91Ó°ÊÓ

Consumer Reports stated that the mean time for a Chrysler Concorde to go from 0 to 60 miles per hour is 8.7 seconds. (a) If you want to set up a statistical test to challenge the claim of 8.7 seconds, what would you use for the null hypothesis? (b) The town of Leadville, Colorado, has an elevation over 10,000 feet. Suppose you wanted to test the claim that the average time to accelerate from 0 to 60 miles per hour is longer in Leadville (because of less oxygen). What would you use for the alternate hypothesis? (c) Suppose you made an engine modification and you think the average time to accelerate from 0 to 60 miles per hour is reduced. What would you use for the alternate hypothesis? (d) For each of the tests in parts (b) and (c), would the \(P\) -value area 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

Understanding the Null Hypothesis for the Claim

The null hypothesis (H_0) is used to state that there is no effect or no difference from what is claimed. In this case, Consumer Reports claims the mean time for a Chrysler Concorde to accelerate from 0 to 60 mph is 8.7 seconds.Thus, the null hypothesis is:\[ H_0: \mu = 8.7 \text{ seconds} \]

02

Defining the Alternate Hypothesis for High Altitude Test

In statistical testing, the alternate hypothesis (H_1) is used to propose an alternative to the null hypothesis. For Leadville, where it is believed that the mean time is longer due to less oxygen, the alternate hypothesis is:\[ H_1: \mu > 8.7 \text{ seconds} \]

03

Defining the Alternate Hypothesis for Engine Modification Test

If you believe that the engine modification results in a faster acceleration, then the alternate hypothesis (H_1) implies the mean time is reduced, which suggests:\[ H_1: \mu < 8.7 \text{ seconds} \]

04

Determining the P-Value Area for Leadville Test

For the test in Leadville (Step 2), the alternate hypothesis is that the mean time is greater than 8.7 seconds. This suggests a one-tailed test where the P -value is the area to the right of 8.7 seconds on the distribution curve.

05

Determining the P-Value Area for Engine Modification Test

For the engine modification test (Step 3), the alternate hypothesis suggests the mean time is less than 8.7 seconds. Therefore, this is a one-tailed test where the P -value is the area to the left of 8.7 seconds on the distribution curve.

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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 is a fundamental part of hypothesis testing in statistics. It is symbolized as \( H_0 \) and represents a statement of no change or no effect.
In many situations, it reflects the status quo or a generally accepted fact.
For instance, when Consumer Reports claims that a Chrysler Concorde accelerates from 0 to 60 miles per hour in 8.7 seconds,
the null hypothesis would be that the mean acceleration time \( \mu \) is equal to 8.7 seconds:

• \( H_0: \mu = 8.7 \text{ seconds} \)

This hypothesis acts as a starting point for statistical testing and is what you aim to challenge or disprove with evidence.

Alternate Hypothesis

The alternate hypothesis is the counterclaim to the null hypothesis. It is expressed as \( H_1 \) and proposes that a deviation or a difference from the null hypothesis condition exists.
In the context of the Chrysler Concorde example, if the town of Leadville is considered, where people believe lower oxygen levels affect the car's performance negatively, the alternate hypothesis would suggest a longer acceleration time:

• \( H_1: \mu > 8.7 \text{ seconds} \)

On the other hand, if modifications are made to the engine to potentially enhance performance and results are anticipated to improve, the alternate hypothesis would propose a shorter acceleration time:

• \( H_1: \mu < 8.7 \text{ seconds} \)

The alternate hypothesis is what you seek evidence for, using data collected during research or experimentation.

P-Value

The \( P \)-Value is a crucial concept in hypothesis testing. It provides a measure of the strength of evidence against the null hypothesis.
Simply put, the \( P \)-value tells you how probable it is to observe the given data, or something more extreme, in the assumption that the null hypothesis is true.
A smaller \( P \)-value indicates stronger evidence against \( H_0 \).In hypothesis tests, if the \( P \)-value is less than the significance level (usually 0.05), the null hypothesis is rejected in favor of the alternate hypothesis.

• For the Leadville test, the \( P \)-value is the area to the right of 8.7 seconds on the distribution curve, since we test if the mean time is greater.
• For the engine modification test, the \( P \)-value is the area to the left of 8.7 seconds, as we test if the mean time is shorter.

Thus, the \( P \)-value helps determine whether results are statistically significant or not.

One-Tailed Test

A one-tailed test in hypothesis testing investigates the possibility of the deviation going in one specific direction, either greater than or less than a certain point.
This is different from a two-tailed test, which checks for deviations in both directions.In the Leadville example, a one-tailed test is performed because the concern is that the acceleration time will be more than 8.7 seconds due to high altitude effects:

• Right-tailed test: Checking if \( \mu > 8.7 \text{ seconds} \).

For the engine modification scenario, another one-tailed test is used since improvements aim to reduce the time:

• Left-tailed test: Testing if \( \mu < 8.7 \text{ seconds} \).

One-tailed tests are powerful tools when the direction of the effect is known and confidence exists regarding its direction.