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Chapter 10: Problem 8

A researcher speculates that because of differences in diet, Japanese children may have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170 . Let \(\mu\) represent the mean blood cholesterol level for all Japa-nese children. What hypotheses should the researcher test?

Short Answer

Expert verified
The null hypothesis \(H_0\) is \(\mu = 170\) (The mean blood cholesterol level for Japanese children is same as the U.S. children). The alternative hypothesis \(H_1\) is \(\mu < 170\) (The mean blood cholesterol level for Japanese children is lower than U.S. children).

Step by step solution

01

Understanding Null and Alternative Hypotheses

In hypothesis testing, the initial claim that is assumed to be true is called the Null Hypothesis. The contradictory statement is called the Alternative Hypothesis. Generally, we denote Null Hypothesis as \(H_0\) and Alternative Hypothesis as \(H_1\) or \(H_a\).
02

Define the Null Hypothesis

In this case, the null hypothesis assumes that the mean blood cholesterol level for Japanese children is the same as that for US children. Therefore, the null hypothesis can be written as \(H_0: \mu = 170\).
03

Define the Alternative Hypothesis

The researcher speculates that the mean blood cholesterol level for Japanese children is lower than 170. Therefore, the alternative hypothesis can be defined as \(H_1: \mu < 170\).

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Key Concepts

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

Null Hypothesis
In hypothesis testing, when we begin our analysis, we start with an assumption called the "null hypothesis." This is a statement of no effect or no difference and serves as the default or status quo assumption. The null hypothesis is symbolized by the notation \(H_0\). In the context of our exercise, the null hypothesis suggests that the mean blood cholesterol level among Japanese children is the same as that among U.S. children. Hence, for this test, the null hypothesis is written as \(H_0: \mu = 170\).
  • The null hypothesis sets a baseline for statistical testing.
  • It assumes any observed effect is due to random chance.
  • Rejecting \(H_0\) indicates there is sufficient evidence pointing towards an effect or difference.
Real-world applications of the null hypothesis help researchers understand whether changes in variables have significant effects, outside of random chance.
Alternative Hypothesis
The alternative hypothesis is crucial in hypothesis testing. It proposes a new idea or theory, challenging the initial assumption, represented by the null hypothesis. In contrast to the null hypothesis, an alternative hypothesis suggests there is an effect or a difference.In our exercise, the researcher suggests that Japanese children may have a lower mean blood cholesterol level than U.S. children. This leads to the alternative hypothesis being \(H_1: \mu < 170\).
  • It represents what the researcher aims to support through testing.
  • The symbol \(H_1\) or \(H_a\) is used to denote this hypothesis.
  • Acceptance of the alternative hypothesis indicates a statistically significant difference.
Research often designs studies to obtain evidence supporting the alternative hypothesis, which indicates a significant finding in their work.
Mean Blood Cholesterol Level
Mean blood cholesterol level is an average measure of cholesterol found in the blood, important for determining an individual's health, particularly in relation to heart disease risk. In the exercise, the average level for U.S. children is specified as 170, which acts as a reference point for testing the mean level in Japanese children.
  • Cholesterol levels can be influenced by factors such as diet and genetics.
  • The average, or mean, level provides a summary statistic for simplifying comparison between groups.
  • Cholesterol is a key indicator of cardiovascular health in demographics.
Understanding and examining the differences in mean blood cholesterol levels can guide actions and policies aimed at improving health and preventing disease across populations.

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Most popular questions from this chapter

Pairs of \(P\) -values and significance levels, \(\alpha\), are given. For each pair, state whether the observed \(P\) -value leads to rejection of \(H_{0}\) at the given significance level. a. \(\quad P\) -value \(=.084, \alpha=.05\) b. \(\quad P\) -value \(=.003, \alpha=.001\) c. \(\quad P\) -value \(=.498, \alpha=.05\) d. \(\quad P\) -value \(=.084, \alpha=.10\) e. \(P\) -value \(=.039, \alpha=.01\) f. \(\quad P\) -value \(=.218, \alpha=.10\)

A manufacturer of hand-held calculators receives large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selccted for inspcction. Information from the samplc is then used to test \(H_{0}: p=.01\) versus \(H_{a}: p>.01\), where \(p\) is the actual proportion of defective circuits in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier because of inferior quality. (A shipment is defined to be of inferior quality if it contains more than \(1 \%\) defective circuits.) a. In this context, define Type I and Type II errors. b. From the calculator manufacturer's point of view, which type of error is considered more serious? ca From the printed circuit supplier's point of view, which type of error is considered more serious?

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 \(150^{\circ} \mathrm{F}\), a scientist will take 50 water samples at randomly selected times and will record the water temperature of each sample. She will then use a \(z\) statistic $$ z=\frac{\bar{x}-150}{\frac{\sigma}{\sqrt{n}}} $$ to decide between the hypotheses \(H_{0}: \mu=150\) and \(H_{a}: \mu>150\), where \(\mu\) is the mean temperature of discharged water. Assume that \(\sigma\) is known to be 10 . a. Explain why use of the \(z\) statistic is appropriate in this setting.

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let \(p\) be the proportion of all apartments that prohibit children. If the city council is convinced that \(p\) is greater than \(0.75\), it will consider appropriate legislation. a. If 102 of the 125 sampled apartments exclude renters with children, would a level \(.05\) test lead you to the conclusion that more than \(75 \%\) of all apartments exclude children? b. What is the power of the test when \(p=.8\) and \(\alpha=.05 ?\)

The paper titled "Music for Pain Relief" (The Cochrane Database of Systematic Reviews, April \(19 .\) 2006) concluded, based on a review of 51 studies of the effect of music on pain intensity, that "Listening to music reduces pain intensity levels .... However, the magnitude of these positive effects is small, the clinical relevance of music for pain relief in clinical practice is unclear." Are the authors of this paper claiming that the pain reduction attributable to listening to music is not statistically significant, not practically significant, or neither statistically nor practically significant? Explain.
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