Reading 11: Hypothesis Testing LOS c习题精选 hypothesis, determine, power, interpret, div

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LOS c: Define and interpret a decision rule and the power of a test, and explain the relation between confidence intervals and hypothesis tests.

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.

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The way the question is worded, this is a two tailed test.The alternative hypothesis is not H_a: 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.

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c

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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) 1.640.

B) 1.697.

C) 2.042.

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

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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) both are correct.

B) only one is 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.

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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%.

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10% +/- 14(1.96) = -17.44% to 37.44%.

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A goal of an “innocent until proven guilty” justice system is to place a higher priority on:

A) avoiding type II errors.

B) avoiding type I errors.

C) the null hypothesis.

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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.

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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) type II error over a type I error.

B) higher level of significance.

C) type I error over a type II error.

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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).

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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) reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea.

C) not reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea.

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H_o: μ = 16; H_a: μ ≠ 16. Do not reject the null since |t| = 1.09 < 1.96 (critical value).

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The power of the test is:

A) the probability of rejecting a true null hypothesis.

B) the probability of rejecting a false null hypothesis.

C) equal to the level of confidence.

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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).