Business Statistics Ch 10

If you conduct a matched-pairs hypothesis test about the mean difference μD, then you calculate the value of the test statistic as...

C)

The confidence interval for the mean difference μD in paired samples is...

C)

For a given α level, what happens to the F values as df1 and df2 increase? Multiple choice question. F decreases F increases. F stays the same. F approaches 0.

F decreases

For a given set of df1 and df2, what happens to the F values as α decreases? Multiple choice question. F decreases F approaches 0. F increases. F stays the same.

F increases.

The test statistic for a one-way ANOVA test follows the Multiple choice question. normal distribution. tdf distribution. χdf2 distribution. Fdf1,df2 distribution.

Fdf1,df2 distribution.

The hypotheses to determine whether the average AAA battery life for Brand 1 differs from Brand 2 are Multiple choice question. H0: μ1 - μ2 ≥ 0 versus HA: μ1 - μ2 < 0 H0: μ1 - μ2 ≤ 0 versus HA: μ1 - μ2 > 0 H0: μ1 - μ2 = 0 versus HA: μ1 - μ2 ≠ 0 H0: x 1- x2 = 0 versus HA: x1 - x2 ≠ 0

H0: μ1 - μ2 = 0 versus HA: μ1 - μ2 ≠ 0

Error sum of squares (SSE)

In ANOVA, a measure of the degree of variability that exists even if all population means are the same. In regression analysis, it measures the unexplained variation in the response variable.

Select all that apply Which of the following is true about the test statistic for a hypothesis test about the mean difference μD? Multiple select question. Its degrees of freedom is n1 + n2 - 2. It follows the t-distribution. It is valid only if the distribution of Dbar is normal.

It follows the t-distribution. It is valid only if the distribution of Dbar is normal.

In general, one-way ANOVA is used for testing c population means under the following assumptions:

The populations are normally distributed. The population variances are unknown but assumed equal. The samples are selected independently.

Independent random samples

Two (or more) random samples are considered independent if the process that generates one sample is completely separate from the process that generates the other sample.

Matched-pairs sample

When a sample is matched or paired in some way.

In one-way ANOVA, between-treatments variability is based on Multiple choice question. the sum of all observations in the data set divided by the total number of observations. a weighted sum of squared differences between the sample means and the grand mean. a weighted sum of the sample variances of each treatment.

a weighted sum of squared differences between the sample means and the grand mean.

Two or more random samples are considered independent Multiple choice question. if the sample from one population is selected in such a way that it is related to the sample from the other. if before and after measurements are collected on the same sample. if the process that generates one sample is completely separate from the process that generates the other sample.

if the process that generates one sample is completely separate from the process that generates the other sample.

When comparing two population means, their hypothesized difference Multiple choice question. may assume any value. must be zero. must be negative. must be positive.

may assume any value. Reason: Using d0 = 0 is most common (just tests whether the means differ) but positive and negative values for d0 can also be used.

A one-way analysis of variance (ANOVA) test compares population ______ based on one categorical variable or factor. Multiple choice question. variances standard deviations means proportions

means

When the population variances are unknown and assumed equal, what are the degrees of freedom? Multiple choice question. n1 + n2 - 2 σ1 + σ2 - 2 μ1 + μ2 - 2 Xbar1 - Xbar2 - 2

n1 + n2 - 2

Which of the following pairs of hypotheses is used to test if the mean of Population 1 is smaller than the mean of Population 2 when using independent random sampling? Multiple choice question. H0: μD ≤ 0 versus HA: μD > 0 H0: μ1 - μ2 ≥ 0 versus HA: μ1 - μ2 < 0 H0: μD ≥ 0 versus HA: μD < 0 H0: μ1 - μ2 ≤ 0 versus HA: μ1 - μ2 > 0

H0: μ1 - μ2 ≥ 0 versus HA: μ1 - μ2 < 0

Within-treatments variance

In ANOVA, a measure of the variability within each sample.

A specific type of dependent sampling when the samples are paired in some way is called Multiple choice question. matched-pairs sampling. difference-paired sampling. equal-pairs sampling.

Matched-pairs sample

When the population variances are known, which distribution does the test statistic follow? Multiple choice question. The t-distribution with the n-1 degrees of freedom. The z-distribution. The z-distribution with n-1 degrees of freedom. The t-distribution with n1 + n2 -2 degrees of freedom.

The z-distribution.

Select all that apply Which of the following is true about matched-pairs problems? Select all that apply. Multiple select question. There is no assumption requiring the population variances to be equal. The sample sizes can be different, but each must be at least 30. Both sample sizes must be equal. Matched-pairs problems are similar to single sample problems because the individual differences are analyzed.

There is no assumption requiring the population variances to be equal. Both sample sizes must be equal. Matched-pairs problems are similar to single sample problems because the individual differences are analyzed.

In order to determine if there is a difference between the means of three or more populations, we use ______. Multiple choice question. a chi-square test of independence histograms or box plots. analysis of variance a goodness-of-fit test

analysis of variance

If we reject the null hypothesis H0: μ1=μ2=μ3 when conducting an ANOVA, we conclude that Multiple choice question. all population means are equal. all population means are different. at least one population mean is different. Only the populations with the largest and smallest sample means are different.

at least one population mean is different.

A confidence interval for the mean difference μD follows the general format of a point estimate ± Multiple choice question. zα/2 or tα/2, df . margin of error. variance. standard error.

margin of error.

For matched-pairs sampling, the parameter of interest is referred to as the Multiple choice question. population variance. mean difference. median. population standard deviation.

mean difference.

The margin of error equals the...

standard error se ( xbar1 - xbar2 ) * za/2 or ta/2,df

The parameter of interest for matched-pairs sampling is denoted as ______. Multiple choice question. p1 - p2 μD μ1 - μ2

μD

Select all that apply Which of the following methods below are types of matched-pairs experiments? Select all that apply. Multiple select question. "Before" and "after" studies A pairing of observations in each sample by some common characteristic Selecting samples from each populations without consideration of the other sample selected.

"Before" and "after" studies A pairing of observations in each sample by some common characteristic

Suppose a test of H0: μ1 = μ2 is being conducted at the 5% significance level. For which of the following 95% confidence intervals for μ1 - μ2 would the null be rejected? Multiple choice question. -1.2 to -0.8 -12 to 25 -1 to 2 -2 to 1

-1.2 to -0.8 Reason: The null cannot be rejected because the interval includes 0 thus both are negative.

Two Types of Matched-pairs sampling

1) "before" and "after" studies 2) pairing of observations, where it is not the same subject that gets sampled twice

The assumption for statistical inference for the mean difference using matched-pairs sampling requires either normality regarding the paired differences D = X1 - X2 or that the sample size n be at least Multiple choice question. 60 15 10 30

30

F distribution

A family of distributions where each distribution depends on two degrees of freedom: the numerator degrees of freedom df1 and the denominator degrees of freedom df2. It is positively skewed, with values ranging from zero to infinity, but becomes increasingly symmetric as df1 and df2 increase.

One-way ANOVA

A statistical technique that analyzes the effect of one categorical variable (factor) on the mean.

Analysis of variance (ANOVA)

A statistical technique used to determine if differences exist between three or more population means.

Which of the following are examples of matched-pairs experiments? Select all that apply. Multiple select question. The sugar content of two soft drink brands is studied sampling bottles of each soft drink. A study of salaries for two cities is conducted by matching employees in the cities according to major and work experience. The same individuals are evaluated before and after a weight loss program.

A study of salaries for two cities is conducted by matching employees in the cities according to major and work experience. The same individuals are evaluated before and after a weight loss program.

Suppose that the competing hypotheses for a test are H0: μD = 0 versus HA: μD ≠ 0. A 95% confidence interval for the mean difference is calculated as [-1, 4]. At the 5% significance level, the correct conclusion to the hypothesis test is: Multiple choice question. Reject H0; the mean difference does not significantly differ from zero. Reject H0; the mean difference significantly differs from zero. Do not reject H0; the mean difference significantly differs from zero. Do not reject H0; the mean difference does not significantly differ from zero.

Do not reject H0; the mean difference does not significantly differ from zero.

A one-way ANOVA test is based on the ______ distribution. Multiple choice question. χ2 z tdf F

F

True or false: A one-way analysis of variance (ANOVA) is used to test for equality of two population variances.

False

True or false: If the populations are normally distributed, the z distribution is used to conduct an ANOVA test.

False

The competing hypotheses for a one-way ANOVA test that compares the means of three populations are defined as Multiple choice question. H0: μ1 = μ2 = μ3 HA: Not all population means are equal H0: μ1 = μ2 = μ3 HA: All population means are different H0: μ1 ≠ μ2 ≠ μ3 HA: μ1 = μ2 = μ3 H0: μ1 = μ2 = μ3 HA: μ1 ≠ μ2 ≠ μ3

H0: μ1 = μ2 = μ3 HA: Not all population means are equal

For a matched-pairs test, the null and alternative hypotheses for a right-tailed test concerning the mean difference μD are constructed as Multiple choice question. H0: μD ≥ d0 versus HA: μD < d0. H0: μD = d0 versus HA: μD ≠ d0. H0: μD ≤ d0 versus HA: μD > d0.

H0: μD ≤ d0 versus HA: μD > d0.

A confidence interval could be used to test for a difference in population means. Under what conditions would the null be rejected? Select all that apply. Multiple select question. If the hypothetical difference, d0, is less than the lower value of the confidence interval. If the hypothetical difference, d0, is greater than the upper value of the confidence interval. If hypothetical difference, d0, is between the lower and upper values of the confidence interval.

If the hypothetical difference, d0, is less than the lower value of the confidence interval. If the hypothetical difference, d0, is greater than the upper value of the confidence interval.

Between-treatments variance

In ANOVA, a measure of the variability between sample means.

Sum of squares due to treatments (SSTR)

In ANOVA, a weighted sum of squared differences between the sample means and the overall mean of the data.

Grand mean

In ANOVA, the sum of all observations in a data set divided by the total number of observations.

Suppose that you want to perform a test to compare the mean lifetime of two brands of batteries. One person purchases 10 Brand A batteries from a store and another person purchases 12 Brand B batteries from another store. What type of sampling is required for this test? Multiple choice question. Dependent sampling using before/after data Independent random samples Dependent sampling using matched pairs

Independent random samples

Select all that apply Which of the following are true about the F distribution? Select all that apply. Multiple select question. The F distribution is a negatively skewed distribution. Like the t distribution, the F distribution is characterized by a family of distributions. Like the t distribution, it's value ranges from -∝ to ∝. It is like the t distribution in that it is based on degrees of freedom.

Like the t distribution, the F distribution is characterized by a family of distributions. It is like the t distribution in that it is based on degrees of freedom.

Within-treatments Estmate of σ2 is denoted as

MSE

Between-treatments Estmate of σ2 is denoted as

MSTR

Suppose the competing hypotheses for a test at the 5% significance level are H0: μD = 0 versus HA: μD ≠ 0. Which of the following would lead to rejecting the null? Multiple choice question. The 95% confidence interval for μD that includes 0. The 95% confidence interval for μD that does not include 0. The 5% confidence interval for μD that includes 0. The 5% confidence interval for μD that does not include 0.

The 95% confidence interval for μD that does not include 0.

Mean square error (MSE)

The average of the error (residual) sum of squares, where the residual is the difference between the observed and the predicted value of a variable.

When the population variances are unknown and assumed equal, we calculate a pooled estimate of the population variance. What are the weights that are used to calculate the pooled variance? Multiple choice question. The degrees of freedom The sample means The sample sizes

The degrees of freedom

Select all that apply Statistical inference concerning the mean difference based on matched-pairs sampling requires one of two conditions. Select the two conditions. Multiple select question. The distribution of differences is normally distributed. np ≥ 5 and n(1 - p) ≥ 5. The sampling distribution of p is normally distributed. The sample size n ≥ 30.

The distribution of differences is normally distributed. The sample size n ≥ 30.

Which of the following is NOT an assumption for performing a one-way ANOVA? Multiple choice question. The population correlation coefficients indicate a strong linear relationship. The populations are normally distributed. The population standard deviations are unknown but assumed equal. The samples are selected independently.

The population correlation coefficients indicate a strong linear relationship.

In one-way ANOVA, two independent estimates of the common population variance σ2 are estimated. These estimates are commonly referred to as ______. Multiple choice question. Type I variability and Type II variability correlated variability and uncorrelated variability between-treatments variability and within-treatments variability biased variability and unbiased variability

between-treatments variability and within-treatments variability

The degrees of freedom for the between-treatments variance is Multiple choice question. n n - 1 c c - 1

c - 1

Matched-pairs sampling is an example of Multiple choice question. independent sampling. correlated sampling. dependent sampling. cluster sampling.

dependent sampling.

The degrees of freedom are...

df = n1 + n2 − 2.

What items are needed to describe a particular value of the F distribution? Select all that apply. Multiple select question. μ1 df2 μ2 df1 α

df2 df1 α

d0

hypothesized difference

We use ANOVA to determine ______. Multiple choice question. if differences exist between two population proportions if differences exist between two sample variances. if differences exist between the means of three or more populations if differences exist between two population variances.

if differences exist between the means of three or more populations

one-way ANOVA

is a test that compares population means based on one categorical variable or factor.

The degrees of freedom for the within-treatments variance is Multiple choice question. c - 1 nT - c nT - 1 c - nT

nT - c

Statistical inference concerning the difference in population means is based on the condition that the sampling distribution of (Xbar1 - Xbar2) follows a(n) ______ distribution. Multiple choice question. binomial tdf normal unknown

normal

Statistical inference concerning the difference in population means is based on the condition that the sampling distribution of (Xbar1 - Xbar2) follows a(n) ______ distribution. Multiple choice question. normal binomial tdf unknown

normal

Performing a one-way ANOVA test, instead of performing a series of two-sample t tests, ______ the risk of incorrectly rejecting the null hypothesis. Multiple choice question. removes increases makes no difference in reduces

reduces

In a one-way ANOVA test, if the amount of variability between treatments is significantly greater than the amount of variability within treatments, then we Multiple choice question. do not reject the null hypothesis of equal population means. reject the null hypothesis of equal population means. reject the alternative hypothesis of at least one population mean being different.

reject the null hypothesis of equal population means.

The notation Fα,(df1,df2) represents a value such that the area in the Multiple choice question. left tail of the distribution is α. left tail of the distribution is α/2. right tail of the distribution is α. right tail of the distribution is α/2.

right tail of the distribution is α.

The one-way ANOVA test is always a Multiple choice question. left-tailed test. two-sided test. right-tailed test.

right-tailed test.

A one-way ANOVA test is better than using a series of two-sample t tests because conducting a Multiple choice question. series of two-sample t tests inflates the risk of committing both Type I and Type II errors. series of two-sample t tests inflates the risk of committing a Type I error. one-way ANOVA test is based on the F distribution which is superior to the t distribution.

series of two-sample t tests inflates the risk of committing a Type I error.

In one-way ANOVA, within-treatments variability is based on the Multiple choice question. variability between sample means. variability between sample medians. variability within each sample.

variability within each sample.

In one-way ANOVA, the independent estimates of the common population variance σ2 are based on which of the following? Select all that apply.

variability within each sample. variability between sample means

The confidence interval for the difference between two population means when the samples are independent and the population standard deviations are known is

B

The competing hypotheses for a left-tailed matched-pairs test concerning the mean difference μD are constructed as Multiple choice question. H0: μD = d0; HA: μD ≠ d0. H0: μD ≤ d0; HA: μD > d0. H0: μD ≥ d0; HA: μD < d0.

H0: μD ≥ d0; HA: μD < d0.

Parameter of interest for matched-pairs sampling for random Variables x1 and x2 is...

μD

The alternative hypothesis HA in one-way ANOVA is Multiple choice question. All means are different. All means are equal. Not all population means are equal.

Not all population means are equal.

If the underlying population cannot be assumed to be normally distributed, then by the central limit theorem, the sampling distribution of Xbar1 - Xbar2 is approximatly normal only if both sample sizes are sufficiently large enough; in other words must be....

n1 >= 30 and n2 >= 30

When examining the difference between two population means, if the populations cannot be assumed normal, then (Xbar1 - Xbar2) is approximately normal if Multiple choice question. n1 ≥ 30 and n2 ≥ 30. n1 ≥ 30 or n2 ≥ 30. n1p ≥ 5 and n2 p ≥ 5. n1 ≥ 5 and n2 ≥ 5.

n1 ≥ 30 and n2 ≥ 30.

In comparing c population means, the ANOVA test assumes...

the population variances are equal.

In most applications, the hypothesized difference between two population means is _____. Multiple choice question. positive zero one negative

zero

Which one of the following is true about the shape of the F distribution? Multiple choice question. It is skewed right or left, depending on the degrees of freedom. It is bell-shaped and it becomes skewed as the degrees of freedom increases. Its values range from 0 to -∝. It is skewed right and approaches the normal distribution as the degrees of freedom increases.

It is skewed right and approaches the normal distribution as the degrees of freedom increases.

Select all that apply Which of the following is a scenario that exists when constructing a confidence interval for the difference in population means? Multiple select question. The population variances are known. The population variances are unknown and assumed equal. The population variances are unknown and assumed unequal. The sample medians are known to estimate the difference in population means.

The population variances are known. The population variances are unknown and assumed equal. The population variances are unknown and assumed unequal.

When constructing a confidence interval for the difference between two population means, the margin of error equals Multiple choice question. the margin of error multiplied by zα/2 or tα/2,df. zα/2 or tα/2,df. the standard error multiplied by zα/2 or tα/2,df. the point estimate multiplied by zα/2 or tα/2,df.

the standard error multiplied by zα/2 or tα/2,df.