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

Define Null Hypothesis for Part (a)

The null hypothesis \( H_0 \) is a statement of no effect or no difference. It is what we assume to be true until we have evidence to suggest otherwise. In this case, we are challenging the claim that the average body weight of a 3-month-old colt is 60 kg. Therefore, the null hypothesis is: \( H_0: \mu = 60 \mathrm{~kg} \).

02

Define Alternate Hypothesis for Part (b)

For part (b), you want to test if the average weight of a wild Nevada colt is less than 60 kg. The alternate hypothesis \( H_1 \) is the statement you wish to test against the null. Thus, the alternate hypothesis for this test is: \( H_1: \mu < 60 \mathrm{~kg} \).

03

Define Alternate Hypothesis for Part (c)

In part (c), you are testing if the average weight of a wild colt is greater than 60 kg. Therefore, the alternate hypothesis for this scenario is: \( H_1: \mu > 60 \mathrm{~kg} \).

04

Define Alternate Hypothesis for Part (d)

For part (d), you are testing if the average weight of a wild colt is different from 60 kg. This is a two-tailed test, so the alternate hypothesis is: \( H_1: \mu eq 60 \mathrm{~kg} \).

05

Determine P-value Regions for Part (e)

- For part (b), \( H_1: \mu < 60 \mathrm{~kg} \) is a left-tailed test, so the P-value area is on the left of the mean.- For part (c), \( H_1: \mu > 60 \mathrm{~kg} \) is a right-tailed test, so the P-value area is on the right of the mean.- For part (d), \( H_1: \mu eq 60 \mathrm{~kg} \) is a two-tailed test, so the P-value areas are 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.

Understanding the Null Hypothesis

In statistical testing, the null hypothesis, denoted as \( H_0 \), represents a statement of no change or no difference. It acts as the starting assumption that there is no effect or relationship in the data we are examining. When you set up a statistical test, you presume the null hypothesis to be true until you have enough evidence to challenge it.
In the context of our exercise, we are evaluating the claim that the average body weight of a healthy 3-month-old colt is \( 60 \ ext{kg} \). Thus, our null hypothesis is:\[ H_0: \mu = 60\ \text{kg} \]
This means we assume the average body weight is exactly \( 60 \ \text{kg} \) unless our data provides strong evidence otherwise. If the data suggests the weight is different, then we challenge this null hypothesis.

What is an Alternate Hypothesis?

The alternate hypothesis, noted as \( H_1 \), is the counterpart to the null hypothesis. It's what you believe to be true as opposed to the status quo. In essence, the alternate hypothesis reflects that there is an effect or a difference. It is this hypothesis that you seek evidence for through testing.
Let's look at scenarios addressed in the exercise:

• For testing if the average weight is less than \( 60 \ \text{kg} \): \( H_1: \mu < 60 \ \text{kg} \)
• For testing if the average weight is greater than \( 60 \ \text{kg} \): \( H_1: \mu > 60 \ \text{kg} \)
• For testing if the weight is different from \( 60 \ \text{kg} \): \( H_1: \mu eq 60 \ \text{kg} \)

The aim is either to provide enough evidence to support any of these alternate hypotheses or to prove them false.
Remember, each alternate hypothesis represents a distinct challenge to the claim made by \( H_0 \).

Exploring the P-value

The P-value is a crucial part of hypothesis testing as it helps determine the strength of the results. It represents the probability of observing data as extreme as the sample data, under the assumption that the null hypothesis is true. A small P-value indicates strong evidence against the null hypothesis.
Using preset significance levels (e.g., 0.05, 0.01), you can make decisions:

• If the P-value is less than the significance level, we reject the null hypothesis.
• If the P-value is greater, we fail to reject the null hypothesis.

In our original exercise's context, the P-value helps decide whether there is sufficient evidence to conclude that the average colt weight differs from \( 60 \ \text{kg} \) by comparing it to hypothetical expectations based on \( H_0 \).
This statistic tells you just how "unusual" your data is, given that the null hypothesis is true.

Understanding Two-tailed Tests

A two-tailed test determines if there is a statistically significant difference in either direction from a specific value. This means you're interested in assessing if the true parameter is either considerably lower or higher than your initial claim.
In hypothesis testing, a two-tailed test is applicable when your alternate hypothesis (\(H_1\)) suggests that the parameter is not equal to the hypothesized value. For example, in the exercise:

• The two-tailed test explores if \( H_1: \mu eq 60 \ \text{kg} \), which means any change from \( 60 \ \text{kg} \) is of interest.

In such tests, the P-value regions lie on both sides of the mean, doubling the critical area you consider for rejecting the null hypothesis.
This type of test is particularly useful when you believe deviations in both directions are equally important or likely.