Question

CHAPTER 10: INTRODUCTION TO HYPOTHESIS TESTING: The z Test

Key Terms

Sampling distribution of z --- the distribution of z values that would be obtained if a value of z was calculated for each sample mean for all possible random samples of a given size from some population.

Z Test for a population mean --- A hypothesis test that evaluates how far the observed sample mean deviates, in standard error term units, from the hypothesized population mean.

Null hypothesis (Ho) --- A statistical hypothesis that usually asserts that nothing special is happening with respect to some characteristic of the underlying population.

Alternative hypothesis (H1) --- The opposite of the null hypothesis.

Research hypothesis --- Usually identified with the alternative hypothesis, this is the informal hypothesis or hunch that inspires the entire investigation.

Decision rule --- Specifies precisely when Ho should be rejected (because the observed z qualifies as a rare outcome).

Critical z score --- A z score that separates common from rare outcome and hence dictates whether Ho should be retained or rejected.

Level of significance (α)--- the degree of rarity required of an observed outcome in order to reject the null hypothesis (Ho)

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Text Review

            A null hypothesis is tentatively assumed to be true. It is tested by determining whether an observed sample mean qualifies as a common outcome or a rare outcome in the hypothesized sampling distribution. An observed sample mean qualifies as a common outcome if the difference between its value and that of the hypothesized population mean is small enough to be viewed as merely another random outcome. A common outcome signifies that nothing special is happening in the underlying population and thus the null hypothesis should be retained(?). An observed sample mean qualifies as a rare outcome if the difference between its value and the hypothesized value is too large to reasonably viewed as merely another random outcome. A rare outcome will be a mean that deviates so far from the hypothesized mean that it would emerge from the sparse concentration of possible sample means in either tail of the sampling distribution.

            For the actual hypothesis test, it is customary to convert the mean to a _____________, the familiar standard score conversion presented in Chapter 6. This conversion yields a sampling distribution that approximates the _________________________________. The conversion is accomplished by the z score formula variation where z equals the observed sample mean minus the hypothesized population mean divided by the standard error.

            The z test is accurate only when (1) the population is normally distributed or the sample size is large enough to satisfy the requirements of the ______________________________ theorem and (2) the population SD is known.

            The most crucial and exciting phase of the research is the formulation of the ______________.

The problem is then translated into the __________________ hypothesis, which asserts that nothing special is happening with respect to some characteristic of the underlying population. The null hypothesis always makes a precise statement about a number, never a range of numbers. This single number actually used in the null hypothesis may be based on available information about a relevant population, or it may be based on some existing standard or theory. The null hypothesis is also always makes a statement about a characteristic of the ________________________, never about a characteristic of the sample.

In general, the alternative hypothesis asserts the opposite of the _______________________ hypothesis and it specifies a range of values about the single number that appears in the null hypothesis. The alternative hypothesis is usually identified with the ____________________ hypothesis, the informal hypothesis or hunch that, by implying the presence of something special in the underlying population, serves as inspiration for the entire investigation.

A __________________________ specifies precisely when Ho should be rejected. Decision rules are based on critical z scores that separate common from rare outcomes and dictate whether Ho should be retained or rejected. We can identify the proportion of the total area under the sampling distribution that is identified with rare outcomes. This proportion is often referred to as ____________. The level of significance indicates the degree of rarity among random outcomes required to reject the null hypothesis. The null hypothesis is rejected if the observed z value equals or exceeds the critical z value because it deviates too far into the tails of the sampling distribution. After a decision has been made to retain or reject the null hypothesis, the decision must be ________________________.

To summarize, the step-by-step procedure for testing a hypothesis is: (1) state the research problem, (2) identify the statistical hypotheses, (3) specify a decision rule, (4) calculate the value of the observed z, (5) make a decision, and (6) interpret the decision.

Answer 1

For the actual hypothesis test, it is customary to convert the mean to a ____z score_________, the familiar standard score conversion presented in Chapter 6. This conversion yields a sampling distribution that approximates the ________standard normal distribution_________________________. The conversion is accomplished by the z score formula variation where z equals the observed sample mean minus the hypothesized population mean divided by the standard error.

            The z test is accurate only when (1) the population is normally distributed or the sample size is large enough to satisfy the requirements of the ______Central Limit________________________ theorem and (2) the population SD is known.

            The most crucial and exciting phase of the research is the formulation of the ____research hypothesis__________.

The problem is then translated into the ____null______________ hypothesis, which asserts that nothing special is happening with respect to some characteristic of the underlying population. The null hypothesis always makes a precise statement about a number, never a range of numbers. This single number actually used in the null hypothesis may be based on available information about a relevant population, or it may be based on some existing standard or theory. The null hypothesis is also always makes a statement about a characteristic of the _____population___________________, never about a characteristic of the sample.

In general, the alternative hypothesis asserts the opposite of the _______________________ hypothesis and it specifies a range of values about the single number that appears in the null hypothesis. The alternative hypothesis is usually identified with the ____________________ hypothesis, the informal hypothesis or hunch that, by implying the presence of something special in the underlying population, serves as inspiration for the entire investigation.

A __________________________ specifies precisely when Ho should be rejected. Decision rules are based on critical z scores that separate common from rare outcomes and dictate whether Ho should be retained or rejected. We can identify the proportion of the total area under the sampling distribution that is identified with rare outcomes. This proportion is often referred to as ____________. The level of significance indicates the degree of rarity among random outcomes required to reject the null hypothesis. The null hypothesis is rejected if the observed z value equals or exceeds the critical z value because it deviates too far into the tails of the sampling distribution. After a decision has been made to retain or reject the null hypothesis, the decision must be ________________________.