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Chapter 9: Problem 5

Low-carbohydrate diet A study plans to have a sample of obese adults follow a proposed low-carbohydrate diet for three months. The diet imposes limited eating of starches (such as bread and pasta) and sweets, but otherwise no limit on calorie intake. Consider the hypothesis, The population mean of the values of weight change \((=\) weight at start of study \(-\) weight at end of study) is a positive number. a. Is this a null or an alternative hypothesis? Explain your reasoning. b. Define a relevant parameter, and express the hypothesis that the diet has no effect in terms of that parameter. Is it a null or alternative hypothesis?

Short Answer

Expert verified
a. Alternative hypothesis; b. \(\mu = 0\), null hypothesis.

Step by step solution

01

Identifying the Hypothesis Type

In hypothesis testing, the null hypothesis (denoted as \(H_0\)) is usually the statement that there is no effect or no difference, and it serves as the default or starting assumption. The alternative hypothesis (denoted as \(H_a\)) is the statement that there is an observed effect or difference. If we have a hypothesis stating that the population mean of weight change is a positive number, it suggests that there is an effect, i.e., the diet is effective. This is an alternative hypothesis \(H_a\).
02

Defining the Parameter

To form a hypothesis about the diet's effect, define a relevant parameter. Let \(\mu\) represent the population mean of weight changes (weight at the start minus weight at the end). The hypothesis that the diet has no effect implies that there is no average weight change. This is equivalent to stating that the population mean \(\mu\) is zero.
03

Expressing the Hypothesis that Diet Has No Effect

Express the hypothesis that the diet has no effect using the parameter defined in Step 2: \(\mu = 0\). This statement represents the null hypothesis \(H_0\), since it suggests no change or no effect, which is typical of null hypotheses.

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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, the null hypothesis is a critical concept to grasp. It is denoted as \( H_0 \) and represents the statement that there is no effect, no change, or no difference. This hypothesis acts as the default position or the baseline assumption when conducting a statistical test. For example, if we are testing a new diet's impact, the null hypothesis might state that the diet leads to zero weight change. In mathematical terms, using a population mean \( \mu \) as our statistical parameter, we would express this hypothesis as \( \mu = 0 \).
  • The null hypothesis is conventionally tested with the belief that it is true until we have enough evidence to reject it.
  • Rejection of the null hypothesis suggests that there is a statistically significant effect or change, leading us to consider other hypotheses.
Alternative Hypothesis
The alternative hypothesis, denoted by \( H_a \), stands in contrast to the null hypothesis. It is the claim that there is an observable effect or difference. It represents the idea that is being tested in hopes of finding evidence to support it. When discussing a low-carbohydrate diet's impact, as in our study, we hypothesize that the diet causes a positive weight change.
So, the alternative hypothesis would assert that the population mean \( \mu \) of weight change is greater than zero \( (\mu > 0) \).
  • Evidence supporting the alternative hypothesis usually requires rejecting the null hypothesis after statistical tests.
  • In practical terms, accepting the alternative hypothesis suggests that the factor tested (e.g., diet) has an effect.
Population Mean
The population mean is a fundamental statistical concept used to summarize the central tendency of a set of values in the population. It is often represented by the Greek letter \( \mu \). In our exercise, the population mean represents the average weight change in the study sample.
By measuring the mean weight loss (or gain) after a diet, researchers can evaluate the diet's efficacy. If \( \mu \) is found to be significantly different from a predetermined value (like zero), this suggests that the diet may have a notable effect.
  • The population mean helps in making inferences about the entire group based on a sample.
  • Changes in population mean can be key indicators in studies evaluating interventions like diets or treatments.
  • Measuring the population mean can determine if the observed effects are due to the intervention or random chance.
Statistical Parameter
In statistics, a parameter is a measurable factor that helps define a model or system. The most common parameters include measures like the mean, variance, or standard deviation. In hypothesis testing, statistical parameters play a vital role in quantifying the elements being studied.
For example, in our scenario, the population mean \( \mu \) is a statistical parameter that denotes the average weight change. It serves as a focal point in framing both null and alternative hypotheses.
  • Parameters help to specify the conditions of the theoretical distribution of variables in a study.
  • They form the basis of predictions about the population based on a sample.
  • Choosing the right parameter is essential for accurate and meaningful statistical analysis.

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Most popular questions from this chapter

Error probability A significance test about a proportion is conducted using a significance level of \(0.05 .\) The test statistic equals \(2.58 .\) The P-value is 0.01 a. If \(\mathrm{H}_{0}\) were true, for what probability of a Type I error was the test designed? b. If this test resulted in a decision error, what type of error was it?

Practice mechanics of a \(t\) test A study has a random sample of 20 subjects. The test statistic for testing \(\mathrm{H}_{0}: \mu=100\) is \(t=2.40\). Find the approximate P-value for the alternative, (a) \(\mathrm{H}_{a}: \mu \neq 100,\) (b) \(\mathrm{H}_{a}: \mu>100,\) and (c) \(\mathrm{H}_{a} ; \mu<100\).

A person who claims to be psychic says that the probability \(p\) that he can correctly predict the outcome of the roll of a die in another room is greater than \(1 / 6,\) the value that applies with random guessing. If we want to test this claim, we could use the data from an experiment in which he predicts the outcomes for \(n\) rolls of the die. State hypotheses for a significance test, letting the alternative hypothesis reflect the psychic's claim.

Errors in the courtroom Consider the test of \(\mathrm{H}_{0}\) : The defendant is not guilty against \(\mathrm{H}_{a}:\) The defendant is guilty. a. Explain in context the conclusion of the test if \(\mathrm{H}_{0}\) is rejected. b. Describe the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe the consequence of a Type II error.

Gender bias in selecting managers Exercise 9.19 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is \(40 \%\) female and \(60 \%\) male. Let \(p\) be the probability of selecting a female for any given selection. For testing \(\mathrm{H}_{0}: p=0.40\) against \(\mathrm{H}_{a}: p<0.40\) based on a random sample of 50 selections, using the 0.05 significance level, verify that: a. A Type II error occurs if the sample proportion falls less than 1.645 standard errors below the null hypothesis value, which means that \(\hat{p}>0.286\). b. When \(p=0.20,\) a Type II error has probability 0.06 .
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