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Chapter 9: Problem 12

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 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 rejection regions is appropriate: \(R_{1}=\\{\bar{x}: \bar{x} \geq 124.80\\}, R_{2}=\) \(\\{\bar{x}: \bar{x} \leq 115.20\\}, R_{3}=\\{\bar{x}:\) 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 \(\\{z: z \leq-2.88\\}\) ?

Short Answer

Expert verified
a) \( \mu < 120 \); b) \( R_2 \); c) \( \alpha \approx 0.0020 \), new region \( \{\bar{x} \leq 114.67\} \); d) \( \beta \approx 0.4522 \); e) \( \alpha = 0.0104, 0.0020 \).

Step by step solution

01

Define the Parameter of Interest and Hypotheses

The parameter of interest is the true average braking distance, denoted by \( \mu \), for the new braking system. The null hypothesis (\(H_0\)) is that the true average braking distance is \( \mu = 120 \) ft, which is the average distance for the current system. The alternative hypothesis (\(H_a\)) is \( \mu < 120 \) ft, indicating that the new design has a reduced average braking distance.
02

Identify the Appropriate Rejection Region

The rejection region should reflect the direction of the alternative hypothesis, \( \mu < 120 \) ft. Among the given regions, \( R_2 = \{ \bar{x} : \bar{x} \leq 115.20 \} \) is appropriate because it specifies a condition for observing a sample mean considerably less than 120, which suggests the average braking distance decreases.
03

Calculate the Significance Level for the Chosen Region

The significance level for region \( R_2 \) is determined by calculating the probability that \( \bar{X} \leq 115.20 \) when \( \mu = 120 \). Using the standard normal distribution: \[ Z = \frac{115.20 - 120}{10 / \sqrt{36}} = -2.88 \] The significance level \( \alpha \) is the probability that \( Z \leq -2.88 \), which is approximately 0.0020. To change \( \alpha \) to 0.001, find the corresponding \( \bar{x} \) such that \( Z = -3.08 \). Thus, the new rejection region will be \( \{ \bar{x} \leq 114.67 \} \).
04

Probability of Type II Error with \( \mu = 115 \)

The probability of not rejecting \( H_0 \) when \( \mu = 115 \) (Type II error, \( \beta \)) is found by calculating \( P(\bar{X} > 115.20 | \mu = 115) \). Using the standard normal distribution: \[ Z = \frac{115.20 - 115}{10/\sqrt{36}} = 0.12 \] Thus, \( \beta = P(Z > 0.12) \approx 0.4522 \).
05

Calculate Significance Level for Given Z-Regions

For the region \( \{z: z \leq -2.33\} \), \( \alpha \) is the probability that \( Z \leq -2.33 \), which is approximately 0.0104. For \( \{z: z \leq -2.88\} \), \( \alpha \) is the probability that \( Z \leq -2.88 \), which is approximately 0.0020.

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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 is the specific characteristic or measure we want to make a decision about. For our exercise, the parameter of interest is denoted by \( \mu \), representing the true average braking distance for the new car braking system design. We want to assess whether this new design results in a reduced average braking distance compared to the current system, which has a known average of 120 feet. By focusing on \( \mu \), we can evaluate if the new design improves braking performance as intended.
Null Hypothesis
The null hypothesis, often symbolized as \( H_0 \), serves as the initial assumption or claim that there is no effect or difference. In the context of the exercise, the null hypothesis is stated as \( \mu = 120 \) feet. This means that we start our investigation with the assumption that the new braking system does not lead to any change in the average braking distance compared to the current baseline set at 120 feet. It's the claim we aim to test against to see if any observed data provides compelling evidence to reject this initial assumption.
Alternative Hypothesis
When conducting hypothesis testing, the alternative hypothesis \( H_a \) is what researchers want to prove. In our exercise, the alternative hypothesis is \( \mu < 120 \), suggesting that the new design results in a shorter average braking distance. This hypothesis is directional, showing interest only in a decrease from the current average of 120 feet. Successfully rejecting the null hypothesis in favor of this alternative means that we have significant evidence to support the claim that the new design improves braking performance by reducing the average braking distance.
Rejection Region
The rejection region in statistical testing identifies the range of values for which the null hypothesis is rejected. For this exercise, we want to determine the correct area where we would reject \( H_0 \). Since the alternative hypothesis is \( \mu < 120 \), we are interested in significantly smaller sample means. Here, we choose the rejection region \( R_2 = \{ \bar{x} : \bar{x} \leq 115.20 \} \), which means if the sample's average braking distance is less than or equal to 115.20, then we will reject \( H_0 \). This threshold aligns with our interest in proving the new system offers a better, reduced average braking distance.
Significance Level
Significance level, denoted as \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true. It measures the risk of a Type I error. For the chosen rejection region \( R_2 \), the significance level is approximately 0.0020, calculated based on the likelihood that the observed sample mean could be as extreme as 115.20 when \( \mu = 120 \). A significance level of 0.0020 implies a very stringent test, allowing only a small 0.2% probability of wrongly rejecting \( H_0 \). If we need a significance level of 0.001, we'd adjust our rejection threshold to \( \bar{x} \leq 114.67 \), making it even more rigorous to ensure strong evidence before implementing the new design.

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Most popular questions from this chapter

A random sample of soil specimens was obtained, and the amount of organic matter \((\%)\) in the soil was determined for each specimen, resulting in the accompanying data (from "Engineering Properties of Soil," Soil Sci., 1998: 93-102). \(\begin{array}{llllllll}1.10 & 5.09 & 0.97 & 1.59 & 4.60 & 0.32 & 0.55 & 1.45 \\\ 0.14 & 4.47 & 1.20 & 3.50 & 5.02 & 4.67 & 5.22 & 2.69 \\ 3.98 & 3.17 & 3.03 & 2.21 & 0.69 & 4.47 & 3.31 & 1.17 \\ 0.76 & 1.17 & 1.57 & 2.62 & 1.66 & 2.05 & & \end{array}\) The values of the sample mean, sample standard deviation, and (estimated) standard error of the mean are \(2.481,1.616\), and \(.295\), respectively. Does this data suggest that the true average percentage of organic matter in such soil is something other than \(3 \%\) ? Carry out a test of the appropriate hypotheses at significance level 10 by first determining the \(P\)-value. Would your conclusion be different if \(\alpha=.05\) had been used? [Note: A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size.]

The error in a measurement is normally distributed with mean \(\mu\) and standard deviation 1 . Consider a random sample of \(n\) errors, and show that the likelihood ratio test for \(H_{0}: \mu=0\) versus \(H_{\mathrm{a}}: \mu \neq 0\) rejects the null hypothesis when either \(\bar{x} \geq c\) or \(\bar{x} \leq-c\). What is \(c\) for a test with \(\alpha=.05\) ? How does the test change if the standard deviation of an error is \(\sigma_{0}\) (known) and the relevant hypotheses are \(H_{0}: \mu=0\) versus \(H_{\mathrm{a}}: \mu \neq \mu_{0}\) ?

A new method for measuring phosphorus levels in soil is described in the article "A Rapid Method to Determine Total Phosphorus in Soils" (Soil Sci. Amer. \(J ., 1988: 1301-1304)\). Suppose a sample of 11 soil specimens, each with a true phosphorus content of \(548 \mathrm{mg} / \mathrm{kg}\), is analyzed using the new method. The resulting sample mean and standard deviation for phosphorus level are 587 and 10 , respectively. a. Is there evidence that the mean phosphorus level reported by the new method differs significantly from the true value of \(548 \mathrm{mg} / \mathrm{kg}\) ? Use \(\alpha=.05\). b. What assumptions must you make for the test in part (a) to be appropriate?

The error \(X\) in a measurement has a normal distribution with mean value 0 and variance \(\sigma^{2}\). Consider testing \(H_{0}: \sigma^{2}=2\) versus \(H_{\mathrm{a}}: \sigma^{2}=3\) based on a random sample \(X_{1}, \ldots, X_{n}\) of errors. a. Show that a most powerful test rejects \(H_{0}\) when \(\sum x_{i}^{2} \geq c\). b. For \(n=10\), find the value of \(c\) for the test in (a) that results in \(\alpha=.05\). c. Is the test of (a) UMP for \(H_{0}: \sigma^{2}=2\) versus \(H_{\mathrm{a}}: \sigma^{2}>2 ?\) Justify your assertion.

On the label, Pepperidge Farm bagels are said to weigh four ounces each ( \(113 \mathrm{~g}\) ). A random sample of six bagels resulted in the following weights (in grams): \(\begin{array}{llllll}117.6 & 109.5 & 111.6 & 109.2 & 119.1 & 110.8\end{array}\) a. Based on this sample, is there any reason to doubt that the population mean is at least \(113 \mathrm{~g}\) ? b. Assume that the population mean is actually \(110 \mathrm{~g}\) and that the distribution is normal with standard deviation \(4 \mathrm{~g}\). In a \(z\) test of \(H_{0}\) : \(\mu=113\) against \(H_{\mathrm{a}}: \mu<113\) with \(\alpha=.05\), find the probability of rejecting \(H_{0}\) with \(\operatorname{six}\) observations. c. Under the conditions of part (b) with \(\alpha=.05\), how many more observations would be needed in order for the power to be at least \(.95 ?\)
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