91Ó°ÊÓ

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at \(40 \mathrm{mph}\) under specified conditions is known to be \(120 \mathrm{ft}\). It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design. a. Define the parameter of interest and state the relevant hypotheses. b. Suppose braking distance for the new system is normally distributed with \(\sigma=10\). Let \(\bar{X}\) denote the sample average braking distance for a random sample of 36 observations. Which of the following three rejection regions is appropriate: \(R_{1}=\\{\bar{x}: \bar{x} \geq 124.80\\}, \quad R_{2}=\\{\bar{x}: \bar{x} \leq 115.20\\}\), \(R_{3}=\\{\bar{x}\) : either \(\bar{x} \geq 125.13\) or \(\bar{x} \leq 114.87\\}\) ? c. What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with \(\alpha=.001\) ? d. What is the probability that the new design is not implemented when its true average braking distance is actually \(115 \mathrm{ft}\) and the appropriate region from part (b) is used? e. Let \(Z=(\bar{X}-120) /(\sigma / \sqrt{n})\). What is the significance level for the rejection region \(\\{z: z \leq-2.33\\}\) ? For the region \(\\{z: z \leq-2.88\\}\) ?

Step by step solution

01

Define the Parameter and State Hypotheses

The parameter of interest here is the true average braking distance for the new system, denoted as \( \mu \). The relevant hypotheses are:- Null Hypothesis (\( H_0 \)): \( \mu = 120 \) feet (The new design does not reduce the average braking distance)- Alternative Hypothesis (\( H_a \)): \( \mu < 120 \) feet (The new design reduces the average braking distance)

02

Identify the Appropriate Rejection Region

Given that we are testing \( H_a: \mu < 120 \), we use a left-tailed test. Therefore, the rejection region should contain smaller values. Among the provided options, \( R_2 = \{ \bar{x} : \bar{x} \leq 115.20 \} \) is the appropriate rejection region for the left-tailed test regarding a decrease in braking distance.

03

Determine Significance Level of the Rejection Region

To find the significance level of the chosen rejection region \( R_2 \), we need the critical value for \( \bar{x} \leq 115.20 \). We calculate:\[ z = \frac{115.20 - 120}{\frac{10}{\sqrt{36}}} = \frac{115.20 - 120}{1.67} \approx -2.88 \]The significance level, \( \alpha \), corresponds to \( P(Z \leq -2.88) \). Consulting the standard normal distribution table, we find \( P(Z \leq -2.88) = 0.002 \).

04

Alter Rejection Region for \(\alpha = 0.001\)

To achieve \( \alpha = 0.001 \), find the new critical value \( z_{\alpha} \) using the standard normal distribution. For \( \alpha = 0.001 \), \( z_{\alpha} = -3.09 \). Then solve for \( x_{critical} \):\[ -3.09 = \frac{\bar{x} - 120}{1.67} \Rightarrow \bar{x} = 120 - 3.09(1.67) \approx 114.83 \]The modified rejection region would be \( \{ \bar{x} : \bar{x} \leq 114.83 \} \).

05

Calculate Probability of Type II Error

If the true mean is 115, we find the probability of not rejecting \( H_0 \) when \( \bar{x} > 115.20 \). Calculate:\[ P(\bar{X} > 115.20 | \mu = 115) = P\left(Z > \frac{115.20 - 115}{1.67}\right) = P(Z > 0.12) \approx 0.4522 \] (using normal distribution table).

06

Significance Level for Given Rejection Region \(\{z: z\leq -2.33\}\)

For \( z \leq -2.33 \), \( \alpha = P(Z \leq -2.33) \approx 0.0099 \) using normal tables. For \( z \leq -2.88 \) previously calculated, \( \alpha = 0.002 \).

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Key Concepts

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

Parameter of Interest

In hypothesis testing, the "parameter of interest" refers to the specific characteristic we are examining in a population. For the problem regarding the braking system, this parameter is the true average braking distance for the new design of the braking system. It is commonly denoted as \( \mu \). The focus is on assessing whether this new design results in a shorter average braking distance compared to the current known average of 120 feet. In this scenario, our hypotheses are structured as follows:

• The **Null Hypothesis** (\( H_0 \)): \( \mu = 120 \) feet, meaning there is no change in the average braking distance.
• The **Alternative Hypothesis** (\( H_a \)): \( \mu < 120 \) feet, indicating that the new design decreases the braking distance.

This setup allows us to perform a statistical test to determine if the parameter of interest for the new braking system is indeed smaller than 120 feet.

Rejection Region

The rejection region in hypothesis testing helps decide whether to reject the null hypothesis. It is the range of values for which we would not accept the null hypothesis. For this situation, we're conducting a left-tailed test since we're looking for evidence that the new braking system reduces the average stopping distance. Therefore, the rejection region consists of smaller average braking distances than the current one, 120 feet.Upon evaluating the provided rejection regions, moving forward with:

• **R₂**: \( \{ \bar{x}: \bar{x} \leq 115.20 \} \)

This choice is based on the idea that if the sample mean (\( \bar{X} \)) is less than or equal to 115.20, it provides strong enough evidence against the null hypothesis, supporting the alternative hypothesis. Thus, this region reflects outcomes that are extremely unlikely under the null hypothesis, allowing us to reject \( H_0 \) when they occur.

Significance Level

The significance level, denoted by \( \alpha \), indicates the probability of rejecting the null hypothesis when it is true. It essentially measures the risk of making a Type I error, which is the incorrect rejection of a true null hypothesis.For the rejection region \( R_2 = \{ \bar{x} : \bar{x} \leq 115.20 \} \), assume:

• \( \alpha = 0.002 \) as calculated from the standard normal distribution table for the critical value \( z = -2.88 \).

Should you need to adjust this test to achieve a significance level of \( \alpha = 0.001 \), you will need to change the rejection region. This involves finding a new critical value where it is even less likely (only 0.1% of the time) to reject the null. The new critical value, using the standard normal distribution for \( \alpha = 0.001 \), would be \( z = -3.09 \). Solving for \( x_{critical} \) gives us:

• \( \{ \bar{x} : \bar{x} \leq 114.83 \} \)

This revised rejection region reinforces stricter testing conditions, reducing the chance of a Type I error.

Type II Error

A Type II error occurs when the null hypothesis is incorrectly accepted, although the alternative hypothesis is true. This is represented by the probability \( \beta \), which is the risk of failing to reject the null hypothesis when the actual mean differs from 120 feet.In this context, we need to consider what happens if the true average braking distance for the new system is indeed 115 feet. For this scenario:

• \( P(\bar{X} > 115.20 | \mu = 115) = P(Z > 0.12) \)

Using the standard normal table, this probability is found to be approximately 0.4522. This value represents the likelihood that we fail to reject the null hypothesis, thereby making a Type II error, despite a true reduction in braking distance existing. By understanding both the significance level \( \alpha \) and the probability of a Type II error \( \beta \), one can gain a comprehensive grasp of the test's reliability in detecting true differences and manage the potential risks of error effectively.