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Chapter 8: Problem 146

Determine the critical region and the critical values used to test the following null hypotheses: a. \(\quad H_{o}: \mu=55(\geq), H_{a}: \mu<55, \alpha=0.02\) b. \(\quad H_{o}: \mu=-86(\geq), H_{a}: \mu<-86, \alpha=0.01\) c. \(\quad H_{o}: \mu=107, H_{a}: \mu \neq 107, \alpha=0.05\) d. \(\quad H_{o}: \mu=17.4(\leq), H_{a}: \mu>17.4, \alpha=0.10\)

Short Answer

Expert verified
For a critical region and values: a. \(Z < -2.05\), b. \(Z < -2.33\), c. \(Z < -1.96\) or \(Z > +1.96\), d. \(Z > +1.28\).

Step by step solution

01

Identify the nature of the test

The type of the test (one-tailed or two-tailed) is determined by the alternative hypothesis. For a test, if Ha: 渭 > value or Ha: 渭 < value, it's a one-tailed test. If Ha: 渭 鈮� value, it's a two-tailed test.
02

Determine the Z critical value

The critical values (z) are found from standard normal tables based on the given significance level (伪), tail, and the direction of inequality.
03

Solve for each hypothesis

For the hypotheses: \n a. The test is one-tailed, so the critical value for 伪 = 0.02 is -2.05. The critical region is \(Z < -2.05\).\n b. The test is one-tailed, so the critical value for 伪 = 0.01 is -2.33. The critical region is \(Z < -2.33\).\n c. The test is two-tailed, so the critical values for 伪 = 0.05 are -1.96 and +1.96. The critical region is \(Z < -1.96\) or \(Z > +1.96\).\n d. The test is one-tailed, so the critical value for 伪 = 0.10 is +1.28. The critical region is \(Z > +1.28\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Region
The critical region of a statistical hypothesis test is the set of all outcomes which, if they occur, will lead us to reject the null hypothesis. In more straightforward terms, it's like drawing a line in the sand鈥攖he critical region represents outcomes that are statistically 'too extreme' to support the status quo assumption. For example, if you've set a significance level (伪) and conducted a one-tailed test where you expect values to be less than a certain threshold, then any result falling below the critical value you calculated lies in the critical region鈥攕ignaling that the null hypothesis should be rejected.
Critical Values
Critical values mark the boundaries of the critical regions. They're the numerical cut-offs that define what results of a test are considered extreme enough to reject the null hypothesis. These values are not pull out of a hat; they're determined with respect to a probability distribution, most typically the standard normal distribution in such hypothesis tests. If the test statistic strays into the critical region that lies beyond the critical values, it suggests the observed data is unlikely under the null hypothesis, favoring the alternative hypothesis instead.
Null Hypothesis
The null hypothesis, often denoted as H0, is essentially a proposition that states there is no effect or no difference, serving as a starting point for statistical testing. It's like saying, 'We assume everything is business as usual until proven otherwise.' This hypothetical stance sets the stage for testing and is only rejected if there's sufficient evidence in the data to support the alternative hypothesis. It's crucial not to mix rejecting the null with 'proving' the alternative鈥攐ne simply accumulates evidence against the null hypothesis.
Alternative Hypothesis
The alternative hypothesis, denoted Ha, is the statement we're trying to find evidence for. It's the opposite of the null hypothesis and suggests there is an effect or a difference. Depending on how it's framed, it can be directional (suggesting that a parameter is greater or less than a certain value) or non-directional (suggesting that the parameter is simply not equal to a certain value). Any data falling within the critical regions help support the alternative hypothesis.
Significance Level
A significance level, labeled 伪, is a threshold chosen by the researcher which determines how much risk of error is acceptable when rejecting the null hypothesis. The smaller the 伪, the more stringent the criterion we set for rejecting the null hypothesis. Common levels are 0.05, 0.01, and 0.10, reflecting a 5%, 1%, and 10% risk of drawing a wrong conclusion (Type I error). This has to be set before running the test and dictates the size of the critical regions.
One-tailed Test
A one-tailed test targets one direction of interest in the hypothesis test. If the research hypothesis states that a value is either 'greater than' or 'less than' but not both, the one-tailed test is the correct choice. This approach has more power to detect an effect in the specified direction since it focuses all the significance level in one tail of the probability distribution. Suppose you suspect an enzyme accelerates a reaction; a one-tailed test could be used to determine if the reaction time is less than a standard value.
Two-tailed Test
In contrast to one-tailed tests, a two-tailed test is non-directional and is used when the alternative hypothesis simply states that a parameter is not equal to a specific value鈥攊t can be either greater or lesser. Here, the critical region is split between the two tails (extremes) of the distribution. If our enzyme from the previous example might either accelerate or decelerate a reaction, a two-tailed test would be needed to detect deviations from the standard time in both directions.
Standard Normal Distribution
The standard normal distribution, which is a bell-shaped curve symmetric around a mean of zero and a standard deviation of one, often serves as a basis for critical values in hypothesis tests. This distribution is incredibly useful because it allows us to assign probabilities to different outcomes under the null hypothesis. Critical values are typically found using Z-scores, which represent the number of standard deviations away from the mean a particular value is. When you look up a critical value in a standard normal table, you're effectively using the properties of this universal distribution to make decisions about your hypothesis test.

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Most popular questions from this chapter

From a population of unknown mean \(\mu\) and a standard deviation \(\sigma=5.0,\) a sample of \(n=100\) is selected and the sample mean 41.5 is found. Compare the concepts of estimation and hypothesis testing by completing the following: a. Determine the \(95 \%\) confidence interval for \(\mu\) b. Complete the hypothesis test involving \(H_{a}: \mu \neq 40\) using the \(p\) -value approach and \(\alpha=0.05\) c. Complete the hypothesis test involving \(H_{a}: \mu \neq 40\) using the classical approach and \(\alpha=0.05\) d. On one sketch of the standard normal curve, locate the interval representing the confidence interval from part a; the \(z \star, p\) -value, and \(\alpha\) from part \(b ;\) and the \(z \\#\) and critical regions from part c. Describe the relationship between these three separate procedures.

a. What decision is reached when the test statistic falls in the critical region? b. What decision is reached when the test statis- tic falls in the noncritical region?

There are only two possible decisions as a result of a hypothesis test. a. State the two possible decisions. b. Describe the conditions that will lead to each of the two decisions identified in part a.

Assume that \(z\) is the test statistic and calculate the value of \(z \star\) for each of the following: a. \(\quad H_{o}: \mu=10, \sigma=3, n=40, \bar{x}=10.6\) b. \(\quad H_{o}: \mu=120, \sigma=23, n=25, \bar{x}=126.2\) c. \(\quad H_{o}: \mu=18.2, \sigma=3.7, n=140, \bar{x}=18.93\) d. \(\quad H_{o}: \mu=81, \sigma=13.3, n=50, \bar{x}=79.6\)

A high-tech company wants to estimate the mean number of years of college education its employees have completed. A good estimate of the standard deviation for the number of years of college is \(1.0 .\) How large a sample needs to be taken to estimate \(\mu\) to within 0.5 of a year with \(99 \%\) confidence?
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