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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 larger than zero. 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 an alternative hypothesis?

Short Answer

Expert verified
a. It's an alternative hypothesis as it suggests a change. b. \( \mu = 0 \) is the null hypothesis indicating no effect.

Step by step solution

01

Understanding the Hypothesis

The hypothesis states that the population mean of weight change is greater than zero. This implies the mean weight after following the low-carbohydrate diet for three months is less than the mean weight at the start, suggesting weight loss.
02

Identification of Hypothesis Type for 'a'

Given that the hypothesis is proposing a specific direction (weight change greater than zero), it is formulated as an alternative hypothesis. In research, the alternative hypothesis reflects the effect or change that the study anticipates observing.
03

Defining the Parameter

Let \( \mu \) represent the population mean of weight change observed through the study. This parameter will measure the average difference between the initial and final weight of the participants.
04

Expressing the Null Hypothesis for 'b'

The null hypothesis typically states that there is no effect or difference. Therefore, the null hypothesis is formulated as \( \mu = 0 \). This hypothesis suggests that the diet does not change the mean weight of the participants, indicating no overall weight loss or gain.
05

Identification of Hypothesis Type for 'b'

As the null hypothesis indicates no effect or change (\( \mu = 0 \)), it is considered the null hypothesis. This contrasts with the alternative hypothesis, which would state \( \mu > 0 \) if weight loss were expected.

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Key Concepts

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

Null Hypothesis
When we talk about the null hypothesis, we refer to a statement that there is no effect or no difference in the situation we are studying. It acts as a starting point in hypothesis testing. In experiments, you assume the null hypothesis is true unless evidence suggests otherwise.
In our exercise, the null hypothesis would be that the low-carbohydrate diet has no impact on the weight of participants. So, in mathematical terms, this is expressed as the population mean of the weight change, denoted by \( \mu \), is equal to zero \( (\mu = 0) \).
This translates to saying that the weight before and after the diet doesn鈥檛 change on average. The goal of testing the null hypothesis is to determine if there is enough statistical evidence in favor of the alternative hypothesis by trying to reject the null hypothesis.
  • A null hypothesis often includes: no change, no difference, or being the same.
  • Expressed mathematically using symbols like \( = \), \( \le \), and \( \ge \).
  • It sets a baseline for testing the effect of an intervention or treatment.
Alternative Hypothesis
The alternative hypothesis represents the statement that the researchers are trying to prove. It is the statement of effect or difference that the study is designed to detect. Unlike the null hypothesis, it suggests there is an effect or a change.
In this exercise, the alternative hypothesis proposes that the population mean of the weight change is greater than zero \( (\mu > 0) \). This means participants are expected to lose weight after following the low-carbohydrate diet for three months.
Testing begins with assuming the null hypothesis is correct. If the data provides enough evidence, the null is rejected in favor of the alternative hypothesis.
  • An alternative hypothesis shows a specific direction of effect (like an increase or decrease).
  • It proposes what researchers anticipate: in this case, weight loss.
  • This hypothesis is not directly tested but is supported when the null hypothesis is rejected.
Population Mean
In statistics, the population mean is a measure of the central tendency of a complete population. It is the average of all individual measurements or counts in the given population.
In our context, the population mean, denoted as \( \mu \), represents the average weight change among all individuals who could potentially follow the diet, including those not in the study sample. It provides a standard point of reference for understanding the effect of the diet at a broader level.
When conducting hypothesis tests, such as in this exercise, we use sample data to make inferences about the population mean, since examining an entire population is usually impractical.
  • The population mean is denoted by the Greek letter \( \mu \).
  • It helps determine how sample data relate to the entire population.
  • Understanding the population mean is crucial for interpreting hypothesis test results in relation to the larger group beyond the study sample.

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Most popular questions from this chapter

Examples of hypotheses Give an example of a null hypothesis and an alternative hypothesis about a (a) population proportion and (b) population mean.

A recent study \(^{4}\) considered whether dogs could be trained to detect whether a person has lung cancer or breast cancer by smelling the subject's breath. The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients from those of 83 healthy controls. A dog gave a correct indication of a cancer sample by sitting in front of that sample when it was randomly placed among four control samples. Once trained, the dogs' ability to distinguish cancer patients from controls was tested using breath samples from subjects not previously encountered by the dogs. (The researchers blinded both dog handlers and experimental observers to the identity of breath samples.) Let \(p\) denote the probability a dog correctly detects a cancer sample placed among five samples when the other four are controls. a. Set up the null hypothesis that the dog's predictions correspond to random guessing. b. Set up the alternative hypothesis to test whether the probability of a correct selection differs from random guessing. c. Set up the alternative hypothesis to test whether the probability of a correct selection is greater than with random guessing. d. In one test with 83 Stage I lung cancer samples, the dogs correctly identified the cancer sample 81 times. The test statistic for the alternative hypothesis in part \(\mathrm{c}\) was \(z=17.7 .\) Report the \(\mathrm{P}\) -value to three decimal places and interpret. (The success of dogs in this study made researchers wonder whether dogs can detect cancer at an earlier stage than conventional methods such as MRI scans.

For a test of \(\mathrm{H}_{0}: p=0.50,\) the \(z\) test statistic equals 1.04 a. Find the P-value for \(\mathrm{H}_{a}: p>0.50\). b. Find the P-value for \(\mathrm{H}_{a}: p \neq 0.50\). c. Find the P-value for \(\mathrm{H}_{a}: p<0.50 .\) (Hint: The P-values for the two possible one-sided tests must sum to \(1 .)\) d. Do any of the P-values in part a, part b, or part c give strong evidence against \(\mathrm{H}_{0}\) ? Explain.

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

\(z\) test statistic To test \(\mathrm{H}_{0}: p=0.50\) that a population proportion equals 0.50 , the test statistic is a \(z\) -score that measures the number of standard errors between the sample proportion and the \(\mathrm{H}_{0}\) value of \(0.50 .\) If \(z=3.6,\) do the data support the null hypothesis, or do they give strong evidence against it? Explain.
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