11. Hypothesis Testing (Web + Sch Note)
If we fail to reject the null hypothesis when it is false, what type of error has occured? A) Type II. B) Type III. C) Type I.
A A Type II error is defined as failing to reject the null hypothesis when it is actually false.
Which of the following statements regarding hypothesis testing is least accurate? A) A type I error is acceptance of a hypothesis that is actually false. B) The significance level is the risk of making a type I error. C) A type II error is the acceptance of a hypothesis that is actually false.
A A type I error is the rejection of a hypothesis that is actually true.
The use of the F-distributed test statistic, F = s12 / s22, to compare the variances of two populations does NOT require which of the following? A) two samples are of the same size. B) populations are normally distributed. C) samples are independent of one another.
A The F-statistic can be computed using samples of different sizes. That is, n1 need not be equal to n2.
An analyst for the entertainment industry theorizes that betas for most firms in the industry are higher after September 11, 2001. She sampled 31 firms comparing their betas for the one-year period before and after this date. Based on this sample, she found that the mean differences in betas were 0.19, with a sample standard deviation of 0.11. Her null hypothesis is that the betas are the same before and after September 11. Based on the results of her sample, can we reject the null hypothesis at a 5% significance level and why? Null is: A) rejected. The t-value exceeds the critical value by 7.58. B) not rejected. The critical value exceeds the t-value by 7.58. C) rejected. The t-value exceeds the critical value by 5.67.
A The t-statistic for paired differences: t = (d - ud 0) / sd and sd = sd / √n t = 9.62 from a table with 30 df, the critical value = 2.042
Kyra Mosby, M.D., has a patient who is complaining of severe abdominal pain. Based on an examination and the results from laboratory tests, Mosby states the following diagnosis hypothesis: Ho: Appendicitis, HA: Not Appendicitis. Dr. Mosby removes the patient's appendix and the patient still complains of pain. Subsequent tests show that the gall bladder was causing the problem. By taking out the patient's appendix, Dr. Mosby: A) made a Type II error. B) is correct. C) made a Type I error.
A This statement is an example of a Type II error, which occurs when you fail to reject a hypothesis when it is actually false (also known as the power of the test). The other statements are incorrect. A Type I error is the rejection of a hypothesis when it is actually true (also known as the significance level of the test).
An analyst wants to determine whether the monthly returns on two stocks over the last year were the same or not. What test should she use if she is willing to assume that the returns are normally distributed? A) A difference in means test only if the variances of monthly returns are equal for the two stocks. B) A paired comparisons test because the samples are not independent. C) A difference in means test with pooled variances from the two samples.
B A paired comparisons test must be used. The difference in means test requires that the samples be independent. Portfolio theory teaches us that returns on two stocks over the same time period are unlikely to be independent since both have some systematic risk.
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 hypothesis and conclude that the population mean is not equal to 100. B) reject the null hypotheses and conclude that the population mean is greater than 100. C) reject the null hypothesis and conclude that the population mean is equal to 100.
B Ho:μ ≤ 100; Ha: μ > 100. Reject the null since z = 3.4 > 1.65 (critical value).
In order to test if Stock A is more volatile than Stock B, prices of both stocks are observed to construct the sample variance of the two stocks. The appropriate test statistics to carry out the test is the: A) Chi-square test. B) F test. C) t test.
B The F test is used to test the differences of variance between two samples.
The test of the equality of the variances of two normally distributed populations requires the use of a test statistic that is: A) z-distributed. B) Chi-squared distributed. C) F-distributed.
C The F-distributed test statistic, F = s12 / s22, is used to compare the variances of two populations.
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.
C 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.
Which of the following statements about hypothesis testing is least accurate? A. The power of test = 1 - P(Type II error). B. If the computed z-statistic = —2 and the critical z-value = —1.96, the null hypothesis is rejected. C. The calculated z-statistic for a test of a sample mean when the population variance is known is: z = (X - u) / (α^2 / √n)
C z = (X - u) / (α / √n) z = (X - u) / (α^2 / √n) is variance
The appropriate test statistic to test the hypothesis that the variance of a normally distributed population is equal to 13 is the: A. t-test. B. A-test. C. x^2 test.
C A test of σ^2 = σ^2 is a x^2 test.
Which of the following statements about hypothesis testing is most accurate? A. A Type II error is rejecting the null when it is actually true. B. The significance level equals one minus the probability of a Type I error. C. A two-tailed test with a significance level of 5% has z-critical values of ±1.96.
C Rejecting the null when it is actually true is a Type I error. A Type II error is failing to reject the null hypothesis when it is false. The significance level equals the probability of a Type I error.
An analyst is conducting a hypothesis test to determine if the mean time spent on investment research is different from three hours per day. The test is performed at the 5% level of significance and uses a random sample of64 portfolio managers, where the mean time spent on research is found to be 2.5 hours. The population standard deviation is 1.5 hours. 1. The appropriate null hypothesis for the described test is: A. H0: p = 3 hours. B. H0: p < 3 hours. C. H„: p > 3 hours. 2. This is a: A. one-tailed test. B. two-tailed test. C. paired comparisons test. 3. The calculated z-statistic is: A. -2.67. B. +0.33. C. +2.67. 4. The critical z-value(s) of the test statistic is (are): A. -1.96. B. +1.96. C. ±1.96. 5. The 95% confidence interval for the population mean is: A. {1.00 < p < 3.50}. B. {0.54 < p < 4.46}. C. {2.13 <p< 2.87}. 6. The analyst should most appropriately: A. reject the null hypothesis. B. fail to reject the null hypothesis. C. reach no conclusion because the sample standard deviation was not given.
1. A H0: u = 3 hours 2. B This is a two-sided (two-tailed) test. We want to test if the mean "differs from" 3 hours (i.e., Ha: u ≠ 3 hours). 3. A A The normally distributed test statistic, z = (2.5-3) / (1.5/√64) = -2.67 4. C At α / 2 = 0.025, the critical z-values are: ±za / 2 = ±z0.025 = ±1.96 5. C The 95% confidence interval is {2.5 ± (1 -96)(0.1 875)} = {2.5 ± 0.3675} ->ÿ {2.1325 < p < 2.8675}. 6. A Decision rule: reject H0 if zcomputed < -1.96 or if zcomputed > +1-96. Since -2.67 < -1.96, reject H0.
Austin Roberts believes that the mean price of houses in the area is greater than $145,000. A random sample of36 houses in the area has a mean price of$149,750. The population standard deviation is $24,000, and Roberts wants to conduct a hypothesis test at a 1% level of significance. 1. The appropriate alternative hypothesis is: A. Ha: u < $145,000. B. Ha: u > $145,000. C. Ha: u > $145,000. 2. The value of the calculated test statistic is closest to: A. 0.67. B. 1.19. C. 4.00. 3. Which of the following most accurately describes the appropriate test structure? A. Two-tailed test. B. One-tailed test. C. Chi-square test. 4. The critical value of the z-statistic is: A. ±1.96. B. +2.33. C. ±2.33. 5. At a 1% level of significance, Roberts should: A. reject the null hypothesis. B. fail to reject the null hypothesis. C. neither reject nor fail to reject the null hypothesis.
1. C Ha: u > $145,000 2. B z = (149,750 - 145,000) / [(24,000) / √36)] = 1.1875 3. B The alternative hypothesis, Ha: p > $145,000, only allows for values greater than the hypothesized value. Thus, this is a one-sided (one-tailed) test. 4. B For a one-tailed z-test at the 1% level of significance, the critical z-value is zQ 01 = 2.33. Since the test is one-tailed on the upper end (i.e., 1 1.(: [j > 145,000), we use a positive z-critical value. 5. B The decision rule is to reject Hfl if z-computed > z-critical. Since 1.1875 < 2.33, Roberts will fail to reject the null.
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.
A The significance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value.
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) The calculated test statistic is 1.291. B) With an unknown population variance and a small sample size, no statistic is available to test Shoffield's hypothesis. C) Shoffield should use a two-tailed Z-test.
A 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.
Of the following explanations, which is least likely to be a valid explanation for divergence between statistical significance and economic significance? A) Data errors. B) Transactions costs. C) Adjustment for risk.
A 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.
Which of the following statements about hypothesis testing is least accurate? A) A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is false. B) The significance level is the probability of making a Type I error. C) A Type II error is the probability of failing to reject a null hypothesis that is not true.
A A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is true.
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) null hypothesis can be rejected at the 5% significance level. B) best estimate of the mean excess return produced by the strategy is 3.4%. C) smallest significance level at which the null hypothesis can be rejected is 6.8%.
A 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.
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.
A 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 statements about parametric and nonparametric tests is least accurate? A) Parametric tests are most appropriate when a population is heavily skewed. B) Nonparametric tests have fewer assumptions than parametric tests. C) Nonparametric tests are often used in conjunction with parametric tests.
A For a distribution that is non-normally distributed, a nonparametric test may be most appropriate. A nonparametric test tends to make minimal assumptions about the population, while parametric tests rely on assumptions regarding the distribution of the population. Both kinds of tests are often used in conjunction with one another.
A goal of an "innocent until proven guilty" justice system is to place a higher priority on: A) avoiding type I errors. B) avoiding type II errors. C) the null hypothesis.
A In an "innocent until proven guilty" justice system, the null hypothesis is that the accused is innocent. The hypothesis can only be rejected by evidence proving guilt beyond a reasonable doubt, favoring the avoidance of type I errors.
A test of the population variance is equal to a hypothesized value requires the use of a test statistic that is: A) Chi-squared distributed. B) F-distributed. C) t-distributed.
A In tests of whether the variance of a population equals a particular value, the chi-squared test statistic is appropriate.
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.
A 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.
Which of the following statements about the variance of a normally distributed population is least accurate? A) The Chi-squared distribution is a symmetric distribution. B) The test of whether the population variance equals σ02 requires the use of a Chi-squared distributed test statistic, [(n ? 1)s2] / σ02. C) A test of whether the variance of a normally distributed population is equal to some value σ02, the hypotheses are: H0: σ2 = σ02, versus Ha: σ2 ≠ σ02.
A The Chi-squared distribution is not symmetrical, which means that the critical values will not be numerically equidistant from the center of the distribution, though the probability on either side of the critical values will be equal (that is, if there is a 5% level of significance and a two-sided test, 2.5% will lie outside each of the two critical values).
Which of the following statements about hypothesis testing is most accurate? 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.
A 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.
William Adams wants to test whether the mean monthly returns over the last five years are the same for two stocks. If he assumes that the returns distributions are normal and have equal variances, the type of test and test statistic are best described as: A. paired comparisons test, t-statistic. B. paired comparisons test, A-statistic. C. difference in means test, r-statistic.
A Since the observations are likely dependent (both related to market returns), a paired comparisons (mean differences) test is appropriate and is based on a t-statistic.
If the significance level of a test is 0.05 and the probability of a Type II error is 0.15, what is the power of the test? A. 0.850. B. 0.950. C. 0.975.
A The power of a test is 1 - P(Type II error) = 1 - 0.15 = 0.85.
Joe Sutton is evaluating the effects of the 1987 market decline on the volume of trading. Specifically, he wants to test whether the decline affected trading volume. He selected a sample of 500 companies and collected data on the total annual volume for one year prior to the decline and for one year following the decline. What is the set of hypotheses that Sutton is testing? A) H0: μd ≠ μd0 versus Ha: μd = μd0. B) H0: μd = μd0 versus Ha: μd ≠ μd0. C) H0: μd = μd0 versus Ha: μd > μd0.
B This is a paired comparison because the sample cases are not independent (i.e., there is a before and an after for each stock). Note that the test is two-tailed, t-test.
John Jenkins, CFA, is performing a study on the behavior of the mean P/E ratio for a sample of small-cap companies. Which of the following statements is most accurate? A) One minus the confidence level of the test represents the probability of making a Type II error. B) The significance level of the test represents the probability of making a Type I error. C) A Type I error represents the failure to reject the null hypothesis when it is, in truth, false.
B A Type I error is the rejection of the null when the null is actually true. The significance level of the test (alpha) (which is one minus the confidence level) is the probability of making a Type I error. A Type II error is the failure to reject the null when it is actually false.
A bottler of iced tea wishes to ensure that an average of 16 ounces of tea is in each bottle. In order to analyze the accuracy of the bottling process, a random sample of 150 bottles is taken. Using a t-distributed test statistic of -1.09 and a 5% level of significance, the bottler should: A) not reject the null hypothesis and conclude that bottles do not contain an average of 16 ounces of tea. B) not reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. C) reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea.
B Ho: μ = 16; Ha: μ ≠ 16. Do not reject the null since |t| = 1.09 < 1.96 (critical value).
Which of the following statements about statistical results is most accurate? A) If a result is statistically significant and economically meaningful, the relationship will continue into the future. B) A result may be statistically significant, but may not be economically meaningful. C) If a result is statistically significant, it must also be economically meaningful.
B 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.
Which of the following statements about parametric and nonparametric tests is least accurate? A) The test of the difference in means is used when you are comparing means from two independent samples. B) Nonparametric tests rely on population parameters. C) The test of the mean of the differences is used when performing a paired comparison.
B Nonparametric tests are not concerned with parameters; they make minimal assumptions about the population from which a sample comes. It is important to distinguish between the test of the difference in the means and the test of the mean of the differences. Also, it is important to understand that parametric tests rely on distributional assumptions, whereas nonparametric tests are not as strict regarding distributional properties.
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.
B 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.
The first step in the process of hypothesis testing is: A) selecting the test statistic. B) to state the hypotheses. C) the collection of the sample.
B 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.
If the null hypothesis is innocence, then the statement "It is better that the guilty go free, than the innocent are punished" is an example of preferring a: A) higher level of significance. B) type II error over a type I error. C) type I error over a type II error.
B The statement shows a preference for accepting the null hypothesis when it is false (a type II error), over rejecting it when it is true (a type I error).
A pitching machine is calibrated to deliver a fastball at a speed of 98 miles per hour. Every day, a technician samples the speed of twenty-five fastballs in order to determine if the machine needs adjustment. Today, the sample showed a mean speed of 99 miles per hour with a standard deviation of 1.75 miles per hour. Assume the population is normally distributed. At a 95% confidence level, what is the t-value in relation to the critical value? A) The critical value exceeds the t-value by 1.3 standard deviations. B) The t-value exceeds the critical value by 0.8 standard deviations. C) The t-value exceeds the critical value by 1.5 standard deviations.
B t = (99 - 98) / (1.75 / √25) = 2.86. The critical value for a two-tailed test at the 95% confidence level with 24 degrees of freedom is ±2.06 standard deviations. Therefore, the t-value exceeds the critical value by 0.8 standard deviations.
The appropriate test statistic for a test of the equality of variances for two normally distributed random variables, based on two independent random samples, is the: A. t-test. B. F-test. c. x^2
B The F-test is the appropriate test.
A study was conducted to determine whether the standard deviation of monthly maintenance costs of a Pepper III aircraft is $300. A sample of 30 Pepper Ills had a mean monthly maintenance cost of $3,025 and a standard deviation of $325. Using a 5% level of significance, which of the following is the most appropriate conclusion regarding the difference between the hypothesized value of the population variance and the sample variance? A. The population and sample variances are significantly different. B. The population and sample variances are not significantly different. C. There are no tests that may be used to test variance differences in small samples.
B The wording of the proposition is a little tricky, but the test structure is H„: a2 = 3002 versus Ha: cr2 3002. The appropriate test is a two-tailed chi-square test. The decision rule is to reject HQ if the test statistic is outside the range defined by the critical chi-square values at a / 2 = 0.025 with df = 29. The test statistic is: X^2 = (n-1)s^2 / σ^2 = (29)(105,625)/90,000 X^2 = = 34.035 The critical chi-square values are 16.047 on the left and 45.722 on the right. Since the x^2 fells between these two values, we fail to reject the null hypothesis. This means the population standard deviation is not significantly different than $300.
Given a mean of 10% and a standard deviation of 14%, what is a 95% confidence interval for the return next year? A) -4.00% to 24.00%. B) -17.00% to 38.00%. C) -17.44% to 37.44%.
C 10% +/- 14(1.96) = -17.44% to 37.44%.
A Type II error: A) fails to reject a true null hypothesis. B) rejects a true null hypothesis. C) fails to reject a false null hypothesis.
C A Type II error is defined as accepting the null hypothesis when it is actually false. The chance of making a Type II error is called beta risk.
The variance of 100 daily stock returns for Stock A is 0.0078. The variance of 90 daily stock returns for Stock B is 0.0083. Using a 5% level of significance, the critical value for this test is 1.61. The most appropriate conclusion regarding whether the variance of Stock A is different from the variance of Stock B is that the: A) variance of Stock B is significantly greater than the variance of Stock A. B) variances are not equal. C) variances are equal.
C A test of the equality of variances requires an F-statistic. The calculated F-statistic is 0.0083/0.0078 = 1.064. Since the calculated F value of 1.064 is less than the critical F value of 1.61, we cannot reject the null hypothesis that the variances of the 2 stocks are equal.
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%.
C 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%.
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.
C 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.
Abby Ness is an analyst for a firm that specializes in evaluating firms involved in mineral extraction. Ness believes that the earnings of copper extracting firms are more volatile than those of bauxite extraction firms. In order to test this, Ness examines the volatility of returns for 31 copper firms and 25 bauxite firms. The standard deviation of earnings for copper firms was $2.69, while the standard deviation of earnings for bauxite firms was $2.92. Ness's Null Hypothesis is σ12 = σ22. Based on the samples, can we reject the null hypothesis at a 95% confidence level using an F-statistic and why? Null is: A) rejected. The F-value exceeds the critical value by 0.849. B) rejected. The F-value exceeds the critical value by 0.71. C) not rejected. The critical value exceeds the F-value by 0.71.
C F = s12 / s22 = $2.922 / $2.692 = 1.18 From an F table, the critical value with numerator df = 24 and denominator df = 30 is 1.89.
Which of the following statements about hypothesis testing is most accurate? A Type I error is the probability of: A) failing to reject a false hypothesis. B) rejecting a true alternative hypothesis. C) rejecting a true null hypothesis.
C The Type I error is the error of rejecting the null hypothesis when, in fact, the null is true.
Which of the following statements about hypothesis testing is least accurate? A) The null hypothesis is a statement about the value of a population parameter. B) A Type II error is failing to reject a false null hypothesis. C) If the alternative hypothesis is Ha: μ > μ0, a two-tailed test is appropriate.
C The hypotheses are always stated in terms of a population parameter. Type I and Type II are the two types of errors you can make - reject a null hypothesis that is true or fail to reject a null hypothesis that is false. The alternative may be one-sided (in which case a > or < sign is used) or two-sided (in which case a ≠ is used).
In the process of hypothesis testing, what is the proper order for these steps? 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.
C 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.
An analyst calculates that the mean of a sample of 200 observations is 5. The analyst wants to determine whether the calculated mean, which has a standard error of the sample statistic of 1, is significantly different from 7 at the 5% level of significance. Which of the following statements is least accurate?: A) The mean observation is significantly different from 7, because the calculated Z-statistic is less than the critical Z-statistic. B) The null hypothesis would be: H0: mean = 7. C) The alternative hypothesis would be Ha: mean > 7.
C The way the question is worded, this is a two tailed test.The alternative hypothesis is not Ha: M > 7 because in a two-tailed test the alternative is =, while < and > indicate one-tailed tests. 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) / (standard error of the sample statistic) = (5 - 7) / (1) = -2. The calculated Z is -2, while the critical value is -1.96. The calculated test statistic of -2 falls to the left of the critical Z-statistic of -1.96, and is in the rejection region. Thus, the null hypothesis is rejected and the conclusion is that the sample mean of 5 is significantly different than 7. What the negative sign shows is that the mean is less than 7; a positive sign would indicate that the mean is more than 7. The way the null hypothesis is written, it makes no difference whether the mean is more or less than 7, just that it is not 7.
The power of the test is: A) the probability of rejecting a true null hypothesis. B) equal to the level of confidence. C) the probability of rejecting a false null hypothesis.
C This is the definition of the power of the test: the probability of correctly rejecting the null hypothesis (rejecting the null hypothesis when it is false).
A munitions manufacturer claims that the standard deviation of the powder packed in its shotgun shells is 0.1% of the stated nominal amount of powder. A sport clay association has reviewed a sample of 51 shotgun shells and found a standard deviation of 0.12%. What is the Chi-squared value, and what are the critical values at a 95% confidence level, respectively? A) 72; 34.764 and 67.505. B) 70; 34.764 and 79.490. C) 72; 32.357 and 71.420.
C To compare standard deviations we use a Chi-square statistic. X2 = (n - 1)s2 / σ02 = 50(0.0144) / 0.01 = 72. With 50 df, the critical values at the 95% confidence level are 32.357 and 71.420. Since the Chi-squared value is outside this range, we can reject the hypothesis that the standard deviations are the same.
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.
C 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.
For a hypothesis test with a probability of a Type II error of 60% and a probability of a Type I error of 5%, which of the following statements is most accurate? A. The power of the test is 40%, and there is a 5% probability that the test statistic will exceed the critical value(s). B. There is a 95% probability that the test statistic will be between the critical values if this is a two-tailed test. C. There is a 5% probability that the null hypothesis will be rejected when actually true, and the probability of rejecting the null when it is false is 40%.
C A Type I error is rejecting the null hypothesis when it's true. The probability of rejecting a false null is [1 - Prob Type II] = [1 - 0.60] = 40%, which is called the power of the test. A and B are not necessarily true, since the null may be false and the probability of rejection unknown.
Two samples were drawn from two normally distributed populations. For the first sample, the mean was $50 and the standard deviation was $5. For the second sample, the mean was $55 and the standard deviation was $6. The first sample consists of25 observations and the second sample consists of36 observations. (Note: In the questions below, the subscripts "1" and "2" indicate the first and second sample, respectively.) 1. Consider the hypotheses structured as H0: u1 = $48 versus Ha: u ≠ $48. At a 1% level of significance, the null hypothesis: A. cannot be rejected. B. should be rejected. C. cannot be tested using this sample information provided. 2. Using a 5% level of significance and a hypothesis test structure of H0: α ^2 ≤ 24 versus Ha: α^2 > 24, the null hypothesis: A. cannot be rejected. B. should be rejected. C. cannot be tested using this sample information provided. 3. Consider the hypotheses structured as H0: u ≤ $48 versus Ha: u > $48. At a 5% level of significance, the null hypothesis: A. cannot be rejected. B. should be rejected. C. cannot be tested using the sample information provided. 4. Using a 5% level of significance for a test of the null of H0: α ^2 = α ^2 (first sample) versus the alternative of Ha: α ^2 ≠ α ^2 (second sample), the null hypothesis: A. cannot be rejected. B. should be rejected. C. cannot be tested using the sample information provided.
1. A 2. A 3. B 4. A
For a two-tailed test of hypothesis involving a z-distributed test statistic and a 5% level of significance, a calculated z-statistic of 1.5 indicates that: A) the null hypothesis cannot be rejected. B) the null hypothesis is rejected. C) the test is inconclusive.
A For a two-tailed test at a 5% level of significance the calculated z-statistic would have to be greater than the critical z value of 1.96 for the null hypothesis to be rejected.
Which of the following statements about the F-distribution and chi-square distribution is least accurate? Both distributions: A. are asymmetrical. B. are bound by zero on the left. C. have means that are less than their standard deviations.
C There is no consistent relationship between the mean and standard deviation of the chi-square distribution or F-distribution.
Which of the following assumptions is least likely required for the difference in means test based on two samples? A. The two samples are independent. B. The two populations are normally distributed. C. The two populations have equal variances.
C When the variances are assumed to be unequal, we just calculate the denominator (standard error) differently and use both sample variances to calculate the t-statistic.
