CH 9 & 10
True or false: Using a z score instead of t statistic to test a mean when sigma is unknown will increase Type II error.
False Reason: Will increase Type I error.
When setting up a two-tailed hypothesis test for comparing two variances, the correct formulation is:
H0: σ12/σ22 = 1 vs H1: σ12/σ22 ≠ 1
A business wants to use sample data to confirm that their average processing times have decreased after installing updated equipment. Their alternative hypothesis would be
H1: μ < μ0
True or false: When testing the difference between two population means it is unlikely that one would know the value of the population variances.
True Reason: Typically we do not know the value of the population variances.
The folded F test requires that we put the _____ sample variance in the numerator of the test statistic.
larger
The hypotheses H0: μ1 - μ2 ≥ D0 & H1: μ1 - μ2 < D0 indicate a
left-tailed test.
A 95% confidence interval could be used to conduct a two-tailed hypothesis test when alpha is _____
.05
For a right-tailed test with zcalc = 1.45, the p-value is _____.
.0735 https://www.omnicalculator.com/statistics/z-test
For a two-tailed test with zcalc = 0.78, the resulting p-value is _____.
.4354 or .4353 https://www.omnicalculator.com/statistics/z-test
For a two-tailed test with zcalc = -1.80, the resulting p-value is _____.
.0718 or .0719 https://www.omnicalculator.com/statistics/z-test
True or false: Excel's paired t test provides the p-value for both a two-tailed and one-tailed test allowing the analyst to choose the appropriate value.
True Reason: Excel does provide two p-values - for both a two-tailed and a one-tailed test.
Which of the following describes a two sample test situation?
-An education study that looked at the change in freshmen GPAs from one year to the next. -An agriculture study that compares the yield of two different crops. -A marketing study that looks at purchasing patterns from two different demographic groups.
Power is represented by
1-β
To find the two-tailed p-value for an Fcritical < 1 one would use
=2*F.DIST(Fcritical, df1, df2, 1)
To calculate a right-tailed probability for a specific F value in Excel
=F.DIST.RT(Fcalc, df1, df2)
To find the lower critical value of the F statistic with alpha = .05, df1 = 12 and df2 = 7, use the Excel function
=F.INV(.025, 12,7)
Match the case number with the correct test statistic formula.
Case 1: Known σ1^2 and σ2^2 matches: z = (x1−x2)−(μ1−μ2)/√(σ1^2/n1)+(σ2^2/n2) Case 2: σ12 = σ22 but unknown matches t = (x1−x2)−(μ1−μ2)/√(sp^2/n1)+(sp^2/n2) Case 3: σ12 ≠ σ22 and unknown matches t = (x1−x2)−(μ1−μ2)/√(s1^2/n1)+(s2^2/n2)
The hypotheses for a right-tailed test for a mean difference μd look like
H0: μd ≤ 0 vs H1: μd > 0
True or false: Smaller values of α make it easier to reject the null.
False Reason: smaller values make it harder, it's harder to make a Type I error
True or false: An F test for two samples can be found under Excel's Data Analysis Toolpak.
True
True or false: If a critical value is set at 1.645 for a right-tailed test, a calculated test statistic of 1.82 would lead to the null hypothesis being rejected.
True Reason: A value of 1.82 would be rejected because it is to the right of the critical value.
If normality of p1 - p2 cannot be assumed then the test must be conducted using the
binomial distribution.
A decision rule states what the value of the test statistic must be in order to reject the null hypothesis. This value is called the _____ value.
critical
The F statistic requires two degrees of freedom: df1 is the degrees of freedom for the _____ and df2 is the degrees of freedom for the _____
numerator denominator
An OC curve stands _____ _____ for curve.
operating characteristic
The _____ method is often preferred to the critical value method because it is a direct measure of the likelihood of observing the sample with the null hypothesis is true.
p-value
Which is the correct formula for the margin of error for a confidence interval for the different between two proportions?
zα/2 sqrt[(p1(1−p1)/n1)+(p2(1−p2)/n2)]
In comparing two population proportions, our parameter of interest is
π1 - π2
All other things being equal, power will increase when the sample size _____.
increases. or goes up
We can generally reduce the probability of Type II errors by
increasing the sample size.
The critical value stated in the decision rule is determined by
level of significance
Increasing the value of alpha in a hypothesis test will
make it less difficult to reject the null hypothesis,
The Fdf1,df2 distribution is
positively skewed.
When using a confidence interval to test for difference in means, using equal or balanced sample sizes will increase the _____ of the test.
power
If we know that 90 out of100 customers in a grocery store chain prefer having their groceries bagged for them, we would express this parameter as a population
proportion
The definition of power is
rejecting the null hypothesis when the null is false.
The hypotheses H0: μ1 - μ2 ≤ D0 & H1: μ1 - μ2 > D0 indicate a
right-tailed test.
The p-value is calculated assuming the
the null hypothesis is true
When testing the difference between two means, the test statistic for cases 2 and 3 will be the same if the
the sample sizes are equal.
The hypotheses H0: μ1 - μ2 = D0 & H1: μ1 - μ2 ≠ D0 indicate a
two-tailed test.
The formula for the test statistic for the difference between two population means when the population variances are known has the format
z = [(xbar1−xbar2)−(μ1−μ2)]/√(σ1^2/n1)+(σ2^2/n2)
We combine, or pool, the sample proportions into one "big" sample when testing for _____ difference between population proportions.
zero or no
As the sample size, n, increases
-power increases. -P(Type II error) decreases.
When testing if population proportions are different, the hypotheses are
H0: π1 - π2 = 0 vs H1: π1 - π2 ≠ 0
A power curve looks like an inverted _____ curve
OC or normal
A right-tailed hypothesis test for a mean resulted in tcalc = 2.23 with df = 16. If α = .05 the correct conclusion would be
Reject the null because the p-value < .05.
Treating paired data as independent samples ignores the variation _____ paired values.
between
When testing a population proportion, if either nπ0 or n(1-π0) are less than 10, one must calculate a p-value using the _________ distribution.
binomial
The p-value method for testing hypotheses is often preferred by statisticians because
-it expresses the strength of your evidence against the null. -it is more flexible than the critical value method.
Two sample tests are used to compare sample results taken from two populations with different _____ values.
parameter
When formulating a hypothesis test for comparing two variances, rather than calculating a difference, we calculate a
ratio
One can approximate the t statistic using the z score (and substituting s12 and s22 for the population variances) if
the populations are not badly skewed. both sample sizes are 30 or more.
If the Excel output for a matched pairs right-tailed test shows the t stat = 3.4551 and the t critical value is shown as 1.721 at α = 0.05 then the decision would be
to reject the null hypothesis because 3.4551 > 1.721.
When testing the difference in means, _____ sample sizes can offer some advantages.
equal
Given that β = .23 for a particular set of hypotheses, power =
.77
If we would like to test whether or not two population proportions differ by at least .30, the null hypothesis will state:
H0: π1 - π2 ≥ .30 vs H1: π1 - π2 < .30
Hypothesis testing is used to test _____ and _____ in business and science.
assumptions theories
The parameter value in the null hypothesis is called a _____ and the value does not come from a _____.
benchmark sample
For a matched-pairs test for the difference in means, the Excel output allows us to use either the p-value approach or the
critical value approach.
The parameter of interest for a matched-pairs sampling is
d = X1 - X2
Hypothesis testing uses sample _____ to test assumptions.
data
When choosing which test statistic to use for testing the difference of two means, which of the following are the three cases that one can choose?
-Population variances are known. -Population variances are unknown and not assumed equal. -Population variances are unknown but assumed equal.
The test statistic for testing equality of proportions
-is a z score -assumes when samples are large that p1 - p2 is normally distributed. -uses a pooled proportion to calculate the standard error.
In order to calculate the power a hypothesis test for a mean, the following information must be known:
-significance level, α -an alternative value for μ -a known or estimated value for σ
The statistic S12/S22 follows what distribution if independent samples are drawn from normal populations with equal variances?
Fdf1,df2 distribution
Choose the correct statement about sample sizes when calculating a confidence interval for the difference in means.
Sample sizes do not need to be equal.
An example of the relevance of making inferences using a population variance is
a coffee vending machine wanting to put a certain amount of beverage in a cup, without being too little or spilling over.
When collecting sample proportion data a success is defined as
any event of interest
If a 90% confidence interval for a store's customer accounts is computed as $850 ± 70, then the null hypothesis that μ = $750 would
be rejected at α = .10.
The formula for calculating the confidence interval for a mean difference is
dbar ± tα/2 sd/sqt(n)
Samples might show statistically significant differences but a company's financial expert can tell you if the difference is
important
Hypothesis testing for comparing population means uses the difference (x1 - x2) where the samples are assumed to be taken from populations with a _____ distribution.
normal
The formula for calculating a pooled sample proportion is
pc=(x1+x2)/(n1+n2)
The _____ the p-value, the more inclined we are to reject the null hypothesis.
smaller or lower
The test statistic for hypothesis tests for a population mean difference is
tcalc = dbar−μd/sd/√n
A confidence interval can be used to test a hypothesis if the hypothesis test is
two-tailed and α stays the same.
A manufacturer may want to compare the reliability of two products by comparing the _____ of their product characteristics.
variance, variances, variation, or variability
Using the critical value method for a two-tailed test, the critical value is determined by a tail area equal to
α/2 Reason: The value of significance has to be divided by 2 for a two-tailed test.
When evaluating the variability of repair costs of a certain automobile, the population _____ is an important component.
variance
The test statistic for testing non-zero difference in proportions would be
zcalc = (p1−p2−D0)/sqrt[(p1(1−p1)/n1)+(p2(1−p2)/n2)]
Identify the ways of increasing power.
-Raising the level of significance -Enlarging the sample size
Match the change in the parameter to the effect on power.
Increase in power matches: -Increase in n, Increase in α Decrease in power matches: -Increase in σ
The values of the chi square distribution range from zero to
Infinity
A confidence interval estimate for the difference in customer satisfaction rates between May and June of last year is -.14 ± .03. Which statement below is most appropriate?
It appears that the customer satisfaction rate decreased from May to June.
True or false: A proportion can be expressed as percentage but not all percentages can be expressed as a proportion.
True Reason: It is possible to describe a decrease in profit as a negative percentage and it is also possible to describe an increase in something as greater than 100%. A proportion can never be negative or greater than 1 (100%).
True or false: When testing the difference between two population means it is unlikely that one would know the value of the population variances
True Reason: Typically we do not know the value of the population variances.
To test the equality of variances the F statistic calculates the ______ of two sample variances.
ratio
_____ testing is used by business managers to guide decision making.
hypothesis
True or false: The F test does not assume the populations being sampled have a normal distribution.
False
True or false: All statistically significant result have practical importance.
False Reason: A statistically significant result may lack practical importance if the difference is small.
A statistically significant result would be important if the mean difference is
large enough.
Order the steps in a two sample test.
1. State the hypothesis 2. Set up the decision rule. 3. Collect sample data and calculate the test statistic. 4. Make a conclusion.
For a two-tailed F test, FL is the left-tailed critical value and FR is the right-tailed critical value where FL =
1/Fdf2,df1
The criteria for assuming normality of p1 - p2 is that both np and n(1-p) for each sample is greater than or equal to
10 or ten
The folded F test is used to conduct a _____-tailed test, simplified with only _____ critical value for F, found using α/2.
2 1
Given the following set of hypotheses: H0: No illegal steroid use H1: Illegal steroid use, which statement describes the consequence of a Type I error?
An athlete is banned from competing when he or she did not use illegal steroids.
When comparing the percentage of satisfied customers between Dell and HP one would analyze the difference between population
proportions
When sample observations can be paired (or we have dependent samples) treating these as independent samples will
reduce the power of the test.
The t test statistic for a mean difference follows the _____ (use a letter) distribution with df = _____ -1.
t;n
The test statistic used for testing μ when the population mean, σ, is unknown is
tcalc =(xbar−μ0)/(s/√n)
A quality control engineer would like to test if the average time of use for AAA batteries is equal to 8.5 hours. She does not know the population standard deviation. If a sample of 32 batteries is tested, the test statistic would be calculated as
tcalc=(xbar - 8.5)/(s/sq root(32)), with df = 31
The difference between the sample statistic and the null hypothesis value is measured by a
test statistic.
There is little difference between critical values of t and z when ___.
the sample size is large (n > 30)
When comparing two population proportions with the following sample results (p1 = .24, p2 = .28, n1 = 100, n2 = 200, and pc = .2667) the z test statistic would be
-.74
To conduct an F test in Excel one can
-Choose Data > Data Analysis > F-Test Two Sample Variances -choose the function =F.TEST(array1, array2)
In hypothesis testing, there are 2 possible incorrect decisions:
-Rejecting the null hypothesis when it is true. -Not rejecting the null hypothesis when it is false.
In hypothesis testing, 2 correct decisions are possible:
-Rejecting the null hypothesis when the null hypothesis is false. -Not rejecting the null hypothesis when it is true.
Which variables below would express proportion data?
-The number of customers who agree that the new product price is acceptable. -The number of employees who reported an accident last month.
For a left-tailed test for the following null hypothesis H0: π1 - π2 ≥ .20, the z test statistic = -.75. The p-value for this test is
.2266
When testing differences in population proportions, one can set up a left-tailed test, a right-tailed test, or a _____ -tailed test.
2 or two
If no information is available about the population variances, one should choose Case ___ when testing the difference between means.
3
When conducting a two-tailed F test for the comparison of means, given that n1 = 10, n2 = 8, and α = .05, FR = _____ and FL _____
4.82 .24 FL=F.INV((.05/2),(10-1),(8-1)) FR=F.INV((1-(.05/2),(10-1),(8-1))
Select the following menu choices for conducting a matched-pairs difference test with unknown variance:
Data > Data Analysis > t-test: Paired Two Sample for Means > OK
Inference regarding the ratio of sample variances uses the _____ distribution which is based on a family of distributions depending on the separate degrees of freedom.
F
For a left-tailed test for the following null hypothesis H0: π1 - π2 ≥ .20, the p-value = .2266. Given that α = .10, the correct conclusion would be
Fail to reject H0. There is not significant evidence to conclude the difference in proportions is less than .20.
True or false: A confidence interval can be used to conduct a right-tailed test provided the α values are the same.
False Reason: Only for a two-tailed test with same α values.
True or false: Sample sizes must be equal when testing the difference between two means
False Reason: Sample sizes do not need to be equal but large differences will decrease the power of the test.
True or false: Sample sizes must be equal when testing the difference between two means.
False Reason: Sample sizes do not need to be equal but large differences will decrease the power of the test.
For unknown variances and large samples, approximation of the t statistic using the z score decreases Type I risk.
False Reason: This is false. For unknown variances and large samples, approximation of the t statistic using the z score may increase Type I risk.
True or false: The steps in a two sample hypothesis test are twice the number of steps in a one sample hypothesis test.
False Reason: The steps are the same.
The test statistic for the hypothesis test of the ratio of two population variances is
Fdf1,df2 = s12/s22
The hypotheses for a left-tailed test for a mean difference μd look like
H0: μd ≥ 0 vs H1: μd < 0
If an analyst believes Fund 1 is riskier than Fund 2, the appropriate hypotheses she would use to verify this are
H0: σ12/σ22 ≤ 1; H1: σ12/σ22 > 1
If the same individuals are evaluated before and after a weight loss program, this is an example of
a matched-pairs sample.
For matched-pairs sampling, the parameter of interest is referred to as the mean
difference
Even though repeated hypothesis tests could result in no strong conflicts between the observed data and the null hypothesis, one would still not state the null has been proved, one would state that they would _____ _____ _____ the null hypothesis.
fail to reject
It is possible to see a statistically significant change in a population parameter even though the difference does not have practical implications
if a very large sample was taken which can illuminate very small changes in a population mean.
Select the method below which is NOT one of the two types of matched-pairs samples
independent samples from two populations
The F distribution assumes populations are
normal
A researcher finds that 15 out of 45 customers of store 1 feel they get good service where 20 out of 40 customers in store 2 feel they get good service. The sample proportions are
p1 = .33 for store 1 and p2 = .50 for store 2
The t test is considered _____ to mild violations of normality.
robust or OK
If the population variances are unknown and not assumed equal, to calculate the test statistic we replace σ1^2 and σ2^2 with the
sample variances
An OC curve for a mean shows P(Type II error) on the _____ axis and various values of the true _____ on the horizontal axis.
vertical or y mean or average
Inference concerning the ratio of 2 population variances is used to determine relative variability or, in the context of finance,
volatility.
The test statistic for testing a mean when sigma is known is computed by the formula
z=(x−μ0)/(σ/√n)
The critical value of z for a right-tailed test with α = .10 is
+1.28 https://www.omnicalculator.com/statistics/critical-value
All business managers need a basic understanding of hypothesis testing because
- they are required to read and understand technical reports. - they must make decisions based on statistical evidence. - they often interact with specialists.
The critical value of z for a left-tailed test with α = .05 is
-1.645
A statistical hypothesis test requires that we
-determine whether it is a one- or a two-tailed test. -identify the population parameter of interest.
Skipping a formal t test and simply using descriptive statistics can be a good choice if
-the populations are heavily skewed -the sample sizes are small. -there are extreme outliers.
The critical z value is
-the z value associated with level of significance. -the z value that separates the rejection region from the non-rejection region.
Given the following hypotheses: H0: π ≤ .25 vs. H1: π > .25, zcalc = 2.30. The p-value is _____.
.0107 https://www.gigacalculator.com/calculators/z-score-to-p-value-calculator.php
For a two-tailed test of a proportion, zcalc = 1.30. The p-value is
.1936 https://www.omnicalculator.com/statistics/z-test
A family of power curves for a two tailed test with μ0 = 12000 will have power values closer to _____ when μ is close 12000 and closer to _____ the further away μ is from 12000.
0; 1
To find the two-tailed p-value for a test statistic tcalc using Excel use the following formula:
=t.dist.2t(tcalc, df)
True or false: A null hypothesis can be proved true.
False Reason: We can only reject the null hypothesis but cannot prove the assumption is true.
A service company would like to know if the proportion of customers satisfied with their service has changed since the last time they surveyed their customers. Their alternative hypothesis would be
H1: π ≠ π0
Suppose a hypothesis test resulted in a p-value = .0456. Match the significance level to the correct conclusion.
Reject H0 matches: - α = .10, α = .05 Fail to Reject H0 matches: -α = .01
The p-value is the probability of obtaining a sample mean as extreme as the one observed, assuming the _____ hypothesis is true.
null
The t statistic assumes that the population is normally distributed. However, t-test results are considered fairly _____ to non-normality as long as there are no _____.
robust outliers
When calculating power for a proportion test one must know
significance level, α the sample size n
If the population variances are unknown but assumed equal, the test is often called the _____ t test.
pooled
When testing a hypothesis about a proportion and π0 = .25, then n ≥_____ in order to assume normality.
forty or 40
The greater the difference between the true value of the population parameter and the null hypothesis benchmark value the _____ the value of power.
greater, higher, or more
A hypothesis, or assumption, can be discarded or reformulated if the sample data is found to be _____ with the hypothesis.
inconsistent or in conflict
For a given sample size, reducing α results in an
increase in β. Reason: α is reduced at the expense of increasing β.
Alpha (α) is known as the significance _____ and defines the rejection _____.
level region or area
When finding a p-value for a two-tailed test, it is important to _____ the tail area associated with zcalc by _____ because the α area is split between the upper and lower tails.
multiply or times two or 2
Given the following set of hypotheses: H0: Defendant is not guilty H1: Defendant is guilty, which statement describes the consequence of a Type II error?
The defendant is not convicted of the crime but was guilty.
A manager tested the following hypotheses about the average days until an invoice was paid: H0: μ ≤ 15 vs.H1: μ > 15. The resulting p-value = .024. The level of significance used was .05. Which of the following is a valid conclusion?
The manager would reject the null hypothesis and conclude that the average number of days to pay an invoice was greater than 15.
True or false: All business managers need a basic understanding of hypothesis testing.
True Reason: Business managers need to interact with data analysts and therefore need knowledge of hypothesis testing.
True or false: If a confidence interval does not contain μ0, we can reject the null hypothesis in a two-tailed test for the same values of α.
True Reason: μ0 needs to be within the confidence interval to not be rejected.
The significance level is the probability of making a
Type I error (α)
A test statistic is
a standardized score of a sample statistic.
Statistical inference regarding a population variance is based on the
chi-square distribution.
If the population variances are unknown but assumed equal, the t test statistic uses a pooled
standard deviation
A p-value is defined as the probability that
we observed this sample mean (or one more extreme) assuming the null is true.
