Chapter 8 & 8 Reading Assignment
For a 95% confidence interval for the population mean, if samples of size n are drawn repeatedly from a given population, then 5% of the sample means will fall outside of the corresponding confidence interval. 2.5% of the sample means will fall within the corresponding confidence interval. 5% of the sample means will fall within the corresponding confidence interval. 2.5% of the sample means will fall outside of the corresponding confidence interval.
5% of the sample means will fall outside of the corresponding confidence interval. +++
When calculating the required sample size for a mean CI, if n=50.13, the sample size we use is .
51 (you should always round up to the nearest whole number)
The null hypothesis is specified by using one of the following signs: ≠, ≤, or ≥ =, ≤, or ≥ ≠, <, or > =, <, or >
=, ≤, or ≥
How does an interval estimator differ from a point estimator? A point estimator may produce very biased results, while an interval estimator never has such a feature. An interval estimator provides a range of values that always include the population parameter whereas a point estimator does not usually include it. An interval estimator provides a range of values for the population parameter whereas a point estimator provides a single value. If a point estimator fails to produce an unbiased result, then an interval estimator secures such a result.
An interval estimator provides a range of values for the population parameter whereas a point estimator provides a single value
Which one of the following is NOT a step we use when formulating the null and alternative hypotheses? Calculate the value of the sample statistic. Determine whether it is a one-tailed or two-tailed test. Identify the population parameter of interest. Include some form of the equality sign in the null hypothesis.
Calculate the value of the sample statistic.
Rejecting a false null hypothesis is a: β. Type II error. Correct decision. Type I error. Type III error. α.
Correct decision.++++++
Which of the following is the correct formula for the margin of error in the interval estimation of p?
E = zα/2 * √(p(1-p)/n)
Given a right-tailed hypothesis test, if the value of the test statistic is 1.82 and the critical value is 1.645, then we do not reject the null hypothesis.
FALSE. Since the value of the test statistic is greater than the critical value for this right-tailed test, 1.82 > 1.645, the decision is to reject the null hypothesis.
The _______ ________approach to hypothesis testing is attractive when a computer is unavailable and all calculations must be done by hand.
Field 1: critical Field 2: value
The two competing hypotheses used in hypothesis testing are called the _______hypothesis and the _______ hypothesis.
Field 1: null Field 2: alternative
Which of the following are correctly configured one-tailed tests?
H0: p > p0 HA: p < p0 H0: μ < μ0 HA: μ > μ0 H0: p < p0 HA: p > p0 H0: μ > μ0 HA: μ < μ0
A researcher for a store chain wants to determine whether the proportion of customers who try out the samples being offered is more than 0.15. The null and alternative hypotheses for this test are H0: p > 0.15 and HA: p ≤ 0.15 H0: p ≤ 0.15 and HA: p > 0.15 H0: pp ≤ 0.15 and HA: pp > 0.15 H0: pp > 0.15 and HA: pp ≤ 0.15
H0: p ≤ 0.15 and HA: p > 0.15
Specify the competing hypotheses that would be used in order to determine whether the population mean differs from 15. H0: μ ≠ 15 versus HA: μ = 15 H0: μ = 15 versus HA: μ ≠ 15 H0: xx = 15 versus HA: xx ≠ 15 H0: xx ≠ 15 versus HA: xx = 15
H0: μ = 15 versus HA: μ ≠ 15
The alternative hypothesis for a one-sided test looks like:
HA: μ < μ0
The alternative hypothesis for a one-sided test looks like: HA: μ < μ0 H0: μ ≥ μ0 H0: μ ≤ μ0 HA: μ ≠ μ0
HA: μ < μ0
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.
What is the most typical form of a calculated confidence interval? Population parameter ± margin of error Point estimate ± margin of error Point estimate ± standard error Population parameter ± standard error
Point estimate ± margin of error
The random variable XX-μ/s| (square root of n) follows the ______ distribution.
Student's t
A 95% confidence interval for μ implies that if numerous samples are taken from a population, 95% of the intervals will contain μ.
TRUE
The alternative hypothesis always states the opposite of the null hypothesis.
TRUE
Consider the following competing hypotheses: H0: μ = 150 versus HA: μ ≠ 150. If a 95% confidence interval is [100, 200], then we cannot reject the null hypothesis at the 5% significance level.
TRUE. Since μ0 = 150 falls in the 95% confidence interval, the value 150 is reasonable or believable.
In the critical value approach, if the value of the test statistic does not fall within the rejection region, then we reject the null hypothesis.
This statement is FALSE in the critical value approach, if the test statistic does not fall within the rejection region, then we fail to reject the null hypothesis.
Rejecting the null hypothesis when the null hypothesis is true. α β Correct decision Type I error Type II error Type III error
Type I error
The test statistic when the population standard deviation is know is z = xx−μ0σ/√nx-μ0σ/n. Match each term to its meaning.
Z The test statistic. xx The sample mean. μ0 The hypothesized mean. σ The population standard deviation. n The sample size.
The population proportion p is the essential descriptive measure for a ______variable.
categorical
An important final conclusion to a statistical test is to...
clearly interpret the results in terms of the initial claim.
The alternative hypothesis typically _____ states the probability of rejecting the null hypothesis when it is true. corresponds to the presumed default state of nature. contests the status quo and may suggest a corrective action if true. states the probability of rejecting the null hypothesis when it is false.
contests the status quo and may suggest a corrective action if true.
The two equivalent methods to solve a hypothesis test are the critical value approach. standard deviation approach. population mean approach. p-value approach.
critical value approach. p-value approach.
Each t distribution is identified by its ______. degrees of freedom point estimate standard deviation significance level
degrees of freedom
Whenever we construct a confidence interval for the population mean, the margin of error includes the standard error of XX and the
desired level of confidence
Consider the following competing hypotheses: H0: μ = 10 versus HA: μ ≠ 10. If a 95% confidence interval is [8.25, 11.55], then at the 5% significance level we
do not reject the null hypothesis and conclude that the population mean does not significantly differ from 10. ++++
A Type II error occurs when we... do not reject the null hypothesis when it is actually true. reject the null hypothesis when it is actually false. do not reject the null hypothesis when it is actually false. reject the null hypothesis when it is actually true.
do not reject the null hypothesis when it is actually false.
The p-value approach to hypothesis testing has _________steps.
four steps
A t distribution has slightly broader tails than the Z distribution. does not have asymptotic tails. has a nonzero mean. is either positively skewed or negatively skewed.
has slightly broader tails than the Z distribution.
We use ______ tests to address conflicts between two competing views on a particular population parameter.
hypothesis
It is not sufficient to end the analysis with a conclusion that you reject the null hypothesis or you do not reject the null hypothesis. You must ______ the results.
interpret
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 allowed probability that an interval estimate of a population mean will not contain μ is referred to as the margin of error. level of significance. confidence level. confidence coefficient.
level of significance.
In a confidence interval, the _____ accounts for the standard error of the estimator and the desired confidence level of the interval. standard deviation margin of error point estimate z-value
margin of error
Which of the following requirements must be met in order to approximate PP using the normal distribution?
n(1-p) > 5 np > 5
When constructing a confidence interval for the population mean when the population standard deviation is unknown, the degrees of freedom for the t distribution are defined as: . α/2, n - 1. n. (n - 1), α/2. n - 1.
n-1
If we reject the null hypothesis when it is actually false we have committed... no error. both a Type I and a Type II error. a Type II error. a Type I error.
no error. Rejecting a null hypothesis when it is false is making a correct decision.
A binomial distribution can be approximated by a _______distribution for large sample sizes.
normal
The formula for a mean confidence interval is valid only if XX (approximately) follows a _______ distribution.
normal
The formula z = p−p0√p0(1−p0)np-p0p0(1-p0)n is valid only if PP follows a ______ distribution.
normal
The test statistic when the population standard deviation is know is z = x−μ0σ/√nx-μ0σ/n. This formula is valid only if XX follows a ___________ distribution.
normal
In order to derive a confidence interval for μ, the estimator XX must have a
normal sampling distribution.
In order to derive a confidence interval for μ, the estimator XX must have a small sample size. small standard deviation. normal sampling distribution. known population standard deviation.
normal sampling distribution.
In order to implement an hypothesis test, it is essential that XX is ________distributed
normally
When testing μ, the p-value is the probability of obtaining a sample mean at least as extreme as the one derived from a given sample, assuming that the ________ hypothesis is true.
null
The p-value is calculated assuming the Type I error equals zero. null hypothesis is true. Type II error equals zero. alternative hypothesis is true.
null hypothesis is true
The basic principle of hypothesis testing is to first assume that the ______ hypothesis is true and then determine if the sample data _______ this assumption. null, require alternative, contradict null, contradict alternative, require
null, contradict
The p-value is referred to as the ______ probability of making a Type I error.
observed
When performing a hypothesis test on μ, the p-value is defined as the allowed probability of making a Type II error. observed probability of making a Type I error. allowed probability of making a Type I error. observed probability of making a Type II error.
observed probability of making a Type I error.
A hypothesis test can be ____-tailed or ____-tailed.
one-tailed or two-tailed
A 100(1 - α)% confidence interval for the population proportion is
p(line over it) ± zα/2 ⋅ √p(1−p)/n
You use sample statistics to make inferences about the unknown population _______
parameters
The two main components of a confidence interval are the population parameter and the sample size. point estimate and the margin of error. population parameter and the margin of error. point estimate and the population parameter.
point estimate and the margin of error.+++
Unlike the mean and standard deviation, the population proportion p is a descriptive summary measure that can be used for data that are ________
qualitative or categorical
The population mean μ describes a _________ variable.
quantitative or numerical
The population mean μ describes a ________variable.
quantitative or numerical
If the minimum and maximum values of the population are available, a rough approximation for the population standard deviation is given by its _______ divided by .________
range divided by four
A confidence interval can be interpreted as a range of values used to estimate an unknown population parameter. margin of error. point estimate used to estimate an unknown parameter.
range of values used to estimate an unknown population parameter.
Consider the following competing hypotheses: H0: μ = 10 versus HA: μ ≠ 10. If a 95% confidence interval is [15, 20], then at the 5% significance level we
reject the null hypothesis and conclude that the population mean appears to differ from 10.+++++
The critical value approach specifies a region of values, called the ______. If the test statistic falls into this region, we reject the ______.
rejection region, null hypothesis
If the population standard deviation is unknown, it can be estimated by using ______
s (sample standard deviation)
A_______ is a subset of the population.
sample
Hypothesis testing enables us to determine if the collected ______ data is inconsistent with what is stated in the null hypothesis.
sample
We always use ________evidence and the chosen significance level α to conduct hypothesis tests.
sample
We would conduct a hypothesis test to determine whether or not population evidence contradicts HA. sample evidence contradicts H0. population evidence contradicts H0. sample evidence contradicts HA.
sample evidence contradicts H0. +++++
As a point estimate of the population proportion, we calculate ______.
sample proportion (p̂)
The ______ level of a hypothesis test is defined as 100α%.
significance
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
When performing a hypothesis test on μ when the value of σ is unknown, the test statistic is computed as x−μ0s/√nx-μ0s/n and it follows the
tdf distribution with (n - 1) degrees of freedom.
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.+++++
A confidence interval narrows if the following is accomplished:
the sample size increases. the chosen confidence level decreases.
A confidence interval narrows if the following is accomplished: the sample size increases. the chosen confidence level decreases. the chosen confidence level increases. the sample size decreases.
the sample size increases. the chosen confidence level decreases.
When the confidence level increases from 95% to 99%, the confidence interval for the population mean ________.
widens
For a given confidence level and sample size, the larger the population standard deviation, the _______ the confidence interval.
wider
Match these terms from a mean confidence interval formula with their meaning.
xx: Sample mean. Zα/2: Value from the standard normal table. n: Sample size. σ: Standard deviation. σ/Square root of n:Standard error.
As df increases, the tdf distribution becomes similar to the _______distribution.
z++++
Suppose you are performing a hypothesis test on μ and the value of σ is known. At the 5% significance level, the critical value(s) for a right-tailed test is (are):
z0.05
For a hypothesis test concerning the population proportion p, the value of the test statistic is calculated as
z=np0(1−p0)p^−p0
The Greek letter ______ denotes the level of significance.
α "alpha"
The confidence coefficient equal to Blank______.
(1 - alpha) or 1 - α
Which of the following statements is NOT correct concerning the p-value and critical value approaches to hypothesis testing? . Both approaches lead to the same conclusion. Both approaches specify and compute the test statistic. Both approaches use the same decision rule concerning when to reject H0. Both approaches state the null and alternative hypotheses.
.Both approaches use the same decision rule concerning when to reject H0.
For a 99% confidence interval, α =
0.01
The probability of error α for a 90% confidence interval is _____
0.10
For the 90% confidence interval, the confidence coefficient equals_____ and α equals _______.
0.90 0.10
Put the following steps in the p-value approach to hypothesis testing in the correct order.
1- specify the null and alternative hypothesis 2- specify the significance level 3-calculate the value of the test statistic and its p-value 4- state the conclusion and interpret results
In general, we follow three steps when formulating the competing hypotheses. Place these steps in the correct sequence.
1. Identify the relevant population parameter of interest2. Determine whether it is a one or two tailed test3. Include some form of equality sign in the null hypothesis and use the alternative hypothesis
The confidence level is equal to ______. 100 x (1 - α)% 100 x (1 - α)/2% 100 x (α/2)% 1 - α (100 x α)%
100 x (1 - α)%
The sampling distribution of estimator XX follows a normal distribution when the underlying population is normally distributed and/or when the sample size is large enough. As a "rule of thumb" we use a sample size of at least ___________
30 or more
