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 values of \(\bar{x}\) are more contradictory to \(H_{0}\) than \(117.2\), what is the \(P\)-value in this case, and what conclusion is appropriate if \(\alpha=.10\) ? c. What is the probability that the new design is not implemented when its true average braking distance is actually \(115 \mathrm{ft}\) and the test from part (b) is used?

Step by step solution

01

Define the Parameter and Hypotheses

The parameter of interest is the true average braking distance \( \mu \) for the new braking system. We state the null hypothesis \( H_0: \mu = 120 \) ft (the current average), and the alternative hypothesis \( H_a: \mu < 120 \) ft (indicating a reduction in average).

02

Calculate the Test Statistic

With the sample mean \( \bar{x} = 117.2 \), sample size \( n = 36 \), and population standard deviation \( \sigma = 10 \), we calculate the Z-score:\[ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{117.2 - 120}{10 / \sqrt{36}} = -1.68 \]

03

Determine the Critical Value and P-value

For a left-tailed test at \( \alpha = 0.10 \), we find the critical Z-value from the standard normal distribution corresponds to \( Z = -1.28 \). The P-value of \( Z = -1.68 \) can be found using a standard normal table or calculator, which gives approximately 0.0465.

04

Make a Decision on Hypotheses

Since \( P = 0.0465 < \alpha = 0.10 \), we reject \( H_0 \) in favor of \( H_a \). This suggests that the sample data provides sufficient evidence to indicate a reduction in the average braking distance.

05

Calculate Probability of Type II Error (\(\beta\))

For true \( \mu = 115 \) and using the critical value \( x_{c} = \mu_0 + Z \times \frac{\sigma}{\sqrt{n}} = 120 + (-1.28) \times \frac{10}{6} \approx 117.87 \), we calculate \( \beta \) as:\[ \beta= P(\bar{x} > 117.87 \mid \mu = 115) = P\left( Z > \frac{117.87 - 115}{10/\sqrt{36}}\right) = P(Z > 1.72) \approx 0.0427 \]

06

Interpret Results

The probability that the new design is not implemented when the true average braking distance is 115 ft is 0.0427, which is low. This means there's a small chance of committing a Type II error.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Type I Error

In hypothesis testing, a Type I error occurs when we reject the null hypothesis when in fact, it is true. It is like a false alarm. Imagine if we claimed the new braking system truly reduces the average braking distance, but in reality, it does not; that's a Type I error.
This type of error is measured by the significance level, denoted as \( \alpha \), which is the threshold for deciding whether to reject the null hypothesis. For instance, in the braking system scenario, we set \( \alpha \) to 0.10, which implies a 10% risk of incorrectly rejecting the current braking system's average distance. To reduce the likelihood of a Type I error, we can opt for a smaller \( \alpha \), but this may increase the chance of another type of mistake, known as a Type II error.

Type II Error

A Type II error happens when we fail to reject the null hypothesis when it is false. It’s essentially missing a real effect. Returning to the braking system example, a Type II error would be concluded if we say there's no improvement in braking distance with the new design when there truly is.
To quantify this, we use \( \beta \), which represents the probability of a Type II error. In our example, with a true average braking distance of 115 ft, the probability of making a Type II error was calculated to be approximately 0.0427. Thus, there's a 4.27% chance we might fail to implement the better braking system even though it provides real benefits. Balancing Type I and Type II errors is crucial in hypothesis testing to make reliable decisions.

P-value

The P-value is a crucial concept in hypothesis testing. It helps us determine the strength of the evidence against the null hypothesis. Essentially, the P-value tells us "how extreme" our sample results are, assuming the null hypothesis is true.
In the context of testing the car's braking distance, if we calculate a P-value of 0.0465, this indicates the probability of observing a sample mean as extreme as 117.2 ft or more, given that the true average is indeed 120 ft, is just 4.65%. When the P-value is less than our significance level \( \alpha = 0.10 \), it suggests sufficient evidence to reject the null hypothesis, implying the new design might actually reduce braking distances.

Z-score

The Z-score is a crucial metric for assessing how far our sample statistic is from the hypothesized population parameter, measured in standard deviations. It’s a way of standardizing our sample data to compare it against a normal distribution.
For our braking system example, the Z-score was calculated to be -1.68. This value tells us that the sample mean was 1.68 standard deviations below the hypothesized mean of 120 ft. Using this Z-score, we can determine the P-value, which helps make a decision on the null hypothesis.
The Z-score helps translate our findings into standard terms, allowing us to utilize standard normal distribution tables to understand whether our results are significantly different from the hypothesized average or not.