91Ó°ÊÓ

The body weight of a healthy 3-month-old colt should be about \(\mu=60 \mathrm{kg}\) (Source: The Merck Veterinary Manual, a standard reference manual used in most veterinary colleges). (a) If you want to set up a statistical test to challenge the claim that \(\mu=60 \mathrm{kg},\) what would you use for the null hypothesis \(H_{0} ?\) (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than \(60 \mathrm{kg}\). What would you use for the alternate hypothesis \(H_{1} ?\) (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than \(60 \mathrm{kg}\). What would you use for the alternate hypothesis? (d) Suppose you want to test the claim that the average weight of such a wild colt is different from \(60 \mathrm{kg}\). What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the \(P\) -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Step by step solution

01

Null Hypothesis

In hypothesis testing, the null hypothesis \( H_0 \) is the statement of no effect or no difference. For part (a), where we challenge the claim \( \mu = 60 \) kg, the null hypothesis is \( H_0: \mu = 60 \) kg.

02

Alternate Hypothesis for Less Than 60 kg

For part (b), when testing if the average weight of a wild Nevada colt is less than 60 kg, the alternate hypothesis would be \( H_1: \mu < 60 \) kg.

03

Alternate Hypothesis for Greater Than 60 kg

For part (c), testing if the average weight of a wild colt is greater than 60 kg, the alternate hypothesis would be \( H_1: \mu > 60 \) kg.

04

Alternate Hypothesis for Different From 60 kg

In part (d), to test if the average weight is different from 60 kg, the alternate hypothesis is \( H_1: \mu eq 60 \) kg.

05

P-value Area for Part (b)

In part (b), \( H_1: \mu < 60 \) kg implies a left-tailed test. So, the \( P \)-value area is on the left side of the mean.

06

P-value Area for Part (c)

For part (c), with \( H_1: \mu > 60 \) kg implying a right-tailed test, the \( P \)-value area is on the right side of the mean.

07

P-value Area for Part (d)

In part (d), \( H_1: \mu eq 60 \) kg implies a two-tailed test. Thus, the \( P \)-value area is on both sides of the mean.

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Key Concepts

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

Null Hypothesis

The null hypothesis, commonly denoted as \( H_0 \), is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference, acting as a default or starting assumption. In the realm of our exercise, the null hypothesis is used to challenge the claim that a typical 3-month-old colt weighs \( \mu = 60 \) kg.
This means that before any testing begins, we assume our null hypothesis to be true:
\[ H_0: \mu = 60 \text{ kg} \]

• The null hypothesis is always formulated to include the "equals" sign, meaning it checks for equality — either "equal to," "greater than or equal to," or "less than or equal to."
• This assumption will guide the statistical testing process, where evidence against \( H_0 \) is sought rather than evidence for \( H_0 \).
• Rejecting the null hypothesis occurs when there is significant statistical evidence suggesting that the hypothesized weight does not hold true.

The null hypothesis sets the stage for possible refutation based on collected data.

Alternate Hypothesis

The alternate hypothesis, denoted \( H_1 \) or \( H_a \), is central to evaluating statistical claims. It often represents what the researcher aims to demonstrate, contrasting with the normality assumed under the null hypothesis. Let's explore how it fits into our colt weight problem.
For any condition where we assert the average weight differs from the standard \( 60 \) kg, the alternate hypothesis is framed accordingly:

• For the claim "less than 60 kg," the alternate hypothesis is:
  \[ H_1: \mu < 60 \text{ kg} \]
• When claiming "greater than 60 kg," the alternate hypothesis becomes:
  \[ H_1: \mu > 60 \text{ kg} \]
• If asserting "different from 60 kg," it results in a two-tailed scenario:
  \[ H_1: \mu e 60 \text{ kg} \]

The formulation of \( H_1 \) directly dictates the direction of the test. It tells us whether to look for evidence in one specific tail of the distribution (left or right-tailed) or both (two-tailed). This hypothesis guides the decision-making process after analyzing the statistical outputs like the \( P \)-value.

P-value

The \( P \)-value is an integral metric in hypothesis testing, offering insight into the strength of the evidence against the null hypothesis. It measures the probability of observing results as extreme as the test results, assuming the null hypothesis is true.
Let's relate this to our scenarios:

• In the case where you test \( H_1: \mu < 60 \) kg, it's a left-tailed test. The \( P \)-value area lies entirely on the left side of the mean.
• For \( H_1: \mu > 60 \) kg, which is a right-tailed test, the \( P \)-value appears on the right side of the mean, indicating more weight than assumed.
• In a two-tailed test with \( H_1: \mu e 60 \) kg, the \( P \)-value extends on both sides of the distribution, capturing the possibility of weights being either significantly lighter or heavier.

A small \( P \)-value (typically \( < 0.05 \)) suggests strong evidence against \( H_0 \), prompting rejection of the null hypothesis. Conversely, a large \( P \)-value leads us to fail to reject \( H_0 \), indicating insufficient evidence to support the alternate hypothesis. Understanding the \( P \)-value's role helps demystify the conclusion drawn from the statistical analysis.