Data Analysis for Managers: Chapters 9-10

The formula for the test statistic for the population variance is χdf2 =

(n−1)s2/σ02

The critical value of z for a right-tailed test with α = .10 is

+1.28

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.

The critical value of z for a left-tailed test with α = .05 is

-1.645

Given the following hypotheses: H0: π ≤ .25 vs. H1: π > .25, zcalc = 2.30. The p-value is

.0107

A left-tailed hypothesis test for a mean resulted in tcalc = -1.50 with df = 29. The p-value would be in what range?

.05 < p-value < .10

For a two-tailed test with zcalc = -1.80, the resulting p-value is

.0718

For a right-tailed test with zcalc = 1.45, the p-value is

.0735

For a two-tailed test of a proportion, zcalc = 1.30. The p-value is

.1936

For a two-tailed test with zcalc = 0.78, the resulting p-value is

.4354

A 95% confidence interval could be used to conduct a two-tailed hypothesis test when alpha is

0.05

The power of a test is measured by

1 - β

Order the steps used in hypothesis testing.

1. state they hypothesis 2. specify the decision rule 3. collect data 4. make decision 5. take action

When testing a hypothesis about a proportion and π0 = .25, then n ≥ _______ in order to assume normality. (Round the value to the nearest whole number.)

40

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.

Suppose a hypothesis test resulted in a p-value = .0456. Match the significance level to the correct conclusion. Reject H0

Choice α = .10 , Choice α = .05

For the following hypotheses and sample result choose the correct conclusion: H0: μ ≥ 67 vs. H1: μ < 67. z critical = - 1.28 and zcalc = -1.04.

Fail to reject the null hypothesis and conclude that there is not significant evidence to state μ < 67.

For the following hypotheses and sample result choose the correct conclusion: H0: μ = 67 vs. H1: μ ≠ 67. z critical = ± 1.645 and zcalc = 1.56.

Fail to reject the null hypothesis and conclude that there is not significant evidence to state μ ≠ 67.

True or false: A Type I error is when we FAIL to reject the null hypothesis when it is actually false.

False

True or false: A null hypothesis can be proved true.

False

True or false: All statistically significant result have practical importance.

False

True or false: Smaller values of α make it easier to reject the null.

False

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

An auditor for a small business wants to test the assumption that the mean value of all accounts receivable is equal to $550 by taking a sample of 40 and computing the sample mean. The auditor will reject the null if the sample mean is clearly different from the null. The null and alternative hypotheses for this test are

H0: μ = $550 and H1: μ ≠ $550

An auditor for a small business wants to test the assumption that the mean value of all accounts receivable is greater than or equal to $550 by taking a sample of 40 and computing the sample mean. The auditor will reject the null only if the sample mean is clearly less than $550. The null and alternative hypotheses for this test are

H0: μ ≥ $550 and H1: μ < $550

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

The alternative hypothesis for a left-tailed test looks like:

H1: μ < μ0

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

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.

Given the following hypotheses: H0: π ≥ .25 vs. H1: π < .25, zcalc = -1.30 and the p-value is .0968. The level of significance is .10. Which is the appropriate conclusion?

Reject the null hypothesis and conclude that the population proportion is less than .25.

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 of the following are NOT options in Excel for calculating a p-value from a t distribution?

T.DIST.LT

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: A proportion can be expressed as percentage but not all percentages can be expressed as a proportion.

True

True or false: All business managers need a basic understanding of hypothesis testing.

True

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

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

True or false: We choose a value for α before conducting a hypothesis test.

True

The significance level is the probability of making a

Type I error (α)

Suppose a hypothesis test resulted in a p-value = .0456. Match the significance level to the correct conclusion. Fail to reject H0

a = .01

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.

A test statistic is

a standardized score of a sample statistic.

Using the critical value method for a two-tailed test, the critical value is determined by a tail area equal to

a/2

A hypothesis is also known as an

assumption

Hypothesis testing is used to test _______ and _______ in business and science.

assumptions; theories

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 parameter value in the null hypothesis is called a ________ and the value does not come from a _______

benchmark; sample

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

We would not reject the null hypothesis when the p-value is

close to 1.0 greater than the level of significance.

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

critical value

Hypothesis testing uses sample ______ to test assumptions.

data

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

__________ testing is used by business managers to guide decision making.

hypothesis

A statistical hypothesis test requires that we

identify the population parameter of interest. determine whether it is a one- or a two-tailed test.

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.

For a given sample size, reducing α results in an

increase in β.

We can generally reduce the probability of Type II errors by

increasing the sample size.

The values of the chi square distribution range from zero to

infinity

The p-value method for testing hypotheses is often preferred by statisticians because

it is more flexible than the critical value method. it expresses the strength of your evidence against the null.

The critical value is determined by the chosen

level of significance

The critical value stated in the decision rule is determined by

level of significance

Alpha (α) is known as the significance _____ and defines the rejection _____ in tailed test.

level; region

Increasing the value of alpha in a hypothesis test will

make it less difficult to reject the null hypothesis,

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; two

A Type II error is made when we fail to reject the _________ hypothesis when it is actually false.

null

The p-value is the probability of obtaining a sample mean as extreme as the one observed, assuming the _______ value is true

null

The hypothesis denoted by H0 is the ________ hypothesis and the hypothesis denoted by H1 is the ________ hypothesis

null; alternative

The normal distribution approximation for a binomial distribution is valid when

nπ0 ≥ 10 and n(1-π0) ≥ 10

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

In using Excel to calculate p-values from a t distribution, we can select Formulas > Insert Function > T.DIST.2T. This command returns the

p-value associated with the relevant t-value for a two-tailed test.

The mean of the sampling distribution of p is the ________ proportion

population

The chi square distribution is ____ skew

positively

A two-tailed test at α = 0.10 results in a p-value of 0.03. The null hypothesis should be

rejected

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

The test statistic used for testing μ when the population mean, σ, is unknown is

tcalc =x−μ0s/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=x−8.5(s32)�-8.5�32, with df = 31

The difference between the sample statistic and the null hypothesis value is measured by a

test statistic.

The p-value is calculated assuming the

the null hypothesis is true.

The power of a test is defined as

the probability of rejecting the null hypothesis when the null hypothesis is false.

For an alternative hypothesis of H1: μ > μ0, we would reject the null hypothesis only when

the sample mean is greater than μ0.

There is little difference between critical values of t and z when ___.

the sample size is large (n > 30)

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.

All business managers need a basic understanding of hypothesis testing because

they must make decisions based on statistical evidence. they are required to read and understand technical reports. they often interact with specialists.

A confidence interval can be used to test a hypothesis if the hypothesis test is

two-tailed and α stays the same.

The test statistic for population variance depends on the degrees of freedom, sample variance, and the hypothesized population

variance

When evaluating the variability of repair costs of a certain automobile, the population ________ is an important component.

variance

A p-value is defined as the probability that

we observed this sample mean (or one more extreme) assuming the null is true.

A Type II error is similar to a false negative for a medical test, that is,

when the null hypothesis states that a patient does not have a virus, the physician fails to reject the null and the patient has the virus.

A Type I error is similar to a false positive for a medical test, that is,

when the null hypothesis states that a patient does not have a virus, the physician rejects the null hypothesis and the patient was healthy.

A sample proportion p is calculated by

x/n

The test statistic for testing a mean when sigma is known is computed by the formula

z=(x−μ0)/(σ/n√)

The test statistic for the hypothesis test of the population proportion p is

zcalc = p−π0π0(1−π0)n

The critical value of z for a two-tailed test with α = .01 is

±2.576

A Type I error is commonly denoted by the symbol

α (alpha)

Which of the following is true?

α = the probability of committing a Type I error; β = the probability of committing a Type II error.

The standard error of a sample proportion p is

√(π0(1−π0)n)