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LOS b: Define and interpret a test statistic, a Type I and a Type II error, and a significance level, and explain how significance levels are used in hypothesis testing.

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.

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.

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.

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.

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: H_o: Appendicitis, H_A: 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:

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).

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 H_a: μ > μ₀, a two-tailed test is appropriate.

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).

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.

The Type I error is the error of rejecting the null hypothesis when, in fact, the null is true.

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 Type I error is the probability of rejecting the null hypothesis when the null hypothesis is true.

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.

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)	fails to reject a true null hypothesis.

B)	rejects a true null hypothesis.

C)	fails to reject a false null hypothesis.

If we fail to reject the null hypothesis when it is false, what type of error has occured?

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)	rejects a false null hypothesis.

B)	fails to reject a false null hypothesis.

C)	rejects a true null hypothesis.

A Type I Error is defined as rejecting the null hypothesis when it is actually true. The probability of committing a Type I error is the significance level or alpha risk.

Which of the following statements regarding Type I and Type II errors is most accurate?

A)	A Type I error is failing to reject the null hypothesis when it is actually false.

B)	A Type I error is rejecting the null hypothesis when it is actually true.

C)	A Type II error is rejecting the alternative hypothesis when it is actually true.

A Type I Error is defined as rejecting the null hypothesis when it is actually true. The probability of committing a Type I error is the risk level or alpha risk.

A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $59,000 per year. What is the test statistic given a sample of 135 newly acquired CFA charterholders with a mean starting salary of $64,000 and a standard deviation of $5,500?

With a large sample size (135) the z-statistic is used. 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) = (64,000 – 59,000) / (5,500 / 135^1/2) = (5,000) / (5,500 / 11.62) = 10.56.

A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $58,500 per year. What is the test statistic given a sample of 175 newly acquired CFA charterholders with a mean starting salary of $67,000 and a standard deviation of $5,200?

With a large sample size (175) the z-statistic is used. 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) = (67,000 – 58,500) / (5,200 / 175^1/2) = (8,500) / (5,200 / 13.22) = 21.62.

A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $54,000 per year. Assuming a normal distribution, what is the test statistic given a sample of 75 newly acquired CFA charterholders with a mean starting salary of $57,000 and a standard deviation of $1,300?

With a large sample size (75) the z-statistic is used. 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) = (57,000 – 54,000) / (1,300 / 75^1/2) = (3,000) / (1,300 / 8.66) = 19.99.

Identify the error type associated with the level of significance and the meaning of a 5 percent significance level.

Error type

α = 0.05 means there is a 5 percent probability of

Type I error

failing to reject a true null hypothesis

Type II error

rejecting a true null hypothesis

Type I error

rejecting a true null hypothesis

The significance level is the risk of making a Type 1 error and rejecting the null hypothesis when it is true.

A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $57,000 per year. Assuming a normal distribution, what is the test statistic given a sample of 115 newly acquired CFA charterholders with a mean starting salary of $65,000 and a standard deviation of $4,500?

With a large sample size (115) the z-statistic is used. 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) = (65,000 – 57,000) / (4,500 / 115^1/2) = (8,000) / (4,500 / 10.72) = 19.06.

If a two-tailed hypothesis test has a 5% probability of rejecting the null hypothesis when the null is true, it is most likely that the:

A)	probability of a Type I error is 2.5%.

B)	power of the test is 95%.

C)	significance level of the test is 5%.

Rejecting the null hypothesis when it is true is a Type I error. The probability of a Type I error is the significance level of the test. The power of a test is one minus the probability of a Type II error, which cannot be calculated from the information given.

Which of the following statements about hypothesis testing is most accurate? A Type II error is the probability of:

A)	rejecting a true alternative hypothesis.

B)	failing to reject a false null hypothesis.

C)	rejecting a true null hypothesis.

The Type II error is the error of failing to reject a null hypothesis that is not true.

A)	a Type II error increases.

B)	incorrectly accepting the null decreases.

C)	incorrectly rejecting the null increases.

If P(Type I error) decreases, then P(Type II error) increases. A null hypothesis is never accepted. We can only fail to reject the null.

A)	A Type I error is rejecting the null hypothesis when it is true, and a Type II error is accepting the alternative hypothesis when it is false.

B)	A hypothesized mean of 3, a sample mean of 6, and a standard error of the sampling means of 2 give a sample Z-statistic of 1.5.

C)	When the critical Z-statistic is greater than the sample Z-statistic in a two-tailed test, reject the null hypothesis and accept the alternative hypothesis.

Z = (6 - 3)/2 = 1.5. A Type II error is wrongly accepting the null hypothesis. The null hypothesis should be rejected when the sample Z-statistic is greater than the critical Z-statistic.

A researcher is testing whether the average age of employees in a large firm is statistically different from 35 years (either above or below). A sample is drawn of 250 employees and the researcher determines that the appropriate critical value for the test statistic is 1.96. The value of the computed test statistic is 4.35. Given this information, which of the following statements is least accurate? The test:

A)	has a significance level of 95%.

B)	indicates that the researcher will reject the null hypothesis.

C)	indicates that the researcher is 95% confident that the average employee age is different than 35 years.

This test has a significance level of 5%. The relationship between confidence and significance is: significance level = 1 ? confidence level. We know that the significance level is 5% because the sample size is large and the critical value of the test statistic is 1.96 (2.5% of probability is in both the upper and lower tails).