Chapter 9

The confidence level equals

1 - α.

The following table lists tdf values for selected upper-tail probabilities and degrees of freedom df. If df=6 and α=0.05, find tα/2,df.

2.447

The following table lists tdf values for selected upper-tail probabilities and degrees of freedom df. If df=5 and α=0.05, find tα/2,df.

2.571

What is the confidence level if α = 0.10?

90%

Regardless of the sample size, the estimator x follows a normal distribution when the underlying population follows a distribution.

normal

The two main components of a confidence interval are the

point estimate and the margin of error.

A 99% confidence interval is a range of values that should include a _______ with 99% confidence.

population parameter

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.

Given that we are sampling from a normal population, a 100(1 - α)% confidence interval for the population mean when the population standard deviation is not known is calculated as:

x±t_(α/2,df) s/√n.

Suppose you are constructing a confidence interval for the mean length of life of AAA batteries. You have the following data: x =250, n=25, σ=0.5, and zα/2=z0.025=1.96. The margin of error for a 95% confidence interval is closest to:

0.2

If α equals 0.01, then the confidence level equals

0.99

The confidence level is equal to

100 x (1 - α)%.

A sample of size 25 is drawn from a normal population with a population standard deviation of 100. Suppose the mean of the sample is x = 35. Recall that z0.025=1.96. A 95% confidence interval for the population mean is equal to

35±1.96 100/√25.

If samples of size n are drawn repeatedly from a given population and each sample is used to construct a 95% confidence interval for μ

5% of the confidence intervals will fail to contain μ.

A 95% confidence interval for the population mean is constructed as 6±2. What is the point estimate of μ?

6

A 95% confidence interval for the population mean is calculated as [40, 80]. The point estimate for μ is _______.

60

Suppose you have a random sample from a population whose standard deviation σ is known. If you construct both a 90% and a 95% confidence interval for μ, which interval will be shorter?

90%

A 95% confidence interval for the population mean implies that if samples are drawn repeatedly and confidence intervals for μ are constructed, then

95% of the confidence intervals computed will contain the population mean.

True or False: A confidence interval is constructed around the population mean and makes inference about the sample mean.

False

Suppose you are constructing a confidence interval for the population mean. For a given sample size and population standard deviation, how will the width of the interval change as the confidence level increases?

It gets larger

Which of the following is a valid form of a confidence interval?

Point estimate ± margin of error

All of the following are characteristics of the z and t distributions EXCEPT:

bimodal.

The standard error of x is NOT affected by the

confidence level.

Each t distribution is identified by its

degrees of freedom.

A t distribution

has slightly broader tails than the z distribution.

A confidence interval is constructed by using the point estimate as a base, to which we add and subtract the

margin of error.

When estimating the population mean, the t distribution is used when the

population variance is unknown.

A confidence interval can be interpreted as a

range of values used to estimate an unknown population parameter.

The standard error is equivalent to ______

s/√n

As the degrees of freedom increase, the t distribution becomes more ______ a standard normal distribution.

similar to

If repeated samples of size n are taken from a normal population with an unknown variance, then the statistic ______ follows the t distribution with n-1 degrees of freedom.

t=(x−μ)/(s/√n)

When the confidence level increases from 95% to 99%, the confidence interval for the population mean ________.

widens

The equation for a confidence interval for μ when the value of σ is known is

x±z_(α/2) σ/n√

A sample of size 25 is drawn from a normal population. Suppose the sample mean x = 50 and that the margin of error for a 95% confidence interval is 10. A 95% confidence interval for the mean is Multiple choice question. 50±1.96(10).

50±10.

A confidence interval narrows if the following is accomplished:

the chosen confidence level decreases. the sample size increases.

When constructing a confidence interval for the population mean, the factors that affect the width of the confidence interval for a given standard deviation are

the confidence level and the sample size.

If we use the same data set to compute both 95% and 99% confidence intervals for μ, the margin of error for the 95% confidence interval would be _____ the margin of error for the 99% confidence interval.

less than

The value of t.50 is

0

For a 95% confidence interval, α =

0.05.

The probability of error α for a 90% confidence interval is ____ and the probability of error α for a 99% confidence interval is ____.

0.10, 0.01

Suppose you choose a sample of size 16 from a normal population and find x =1.55 and s =0.44. The standard error of x equals

0.11

Suppose we wish to derive a confidence interval for the mean of a right-skewed population. In order to derive a valid confidence interval for μ, x must be based on a sample which

has size n≥30.

AAA batteries are advertised to have a life of about 9 hours of use. With a certain level of confidence, it is advertised that the life is between 8-10 hours. If 9 hours is the point estimate, then the margin of error is

1 hour.

True or false: The t distribution is bell-shaped and symmetric around 0.

True

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.

The sampling distribution of estimator x follows a normal distribution when the sample size is large enough. As a rule-of-thumb, we use the following:

n ≥30.

Suppose we wish to derive a confidence interval for the mean of a left-skewed population. In order to derive a valid confidence interval for μ, we must rely on ___________

the Central Limit Theorem