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LOS a: Define a hypothesis, describe the steps of hypothesis testing, interpret and discuss the choice of the null hypothesis and alternative hypothesis, and distinguish between one-tailed and two-tailed tests of hypotheses.

Which of the following statements about testing a hypothesis using a Z-test is least accurate?

A)	The calculated Z-statistic determines the appropriate significance level to use.

B)	If the calculated Z-statistic lies outside the critical Z-statistic range, the null hypothesis can be rejected.

C)	The confidence interval for a two-tailed test of a population mean at the 5% level of significance is that the sample mean falls between ±1.96 σ/√n of the null hypothesis value.

Susan Bellows is comparing the return on equity for two industries. She is convinced that the return on equity for the discount retail industry (DR) is greater than that of the luxury retail (LR) industry. What are the hypotheses for a test of her comparison of return on equity?

A)	H₀: μ_DR = μ_LR versus H_a: μ_DR ≠ μ_LR.

B)	H₀: μ_DR ≤ μ_LR versus H_a: μ_DR > μ_LR.

C)	H₀: μ_DR = μ_LR versus H_a: μ_DR < μ_LR.

The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that she wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the “greater than” belief.

A)	Collect the sample and calculate the sample statistics. State the hypotheses. Specify the level of significance. Make a decision.

B)	Specify the level of significance. State the hypotheses. Make a decision. Collect the sample and calculate the sample statistics.

C)	State the hypotheses. Specify the level of significance. Collect the sample and calculate the test statistics. Make a decision.

The hypotheses must be established first. Then the test statistic is chosen and the level of significance is determined. Following these steps, the sample is collected, the test statistic is calculated, and the decision is made.

A)	selecting the test statistic.

B)	to state the hypotheses.

C)	the collection of the sample.

The researcher must state the hypotheses prior to the collection and analysis of the data. More importantly, it is necessary to know the hypotheses before selecting the appropriate test statistic.

Which of the following statements least describes the procedure for testing a hypothesis?

A)	Compute the sample value of the test statistic, set up a rejection (critical) region, and make a decision.

B)	Develop a hypothesis, compute the test statistic, and make a decision.

C)	Select the level of significance, formulate the decision rule, and make a decision.

Depending upon the author there can be as many as seven steps in hypothesis testing which are:

Stating the hypotheses.
Identifying the test statistic and its probability distribution.
Specifying the significance level.
Stating the decision rule.
Collecting the data and performing the calculations.
Making the statistical decision.
Making the economic or investment decision.

Which of the following is the correct sequence of events for testing a hypothesis?

A)	State the hypothesis, select the level of significance, compute the test statistic, formulate the decision rule, and make a decision.

B)	State the hypothesis, formulate the decision rule, select the level of significance, compute the test statistic, and make a decision.

C)	State the hypothesis, select the level of significance, formulate the decision rule, compute the test statistic, and make a decision.

Depending upon the author there can be as many as seven steps in hypothesis testing which are:

Stating the hypotheses.
Identifying the test statistic and its probability distribution.
Specifying the significance level.
Stating the decision rule.
Collecting the data and performing the calculations.
Making the statistical decision.
Making the economic or investment decision.

A)	The probability of a Type I error is equal to the significance level of the test.

B)	If you can disprove the null hypothesis, then you have proven the alternative hypothesis.

C)	The power of a test is one minus the probability of a Type I error.

The probability of getting a test statistic outside the critical value(s) when the null is true is the level of significance and is the probability of a Type I error. The power of a test is 1 minus the probability of a Type II error. Hypothesis testing does not prove a hypothesis, we either reject the null or fail to reject it.

An analyst conducts a two-tailed z-test to determine if small cap returns are significantly different from 10%. The sample size was 200. The computed z-statistic is 2.3. Using a 5% level of significance, which statement is most accurate?

A)	You cannot determine what to do with the information given.

B)	Fail to reject the null hypothesis and conclude that small cap returns are close enough to 10% that we cannot say they are significantly different from 10%.

C)	Reject the null hypothesis and conclude that small cap returns are significantly different from 10%.

At the 5% level of significance the critical z-statistic for a two-tailed test is 1.96 (assuming a large sample size). The null hypothesis is H₀: x = 10%. The alternative hypothesis is H_A: x ≠ 10%. Because the computed z-statistic is greater than the critical z-statistic (2.33 > 1.96), we reject the null hypothesis and we conclude that small cap returns are significantly different than 10%.

An analyst conducts a two-tailed test to determine if mean earnings estimates are significantly different from reported earnings. The sample size is greater than 25 and the computed test statistic is 1.25. Using a 5% significance level, which of the following statements is most accurate?

A)	The analyst should reject the null hypothesis and conclude that the earnings estimates are significantly different from reported earnings.

B)	To test the null hypothesis, the analyst must determine the exact sample size and calculate the degrees of freedom for the test.

C)	The analyst should fail to reject the null hypothesis and conclude that the earnings estimates are not significantly different from reported earnings.

The null hypothesis is that earnings estimates are equal to reported earnings. To reject the null hypothesis, the calculated test statistic must fall outside the two critical values. IF the analyst tests the null hypothesis with a z-statistic, the crtical values at a 5% confidence level are ±1.96. Because the calculated test statistic, 1.25, lies between the two critical values, the analyst should fail to reject the null hypothesis and conclude that earnings estimates are not significantly different from reported earnings. If the analyst uses a t-statistic, the upper critical value will be even greater than 1.96, never less, so even without the exact degrees of freedom the analyst knows any t-test would fail to reject the null.

An analyst is testing to see if the mean of a population is less than 133. A random sample of 50 observations had a mean of 130. Assume a standard deviation of 5. The test is to be made at the 1% level of significance.

The null hypothesis is the hypothesis that the researcher wants to reject. Here the hypothesis that is being looked for is that the mean of a population is less than 133. The null hypothesis is that the mean is greater than or equal to 133. The question is whether the null hypothesis will be rejected in favor of the alternative hypothesis that the mean is less than 133.

A test statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean – hypothesized mean) / ((sample standard deviation / (sample size)^1/2)) = (130 – 133) / (5 / 50^1/2) = (-3) / (5 / 7.0711) = -4.24.

A)	reject the alternative hypothesis.

B)	reject the null hypothesis.

C)	accept the null hypothesis.

The calculated test statistic of -4.24 falls to the left of the z-statistic of -2.33, and is in the rejection region. Thus, the null hypothesis is rejected and the conclusion is that the population mean is less than 133.

The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean ? hypothesized mean) / (population standard deviation / (sample size)^1/2 = (X ? μ) / (σ / n^1/2) = (7 ? 5) / (2 / 17^1/2) = (2) / (2 / 4.1231) = 4.12.

What kind of test is being used for the following hypothesis and what would a z-statistic of 1.68 tell us about a hypothesis with the appropriate test and a level of significance of 5%, respectively?

A)	One-tailed test; fail to reject the null.

B)	Two-tailed test; fail to reject the null.

C)	One-tailed test; reject the null.

The way the alternative hypothesis is written you are only looking at the right side of the distribution. You are only interested in showing that B is greater than 0. You don't care if it is less than zero. For a one-tailed test at the 5% level of significance, the critical z value is 1.645. Since the test statistic of 1.68 is greater than the critical value we would reject the null hypothesis.

In a two-tailed test of a hypothesis concerning whether a population mean is zero, Jack Olson computes a t-statistic of 2.7 based on a sample of 20 observations where the distribution is normal. If a 5% significance level is chosen, Olson should:

A)	reject the null hypothesis and conclude that the population mean is significantly different from zero.

B)	reject the null hypothesis and conclude that the population mean is not significantly different from zero.

C)	fail to reject the null hypothesis that the population mean is not significantly different from zero.

At a 5% significance level, the critical t-statistic using the Student’s t-distribution table for a two-tailed test and 19 degrees of freedom (sample size of 20 less 1) is ± 2.093 (with a large sample size the critical z-statistic of 1.960 may be used). Because the critical t-statistic of 2.093 is to the left of the calculated t-statistic of 2.7, meaning that the calculated t-statistic is in the rejection range, we reject the null hypothesis and we conclude that the population mean is significantly different from zero.

In order to test whether the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 3.4. The null and alternative hypotheses are:

A)	H₀: μ = 100; H_a: μ ≠ 100.

B)	H₀: X ≤ 100; H_a: X > 100.

C)	H₀: μ ≤ 100; H_a: μ > 100.

The null hypothesis is that the theoretical mean is not significantly different from zero. The alternative hypothesis is that the theoretical mean is greater than zero.

In order to test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 1.2. If you choose a 5% significance level you should:

A)	fail to reject the null hypothesis and conclude that the population mean is not greater than 100.

B)	reject the null hypothesis and conclude that the population mean is greater than 100.

C)	fail to reject the null hypothesis and conclude that the population mean is greater than 100.

At a 5% significance level, the critical t-statistic using the Student’s t distribution table for a one-tailed test and 29 degrees of freedom (sample size of 30 less 1) is 1.699 (with a large sample size the critical z-statistic of 1.645 may be used). Because the critical t-statistic is greater than the calculated t-statistic, meaning that the calculated t-statistic is not in the rejection range, we fail to reject the null hypothesis and we conclude that the population mean is not significantly greater than 100.

If the null hypothesis is H₀: ρ ≤ 0, what is the appropriate alternative hypothesis?

Jo Su believes that there should be a negative relation between returns and systematic risk. She intends to collect data on returns and systematic risk to test this theory. What is the appropriate alternative hypothesis?

The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that she wishes to reject (in favor of the alternative). The theory in this case is that the correlation is negative.

Jill Woodall believes that the average return on equity in the retail industry, μ, is less than 15%. What are the null (H₀) and alternative (H_a) hypotheses for her study?

A)	H₀: μ ≤ 0.15 versus H_a: μ > 0.15.

B)	H₀: μ ≥ 0.15 versus H_a: μ < 0.15.

C)	H₀: μ < 0.15 versus H_a: μ ≥ 0.15.

Brian Ci believes that the average return on equity in the airline industry, μ, is less than 5%. What are the appropriate null (H₀) and alternative (H_a) hypotheses to test this belief?

A)	H₀: μ < 0.05 versus H_a: μ ≥ 0.05.

B)	H₀: μ ≥ 0.05 versus H_a: μ < 0.05.

C)	H₀: μ < 0.05 versus H_a: μ > 0.05.

The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that he wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the "less than" belief.

George Appleton believes that the average return on equity in the amusement industry, μ, is greater than 10%. What is the null (H₀) and alternative (H_a) hypothesis for his study?

A)	H₀: > 0.10 versus H_a: ≤ 0.10.

B)	H₀: > 0.10 versus H_a: < 0.10.

C)	H₀: ≤ 0.10 versus H_a: > 0.10.

A researcher is testing the hypothesis that a population mean is equal to zero. From a sample with 64 observations, the researcher calculates a sample mean of -2.5 and a sample standard deviation of 8.0. At which levels of significance should the researcher reject the hypothesis?

Fail to reject

Reject

Fail to reject

Reject

Fail to reject

This is a two-tailed test. With a sample size greater than 30, using a z-test is acceptable. The test statistic = = ?2.5. For a two-tailed z-test, the critical values are ±1.645 for a 10% significance level, ±1.96 for a 5% significance level, and ±2.58 for a 1% significance level. The researcher should reject the hypothesis at the 10% and 5% significance levels, but fail to reject the hypothesis at the 1% significance level.

Using Student's t-distribution, the critical values for 60 degrees of freedom (the closest available in a typical table) are ±1.671 for a 10% significance level, ±2.00 for a 5% significance level, and ±2.66 for a 1% significance level. The researcher should reject the hypothesis at the 10% and 5% significance levels, but fail to reject the hypothesis at the 1% significance level.

James Ambercrombie believes that the average return on equity in the utility industry, μ, is greater than 10%. What are the null (H₀) and alternative (H_a) hypotheses for his study?

A)	H₀: μ ≤ 0.10 versus H_a: μ > 0.10.

B)	H₀: μ < 0.10 versus H_a: μ > 0.10.

C)	H₀: μ > 0.10 versus H_a: μ < 0.10.

Which one of the following is the most appropriate set of hypotheses to use when a researcher is trying to demonstrate that a return is greater than the risk-free rate? The null hypothesis is framed as a:

A)	less than statement and the alternative hypothesis is framed as a greater than or equal to statement.

B)	less than or equal to statement and the alternative hypothesis is framed as a greater than statement.

C)	greater than statement and the alternative hypothesis is framed as a less than or equal to statement.

If a researcher is trying to show that a return is greater than the risk-free rate then this should be the alternative hypothesis. The null hypothesis would then take the form of a less than or equal to statement.

Which one of the following best characterizes the alternative hypothesis? The alternative hypothesis is usually the:

A)	hypothesis to be proved through statistical testing.

B)	hoped-for outcome.

C)	hypothesis that is accepted after a statistical test is conducted.

The alternative hypothesis is typically the hypothesis that a researcher hopes to support after a statistical test is carried out. We can reject or fail to reject the null, not 'prove' a hypothesis.

Jill Woodall believes that the average return on equity in the retail industry, μ, is less than 15%. What is null (H₀) and alternative (H_a) hypothesis for her study?

A)	H₀: μ < 0.15 versus H_a: μ = 0.15.

B)	H₀: μ ≥ 0.15 versus H_a: μ < 0.15.

C)	H₀: μ = 0.15 versus H_a: μ ≠ 0.15.

James Ambercrombie believes that the average return on equity in the utility industry, μ, is greater than 10%. What is null (H₀) and alternative (H_a) hypothesis for his study?

A)	H₀: μ ≤ 0.10 versus H_a: μ > 0.10.

B)	H₀: μ = 0.10 versus H_a: μ ≠ 0.10.

C)	H₀: μ ≥ 0.10 versus H_a: μ < 0.10.

This is a one-sided alternative because of the “greater than” belief. We expect to reject the null.

A)	Less than for the null and greater than for the alternative.

B)	Greater than or equal to for the null and less than for the alternative.

C)	Equal to for the null and not equal to for the alternative.

Robert Patterson, an options trader, believes that the return on options trading is higher on Mondays than on other days. In order to test his theory, he formulates a null hypothesis. Which of the following would be an appropriate null hypothesis? Returns on Mondays are:

A)	greater than returns on other days.

B)	less than returns on other days.

C)	not greater than returns on other days.

An appropriate null hypothesis is one that the researcher wants to reject. If Patterson believes that the returns on Mondays are greater than on other days, he would like to reject the hypothesis that the opposite is true–that returns on Mondays are not greater than returns on other days.

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06
0.0	0.0000	0.0040	0.0080	0.0120	0.0160	0.0199	0.0239
0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636
0.2	0.0793	0.0832	0.0871	0.0910	0.0948	0.0987	0.1026
0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406
\|	\|	\|	\|	\|	\|	\|	\|
1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608
1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686
1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.4750
2.0	0.4772	0.4778	0.4783	0.4788	0.4793	0.4798	0.4803
2.1	0.4821	0.4826	0.4830	0.4834	0.4838	0.4842	0.4846
2.2	0.4861	0.4864	0.4868	0.4871	0.4875	0.4878	0.4881
2.3	0.4893	0.4896	0.4898	0.4901	0.4904	0.4906	0.4909
2.4	0.4918	0.4920	0.4922	0.4925	0.4927	0.4929	0.4931