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Chapter 9: 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
Assertions (a), (d), and (f) are legitimate; assertions (b), (c), and (e) are not.

Step by step solution

01

Understanding Statistical Hypotheses

A statistical hypothesis is a statement regarding a population parameter that can be tested using sample data. It must specify a particular population parameter and propose a testable condition.
02

Evaluating Assertion A

Assertion (a) is "\(H: \sigma > 100\)". This statement proposes a test about a population standard deviation, \(\sigma\). It is legitimate because it hypothesizes a condition about a parameter that can be tested through statistical methods.
03

Evaluating Assertion B

Assertion (b) is "\(H: \tilde{x} = 45\)". Here, \(\tilde{x}\) represents a sample median, not a population parameter. Statistical hypotheses should relate to population parameters like mean or variance. Therefore, this is not a legitimate statistical hypothesis.
04

Evaluating Assertion C

Assertion (c) is "\(H: s \leq 0.20\)". The symbol \(s\) represents the sample standard deviation, which is not a population parameter. Hypotheses must concern population parameters, not sample statistics, thus making this not a legitimate hypothesis.
05

Evaluating Assertion D

Assertion (d) is "\(H: \sigma_{1} / \sigma_{2} < 1\)". This compares the standard deviations of two populations. Since it relates to population parameters, it is a legitimate hypothesis.
06

Evaluating Assertion E

Assertion (e) is "\(H: \bar{X} - \bar{Y} = 5\)". This statement compares sample means, \(\bar{X}\) and \(\bar{Y}\), rather than population parameters. Hypotheses must refer to population parameters, so this is not a legitimate hypothesis.
07

Evaluating Assertion F

Assertion (f) is "\(H: \lambda \leq 0.01\)", where \(\lambda\) is the parameter of an exponential distribution for component lifetime. \(\lambda\) is a population parameter, and this correctly forms a hypothesis that can be tested, making it legitimate.

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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 statistical measures that describe characteristics of an entire population. They are often constants, meaning their values remain the same unless the entire population changes.
For instance, suppose we are interested in the mean height of all adults in a city. The population mean, denoted as \( \mu \), is a parameter in this context.
Key points about population parameters include:
  • Parameters such as the mean (\( \mu \)), variance (\( \sigma^2 \)), and standard deviation (\( \sigma \)) provide insights into the population's behavior or characteristics.
  • These parameters differ from sample statistics, which are calculated from collected data.
  • When forming a legitimate statistical hypothesis, it should involve these population parameters.
Understanding and correctly identifying population parameters is crucial when crafting statistical hypotheses that can provide meaningful results when tested.
Sample Statistics
Sample statistics are measures that provide insights into the population based on a smaller, more manageable set of observations, known as a sample.
Examples of sample statistics include the sample mean (\( \bar{x} \)), sample variance (\( s^2 \)), and sample standard deviation (\( s \)).
These points are key to sample statistics:
  • Sample statistics are not fixed and can vary with different samples selected from the population.
  • They are used to estimate or infer the population parameters.
  • When crafting hypotheses, they don't typically serve as the statement's direct values but rather help estimate the population parameters.
By understanding and using sample statistics wisely, you can make informed decisions about the population without the cumbersome process of surveying every member within it.
Exponential Distribution
The exponential distribution is a continuous probability distribution that is commonly used to model the time between events in a process where events occur continuously and independently at a constant average rate.
One prominent parameter of the exponential distribution is \( \lambda \), which represents the rate of occurrences.
Key characteristics include:
  • The probability density function (PDF) is given by \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \).
  • It has a constant hazard rate, meaning the likelihood of an event happening in a given interval remains the same over time.
  • Commonly used in reliability engineering, queuing theory, and for modeling component lifetimes.
The exponential distribution's unique property of having no memory makes it ideal for modeling the lifetime of products or time between occurrences of events.
Standard Deviation Comparison
Comparing standard deviations between populations allows researchers to understand variability differences. Standard deviation (\( \sigma \)) measures how spread out the numbers in a data set are.
In hypotheses like "\( H: \sigma_1 / \sigma_2 < 1 \)", we compare the standard deviations of two populations.
Here are some essential aspects:
  • A higher standard deviation indicates greater variability in the data.
  • While forming hypotheses, comparing standard deviations helps determine if one population is more consistent than another.
  • Such comparisons are frequently used in quality control and experimental settings to ensure adherence to standards.
In essence, comparing standard deviations can tell us much about the data's consistency, aiding in verifying underlying assumptions in statistical testing.

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Most popular questions from this chapter

A random sample of soil specimens was obtained, and the amount of organic matter \((\%)\) in the soil was determined for each specimen, resulting in the accompanying data (from "Engineering Properties of Soil," Soil Sci., 1998: 93-102). \(\begin{array}{llllllll}1.10 & 5.09 & 0.97 & 1.59 & 4.60 & 0.32 & 0.55 & 1.45 \\\ 0.14 & 4.47 & 1.20 & 3.50 & 5.02 & 4.67 & 5.22 & 2.69 \\ 3.98 & 3.17 & 3.03 & 2.21 & 0.69 & 4.47 & 3.31 & 1.17 \\ 0.76 & 1.17 & 1.57 & 2.62 & 1.66 & 2.05 & & \end{array}\) The values of the sample mean, sample standard deviation, and (estimated) standard error of the mean are \(2.481,1.616\), and \(.295\), respectively. Does this data suggest that the true average percentage of organic matter in such soil is something other than \(3 \%\) ? Carry out a test of the appropriate hypotheses at significance level 10 by first determining the \(P\)-value. Would your conclusion be different if \(\alpha=.05\) had been used? [Note: A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size.]

A new method for measuring phosphorus levels in soil is described in the article "A Rapid Method to Determine Total Phosphorus in Soils" (Soil Sci. Amer. \(J ., 1988: 1301-1304)\). Suppose a sample of 11 soil specimens, each with a true phosphorus content of \(548 \mathrm{mg} / \mathrm{kg}\), is analyzed using the new method. The resulting sample mean and standard deviation for phosphorus level are 587 and 10 , respectively. a. Is there evidence that the mean phosphorus level reported by the new method differs significantly from the true value of \(548 \mathrm{mg} / \mathrm{kg}\) ? Use \(\alpha=.05\). b. What assumptions must you make for the test in part (a) to be appropriate?

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge-water temperature above \(150^{\circ}, 50\) water samples will be taken at randomly selected times, and the temperature of each sample recorded. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ}\) versus \(H_{\mathrm{a}}: \mu>150^{\circ}\). In the context of this situation, describe type I and type II errors. Which type of error would you consider more serious? Explain.

Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let \(p\) denote the true proportion of samples that yield before their theoretical yielding point. If on the basis of a sample it can be concluded that more than \(20 \%\) of all specimens yield before the theoretical point, the production process will have to be modified. a. If 15 of 60 specimens yield before the theoretical point, what is the \(P\)-value when the appropriate test is used, and what would you advise the company to do? b. If the true percentage of "early yields" is actually \(50 \%\) (so that the theoretical point is the median of the yield distribution) and a level \(.01\) test is used, what is the probability that the company concludes a modification of the process is necessary?

Scientists have recently become concerned about the safety of Teflon cookware and various food containers because perfluorooctanoic acid (PFOA) is used in the manufacturing process. An article in the July 27, 2005, New York Times reported that of 600 children tested, \(96 \%\) had PFOA in their blood. According to the FDA, \(90 \%\) of all Americans have PFOA in their blood. a. Does the data on PFOA incidence among children suggest that the percentage of all children who have PFOA in their blood exceeds the FDA percentage for all Americans? Carry out an appropriate test of hypotheses. b. If \(95 \%\) of all children have PFOA in their blood, how likely is it that the null hypothesis tested in (a) will be rejected when a significance level of \(.01\) is employed? c. Referring back to (b), what sample size would be necessary for the relevant probability to be \(.10\) ?
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