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Chapter 8: Problem 1

For each of the following assertions, state whether it is a legitimate statistical hypothesis and why: a. \(H: \sigma>100\) b. \(H: \tilde{x}=45\) c. \(H: s \leq .20\) d. \(H: \sigma_{1} / \sigma_{2}<1\) e. \(H: \bar{X}-\bar{Y}=5\) f. \(H: \lambda \leq .01\), where \(\lambda\) is the parameter of an exponential distribution used to model component lifetime

Short Answer

Expert verified
(a), (d), and (f) are legitimate hypotheses; (b), (c), and (e) are not.

Step by step solution

01

Understanding Statistical Hypotheses

A statistical hypothesis is a statement about a population parameter that can be tested using statistical methods. Generally, it should involve population parameters, such as means (\(\mu\)), proportions, variances (\(\sigma^2\)), or rates. Hypotheses are generally structured as null (\(H_0\)) and alternative (\(H_1\)) hypotheses.
02

Evaluating Hypothesis (a)

The hypothesis is \(H: \sigma>100\). This is a legitimate statistical hypothesis because it concerns the population standard deviation (\(\sigma\)), which is a parameter that can be tested statistically.
03

Evaluating Hypothesis (b)

The hypothesis is \(H: \tilde{x}=45\). This is not a legitimate statistical hypothesis because it refers to the sample median (\(\tilde{x}\)), not a population parameter. Statistical hypotheses should reference population parameters.
04

Evaluating Hypothesis (c)

The hypothesis is \(H: s \leq 0.20\). This is not a legitimate statistical hypothesis because it concerns the sample standard deviation (\(s\)), not the population standard deviation (\(\sigma\)). Hypotheses should be about population parameters.
05

Evaluating Hypothesis (d)

The hypothesis is \(H: \sigma_{1} / \sigma_{2}<1\). This is a legitimate statistical hypothesis because it involves the ratio of two population standard deviations (\(\sigma_1\) and \(\sigma_2\)), which are interpretable as parameters in a hypothesis test for variance among groups.
06

Evaluating Hypothesis (e)

The hypothesis is \(H: \bar{X}-\bar{Y}=5\). This is not a legitimate hypothesis in its current form because it should focus on the difference of two population means (\(\mu_X\) and \(\mu_Y\)), not sample means (\(\bar{X}\) and \(\bar{Y}\)). A correct format might be \( \mu_X - \mu_Y = 5 \).
07

Evaluating Hypothesis (f)

The hypothesis is \(H: \lambda \leq 0.01\). This is a legitimate statistical hypothesis since \(\lambda\) is a parameter of an exponential distribution, which can be statistically tested to model component lifetime.

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Key Concepts

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

Population Parameters
Population parameters are critical in understanding the behavior of a statistical hypothesis. These parameters refer to numerical characteristics of a population. They are like summaries of an entire data set. Parameters can include:
  • Means ( \(\mu\))
  • Proportions
  • Variances ( \(\sigma^2\))
  • Rates
It is important to know that these parameters hold true for the entire population, not just a sample. In statistical hypothesis testing, we use samples to make inferences about these population parameters. By testing hypotheses about parameters, we can decide if our assumptions hold true for the population.
Null and Alternative Hypotheses
Null and alternative hypotheses are pivotal in statistical hypothesis testing. The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. It's like assuming "the old school" method, usually a statement of equality.On the other hand, the alternative hypothesis, represented by \(H_1\), suggests a new theory or an effect. This might involve inequality ( \(>, <, eq\)). Here are quick, clear distinctions:
  • Null Hypothesis ( \(H_0\)): Generally implies no effect or status quo
  • Alternative Hypothesis ( \(H_1\)): Implies some effect or change
The process of hypothesis testing involves deciding whether to reject \(H_0\) in favor of \(H_1\), based on sample data. It's akin to a legal trial where \(H_0\) is "innocent until proven guilty."
Population Variance
Population variance measures the variability of a population data set. It's much like describing how scattered the data points are around the mean. Statistically speaking:
  • Variance ( \(\sigma^2\)) is the square of the standard deviation ( \(\sigma\))
  • It helps in understanding how far individuals in a group differ from the group mean
Why do we care about variance? Because it's crucial in hypothesis testing, especially for tests regarding population dispersion. Lower variance indicates that data points are closer to the mean, whereas higher variance shows more spread-out data points.
Exponential Distribution Parameter
The exponential distribution is a type of continuous probability distribution. It's often used to model time until an event, like the lifespan of a product. The parameter of this distribution, \(\lambda\), plays a vital role:
  • \(\lambda\) represents the rate parameter, indicating how frequently an event occurs
  • A higher \(\lambda\) suggests events happen more frequently
  • A lower \(\lambda\) implies events occur less often
In hypothesis testing, testing for the value of \(\lambda\) can provide insights into the expected duration before an event happens. For example, if you are working with component lifetimes, \(\lambda\) can help predict failure times or assess reliability.

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Most popular questions from this chapter

It is known that roughly \(2 / 3\) of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behavior? The article "Human Behavior: Adult Persistence of Head-Turning Asymmetry" (Nature, 2003: 771) reported that in a random sample of 124 kissing couples, both people in 80 of the couples tended to lean more to the right than to the left. a. If \(2 / 3\) of all kissing couples exhibit this right-leaning behavior, what is the probability that the number in a sample of 124 who do so differs from the expected value by at least as much as what was actually observed? b. Does the result of the experiment suggest that the \(2 / 3\) figure is implausible for kissing behavior? State and test the appropriate hypotheses.

The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article "Experimental Study of Recycled Rubber-Filled HighStrength Concrete" (Magazine of Concrete Res., 2009: \(549-556)\) \(\begin{array}{rrrrr}112.3 & 97.0 & 92.7 & 86.0 & 102.0 \\ 99.2 & 95.8 & 103.5 & 89.0 & 86.7\end{array}\) a. Is it plausible that the compressive strength for this type of concrete is normally distributed? b. Suppose the concrete will be used for a particular application unless there is strong evidence that true average strength is less than \(100 \mathrm{MPa}\). Should the concrete be used? Carry out a test of appropriate hypotheses using the \(P\)-value method.

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at \(40 \mathrm{mph}\) under specified conditions is known to be \(120 \mathrm{ft}\). It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design. a. Define the parameter of interest and state the relevant hypotheses. b. Suppose braking distance for the new system is normally distributed with \(\sigma=10\). Let \(\bar{X}\) denote the sample average braking distance for a random sample of 36 observations. Which of the following three rejection regions is appropriate: \(R_{1}=\\{\bar{x}: \bar{x} \geq 124.80\\}, \quad R_{2}=\\{\bar{x}: \bar{x} \leq 115.20\\}\), \(R_{3}=\\{\bar{x}\) : either \(\bar{x} \geq 125.13\) or \(\bar{x} \leq 114.87\\}\) ? c. What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with \(\alpha=.001\) ? d. What is the probability that the new design is not implemented when its true average braking distance is actually \(115 \mathrm{ft}\) and the appropriate region from part (b) is used? e. Let \(Z=(\bar{X}-120) /(\sigma / \sqrt{n})\). What is the significance level for the rejection region \(\\{z: z \leq-2.33\\}\) ? For the region \(\\{z: z \leq-2.88\\}\) ?

For a fixed alternative value \(\mu^{\prime}\), show that \(\beta\left(\mu^{\prime}\right) \rightarrow 0\) as \(n \rightarrow \infty\) for either a one-tailed or a two-tailed \(z\) test in the case of a normal population distribution with known \(\sigma\).

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than \(.05 \mathrm{~mm}\). What hypotheses should be tested, and why? In this context, what are the type I and type II errors?
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