample size less than 30 might be sufficient for the sampling distribution on x to be considered approximately normal if

the population distribution is symmetric.

If we were to sample (size = n) repeatedly from a given population, the average value of the sample means will equal

the population mean.

A confidence interval provides a range of values that for a given level of confidence contains

the population parameter.

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

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

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

variability

The equation for a confidence interval for μ when the population standard deviation is known is

x± zα/2 σn

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

Quality Control studies

A sample of 120 customers were asked if they were satisfied with their service. 75 responded Yes. The margin of error for a 95% confidence interval for π is

.087

For a 90% confidence interval the value for zα/2 is _____. (Round to three decimals.)

1.645

For a population with σ = 10, an x calculated from a sample of 36 observations will have a standard error =

1.67

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

1.796

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

When applying the CLT to define an interval within which we expect 95% of all sample means to fall we would use a z = _______.

1.96

Given a confidence level of 95% and n = 11, if σ is known use z = . If σ is unknown use t = . (Round z to 2 decimals and t to 3 decimals.)

Blank 1: 1.96 Blank 2: 2.228

The t statistic should be used instead of z to estimate μ when the population is unknown.

Blank 1: standard Blank 2: deviation

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

Counting

The width of a confidence interval for π depends on

confidence level sample proportion sample size

A 95% confidence level means that

for repeated samples of size n, 95% of the confidence intervals will contain the population mean.

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

greater

If we want a more precise interval estimate for π we could

increase the sample size decrease the confidence level.

For a given population proportion π, the sampling distribution for p is approximately normal when the following 2 requirements are satisfied:

nπ ≥ 10 n(1-π) ≥ 10

The confidence interval for a population proportion, π, is calculated using the formula

p ± Zα/2√p(1−p)n