chunty

In a test of the mean of a population, if the population variance is:

A) known, a z-distributed test statistic is appropriate.

B) known, a t-distributed test statistic is appropriate.

C) unknown, a z-distributed test statistic is appropriate.

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If the population sampled has a known variance, the z-test is the correct test to use. In general, a t-test is used to test the mean of a population when the population variance is unknown. Note that in special cases when the sample is extremely large, the z-test may be used in place of the t-test, but the t-test is considered to be the test of choice when the population variance is unknown.

chunty

Which of the following statements about test statistics is least accurate?

A) In a test of the population mean, if the population variance is unknown and the sample is small, we should use a z-distributed test statistic.

B) In the case of a test of the difference in means of two independent samples, we use a t-distributed test statistic.

C) In a test of the population mean, if the population variance is unknown, we should use a t-distributed test statistic.

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If the population sampled has a known variance, the z-test is the correct test to use. In general, a t-test is used to test the mean of a population when the population is unknown. Note that in special cases when the sample is extremely large, the z-test may be used in place of the t-test, but the t-test is considered to be the test of choice when the population variance is unknown. A t-test is also used to test the difference between two population means while an F-test is used to compare differences between the variances of two populations.

chunty

Maria Huffman is the Vice President of Human Resources for a large regional car rental company. Last year, she hired Graham Brickley as Manager of Employee Retention. Part of the compensation package was the chance to earn one of the following two bonuses: if Brickley can reduce turnover to less than 30%, he will receive a 25% bonus. If he can reduce turnover to less than 25%, he will receive a 50% bonus (using a significance level of 10%). The population of turnover rates is normally distributed. The population standard deviation of turnover rates is 1.5%. A recent sample of 100 branch offices resulted in an average turnover rate of 24.2%. Which of the following statements is most accurate?

A) For the 50% bonus level, the critical value is -1.65 and Huffman should give Brickley a 50% bonus.

B) Brickley should not receive either bonus.

C) For the 50% bonus level, the test statistic is -5.33 and Huffman should give Brickley a 50% bonus.

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Using the process of Hypothesis testing:
Step 1: State the Hypothesis. For 25% bonus level - Ho: m ≥ 30% Ha: m < 30%; For 50% bonus level - Ho: m ≥ 25% Ha: m < 25%.
Step 2: Select Appropriate Test Statistic. Here, we have a normally distributed population with a known variance (standard deviation is the square root of the variance) and a large sample size (greater than 30.) Thus, we will use the z-statistic.
Step 3: Specify the Level of Significance. α = 0.10.
Step 4: State the Decision Rule. This is a one-tailed test. The critical value for this question will be the z-statistic that corresponds to an α of 0.10, or an area to the left of the mean of 40% (with 50% to the right of the mean). Using the z-table (normal table), we determine that the appropriate critical value = -1.28 (Remember that we highly recommend that you have the “common” z-statistics memorized!) Thus, we will reject the null hypothesis if the calculated test statistic is less than -1.28.
Step 5: Calculate sample (test) statistics. Z (for 50% bonus) = (24.2 – 25) / (1.5 / √ 100) = −5.333. Z (for 25% bonus) = (24.2 – 30) / (1.5 / √ 100) = −38.67.
Step 6: Make a decision. Reject the null hypothesis for both the 25% and 50% bonus level because the test statistic is less than the critical value. Thus, Huffman should give Soberg a 50% bonus.
The other statements are false. The critical value of –1.28 is based on the significance level, and is thus the same for both the 50% and 25% bonus levels.

chunty

In order to test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken. The sample value of the computed z-statistic = 3.4. The appropriate decision at a 5% significance level is to:

A) reject the null hypotheses and conclude that the population mean is greater than 100.

B) reject the null hypothesis and conclude that the population mean is not equal to 100.

C) reject the null hypothesis and conclude that the population mean is equal to 100.

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Ho:µ ≤ 100; Ha: µ > 100. Reject the null since z = 3.4 > 1.65 (critical value).

chunty

Brandee Shoffield is the public relations manager for Night Train Express, a local sports team. Shoffield is trying to sell advertising spots and wants to know if she can say with 90% confidence that average home game attendance is greater than 3,000. Attendance is approximately normally distributed. A sample of the attendance at 15 home games results in a mean of 3,150 and a standard deviation of 450. Which of the following statements is most accurate?

A) With an unknown population variance and a small sample size, no statistic is available to test Shoffield's hypothesis.

B) The calculated test statistic is 1.291.

C) Shoffield should use a two-tailed Z-test.

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We will use the process of Hypothesis testing to determine whether Shoffield should reject Ho:

            Step 1: State the Hypothesis

                        Ho: μ ≤ 3,000

                        Ha: μ > 3,000

            Step 2: Select Appropriate Test Statistic

Here, we have a normally distributed population with an unknown variance (we are given only the sample standard deviation) and a small sample size (less than 30.) Thus, we will use the t-statistic.

Step 3: Specify the Level of Significance

Here, the confidence level is 90%, or 0.90, which translates to a 0.10 significance level.

Step 4: State the Decision Rule

This is a one-tailed test. The critical value for this question will be the t-statistic that corresponds to an α of 0.10, and 14 (n-1) degrees of freedom. Using the t-table , we determine that the appropriate critical value = 1.345. Thus, we will reject the null hypothesis if the calculated test statistic is greater than 1.345.

Step 5: Calculate sample (test) statistic

The test statistic = t = (3,150 – 3,000) / (450 / √ 15) = 1.291

Step 6: Make a decision

Fail to reject the null hypothesis because the calculated statistic is less than the critical value. Shoffield cannot state with 90% certainty that the home game attendance exceeds 3,000.

The other statements are false. As shown above, the appropriate test is a t-test, not a Z-test. There is a test statistic for an normally distributed population, an unknown variance and a small sample size – the t-statistic. There is no test for a non-normal population with unknown variance and small sample size.

chunty

An analyst is testing the hypothesis that the mean excess return from a trading strategy is less than or equal to zero. The analyst reports that this hypothesis test produces a p-value of 0.034. This result most likely suggests that the:

A) best estimate of the mean excess return produced by the strategy is 3.4%.

B) smallest significance level at which the null hypothesis can be rejected is 6.8%.

C) null hypothesis can be rejected at the 5% significance level.

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A p-value of 0.035 means the hypothesis can be rejected at a significance level of 3.5% or higher. Thus, the hypothesis can be rejected at the 10% or 5% significance level, but cannot be rejected at the 1% significance level.

chunty

Which of the following statements about statistical results is most accurate?

A) A result may be statistically significant, but may not be economically meaningful.

B) If a result is statistically significant and economically meaningful, the relationship will continue into the future.

C) If a result is statistically significant, it must also be economically meaningful.

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It is possible for an investigation to determine that something is both statistically and economically significant. However, statistical significance does not ensure economic significance. Even if a result is both statistically significant and economically meaningful, the analyst needs to examine the reasons why the economic relationship exists to discern whether it is likely to be sustained in the future.

chunty

Of the following explanations, which is least likely to be a valid explanation for divergence between statistical significance and economic significance?

A) Transactions costs.

B) Data errors.

C) Adjustment for risk.

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While data errors would certainly come to bear on the analysis, in their presence we would not be able to assert either statistical or economic significance. In other words, data errors are not a valid explanation. The others are all mitigating factors that can cause statistically significant results to be less than economically significant.

chunty

Ryan McKeeler and Howard Hu, two junior statisticians, were discussing the relation between confidence intervals and hypothesis tests. During their discussion each of them made the following statement:

McKeeler: A confidence interval for a two-tailed hypothesis test is calculated as adding and subtracting the product of the standard error and the critical value from the sample statistic. For example, for a level of confidence of 68%, there is a 32% probability that the true population parameter is contained in the interval.
Hu: A 99% confidence interval uses a critical value associated with a given distribution at the 1% level of significance. A hypothesis test would compare a calculated test statistic to that critical value. As such, the confidence interval is the range for the test statistic within which a researcher would not reject the null hypothesis for a two-tailed hypothesis test about the value of the population mean of the random variable.

With respect to the statements made by McKeeler and Hu:

A) only one is correct.

B) both are correct.

C) both are incorrect.

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McKeeler’s statement is incorrect. Specifically, for a level of confidence of say, 68%, there is a 68% probability that the true population parameter is contained in the interval. Therefore, there is a 32% probability that the true population parameter is not contained in the interval. Hu’s statement is correct.

chunty

For a t-distributed test statistic with 30 degrees of freedom, a one-tailed test specifying the parameter greater than some value and a 95% confidence level, the critical value is:

A) 1.697.

B) 1.640.

C) 2.042.

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This is the critical value for a one-tailed probability of 5% and 30 degrees of freedom.