91Ó°ÊÓ

The body weight of a healthy 3 -month-old colt should be about \(\mu=60 \mathrm{~kg}\) (Source: The Merck Veterinary Mamual, 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

Determine the Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a statement of no effect or no difference and is usually taken to be the status quo. In this context, the null hypothesis makes the assumption that the average weight of a 3-month-old colt is exactly \( \mu = 60 \mathrm{~kg} \). Therefore, the null hypothesis for this situation is \( H_0: \mu = 60 \mathrm{~kg} \).

02

Define the Alternate Hypothesis for Case (b)

The alternate hypothesis reflects the claim that is contrary to the null hypothesis. In part (b), we want to test the claim that the average weight of a wild Nevada colt is less than \( 60 \mathrm{~kg} \). Hence, the alternate hypothesis is \( H_1: \mu < 60 \mathrm{~kg} \).

03

Define the Alternate Hypothesis for Case (c)

In part (c), we're testing the claim that the average weight of such a wild colt is greater than \( 60 \mathrm{~kg} \). The alternate hypothesis for this claim is \( H_1: \mu > 60 \mathrm{~kg} \).

04

Define the Alternate Hypothesis for Case (d)

In part (d), the claim is that the average weight of such a wild colt is different from \( 60 \mathrm{~kg} \). Therefore, the alternate hypothesis should be \( H_1: \mu eq 60 \mathrm{~kg} \).

05

Determine the Position of the P-value Area for Part (b)

For the test in part (b), where the alternate hypothesis is \( H_1: \mu < 60 \mathrm{~kg} \), the test is a left-tailed test. Therefore, the area corresponding to the \( P \)-value is on the left side of the mean.

06

Determine the Position of the P-value Area for Part (c)

In part (c), where we test \( H_1: \mu > 60 \mathrm{~kg} \), the alternate hypothesis indicates a right-tailed test. Hence, the area corresponding to the \( P \)-value is on the right side of the mean.

07

Determine the Position of the P-value Area for Part (d)

For part (d), where the test assesses \( H_1: \mu eq 60 \mathrm{~kg} \), this is a two-tailed test. Therefore, the area for the \( P \)-value is on both sides of the mean, encompassing both tails of the distribution.

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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, noted as \( H_0 \), is a foundational concept in hypothesis testing. It represents a statement of no effect, no difference, or the status quo. Researchers use it as a starting assumption for their tests. In simpler terms, the null hypothesis suggests that any observed effect in the data is due to random chance.In our exercise, the null hypothesis proposes that the average weight of a healthy 3-month-old colt is exactly \( \mu = 60 \mathrm{~kg} \). This claim comes from a reputable source, and serves as a baseline for comparison. When conducting tests, scientists seek evidence that is strong enough to reject \( H_0 \). Only then would they consider an alternative scenario.

Alternate Hypothesis

The alternate hypothesis, symbolized as \( H_1 \), is what you might call the challenger. It suggests an effect, a difference, or a deviation from the status quo as posited by the null hypothesis. If evidence supports \( H_1 \), researchers can argue for a new understanding or conclusion.In scenarios where we suspect that the average weight of wild Nevada colts differs from 60 kg, several alternate hypotheses can be outlined:

• For a claim that the weight is less than 60 kg: \( H_1: \mu < 60 \mathrm{~kg} \).
• If one suspects it to be more: \( H_1: \mu > 60 \mathrm{~kg} \).
• For different weights without specifying direction: \( H_1: \mu eq 60 \mathrm{~kg} \).

The selection of \( H_1 \) impacts how we interpret our data and determine the significance paths.

P-value

The \( P \)-value is a pivotal component in the realm of hypothesis testing. It quantifies the probability of acquiring test results at least as extreme as observed, under the assumption that the null hypothesis is correct. Put simply, the \( P \)-value tells us how likely our data, or more extreme data, would occur if the null hypothesis were true.In hypothesis testing:

• A small \( P \)-value (typically \(< 0.05 \)) suggests the null hypothesis might not hold, giving credence to the alternate hypothesis.
• Conversely, a large \( P \)-value indicates insufficient evidence to cast doubt on the null hypothesis, and we 'fail to reject' it.

Positioning of the \( P \)-value area varies with the nature of the alternate hypothesis:- **Left-tailed test** (for \( H_1: \mu < 60 \)) aligns the \( P \)-value area to the left.- **Right-tailed test** (for \( H_1: \mu > 60 \)) aligns it to the right. - **Two-tailed test** (for \( H_1: \mu eq 60 \)) spreads the \( P \)-value across both extremes.These placements guide researchers in interpreting which deviations from the null hypothesis are statistically significant.