Quantitative Methods 【Reading 11】Sample statements, testing, following, hypothesis

Mechanic

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

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

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

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.

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The significance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value.

Mechanic

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: µDR = µLR versus Ha: µDR ≠ µLR.

B) H0: µDR ≤ µLR versus Ha: µDR > µLR.

C) H0: µDR = µLR versus Ha: µDR < µLR.

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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.

Mechanic

In the process of hypothesis testing, what is the proper order for these steps?

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

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

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

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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.

Mechanic

The first step in the process of hypothesis testing is:

A) to state the hypotheses.

B) selecting the test statistic.

C) the collection of the sample.

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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.

Mechanic

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.

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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.

Mechanic

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

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

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

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

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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.

Mechanic

Which of the following statements about hypothesis testing is most accurate?

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

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

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

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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.

Mechanic

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

C) 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%.

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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 HA: 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%.

Mechanic

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

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

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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.

Mechanic

Given the following hypothesis:

• The null hypothesis is H0 : µ = 5
• The alternative is H1 : µ ≠ 5
• The mean of a sample of 17 is 7
• The population standard deviation is 2.0

What is the calculated z-statistic?

A) 4.00.

B) 4.12.

C) 8.00.

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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 / n1/2) = (7 − 5) / (2 / 171/2) = (2) / (2 / 4.1231) = 4.12.