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Chapter 9: Problem 73

Every road has one at some point - construction zones that have much lower speed limits. To see if drivers obey these lower speed limits, a police officer uses a radar gun to measure the speed (in miles per hours, or mph) of a random sample of 10 drivers in a 25 mph construction zone. Here are the data: \(\begin{array}{llllllllll}27 & 33 & 32 & 21 & 30 & 30 & 29 & 25 & 27 & 34\end{array}\) (a) Is there convincing evidence that the average speed of drivers in this construction zone is greater than the posted speed limit? (b) Given your conclusion in part (a), which kind of mistake - a Type I error or a Type II error - could you have made? Explain what this mistake would mean in context.

Short Answer

Expert verified
(a) There is evidence that the average speed is greater than 25 mph. (b) A Type I error could occur, which means concluding speeds exceed 25 mph when they do not.

Step by step solution

01

Define the Hypotheses

The first step is to define the null and the alternative hypothesis. Here, the null hypothesis \( H_0 \) is that the average speed of drivers in the construction zone is equal to 25 mph. So, \( H_0: \mu = 25 \). The alternative hypothesis \( H_a \) is that the average speed of drivers is greater than 25 mph, so \( H_a: \mu > 25 \).
02

Calculate the Sample Mean

Calculate the mean of the given speeds. \[ \text{Mean} = \frac{27 + 33 + 32 + 21 + 30 + 30 + 29 + 25 + 27 + 34}{10} = 28.8 \text{ mph} \].
03

Calculate the Standard Deviation and Standard Error

First, calculate the standard deviation (\( s \)) of the sample. Then calculate the standard error (SE) using \( SE = \frac{s}{\sqrt{n}} \), where \( n = 10 \). Use the formula for standard deviation: \[ s = \sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}} \].For this dataset, \( s \approx 4.08 \). Thus, \[ SE = \frac{4.08}{\sqrt{10}} \approx 1.29 \].
04

Calculate the Test Statistic

Use the formula for the t-statistic: \[ t = \frac{\text{sample mean} - \text{hypothesized mean}}{SE} = \frac{28.8 - 25}{1.29} \approx 2.95 \].
05

Find the P-value

Refer to the t-distribution table with \( n-1 = 9 \) degrees of freedom for \( t = 2.95 \). This value yields a p-value less than 0.01.
06

Make a Decision

If the p-value is less than the significance level \( \alpha = 0.05 \), we reject the null hypothesis. Since the p-value is less than 0.01, we reject \( H_0 \) and conclude there is convincing evidence that the average speed is greater than 25 mph.
07

Evaluate Potential Errors

Since we rejected the null hypothesis, a Type I error could have been made. A Type I error occurs when we incorrectly reject a true null hypothesis. In context, this means that we concluded the driver's average speed is greater than 25 mph when it actually is not.

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Key Concepts

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

Type I and Type II errors
When conducting hypothesis testing, understanding errors is crucial. There are two primary types: Type I errors and Type II errors.
- **Type I Error**: This occurs when we incorrectly reject a true null hypothesis. In this specific context, a Type I error would mean concluding that the average speed in the construction zone is greater than 25 mph, when in fact, it is not. The consequences could involve unnecessary changes in enforcement or policies based on incorrect assumptions.
- **Type II Error**: This happens when we fail to reject a false null hypothesis. In the construction zone example, a Type II error would mean not concluding that the average speed is greater than 25 mph when it actually is. This could lead to inadequate safety measures being applied.
Understanding these errors helps in interpreting the results of statistical tests more effectively and deciding on appropriate actions.
Statistical Significance
Statistical significance helps us understand whether our results can be considered substantial or occurred by random chance. A result is statistically significant when the p-value is less than a pre-determined significance level, often denoted as \( \alpha \).
In this exercise, the significance level is set at 0.05. With a p-value less than 0.01, the result is statistically significant. This indicates strong evidence against the null hypothesis that the average speed is 25 mph. We can be confident that the observed average speed of 28.8 mph is likely greater than 25 mph and not due to random chance.
Statistical significance provides the backbone for scientific evidence, allowing researchers and analysts to make informed conclusions based on data.
t-test
A t-test is used to determine if there is a significant difference between the means of two groups, or a group's mean and a known value. Here, we're using a one-sample t-test to check if the sample mean differs significantly from the hypothesized population mean.
- **Test Statistic Calculation**: The t-statistic is calculated as \( t = \frac{\text{sample mean} - \text{hypothesized mean}}{SE} \). In our example, the t-statistic is approximately 2.95.
- **Degrees of Freedom**: It's important to consider degrees of freedom, which for a one-sample t-test is \( n-1 \), where \( n \) is the sample size. Here, it is 9.
The t-test helps determine whether the observed differences are statistically significant, considering sample variability and size.
Standard Deviation
Standard deviation quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates a wider spread.
To calculate standard deviation, you use the formula \( s = \sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}} \). For our speeds, \( s \approx 4.08 \).
- **Standard Error (SE)**: This is the standard deviation of the sample mean's distribution. It's calculated as \( SE = \frac{s}{\sqrt{n}} \). SE plays a crucial role in hypothesis testing and helps determine the precision of the sample mean estimate.
Understanding standard deviation enhances our comprehension of data variability and the reliability of statistical conclusions.

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Most popular questions from this chapter

Exercises 21 and 22 refer to the following setting. Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was \(\mu=6.7\) minutes. Emergency personnel arrived within 8 minutes on \(78 \%\) of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to "do better." At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. (a) State hypotheses for a significance test to determine whether the average response time has decreased. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

Multiple choice: Select the best answer for Exercises 95 to 102 . The reason we use \(t\) procedures instead of \(z\) procedures when carrying out a test about a population mean is that (a) \(z\) requires that the sample size be large. (b) \(z\) requires that you know the population standard deviation \(\sigma\). (c) \(z\) requires that the data come from a random sample or randomized experiment. (d) \(z\) requires that the population distribution be perfectly Normal. (e) \(z\) can only be used for proportions.

Your company markets a computerized device for detecting high blood pressure. The device measures an individual's blood pressure once per hour at a randomly selected time throughout a 12 -hour period. Then it calculates the mean systolic (top number) pressure for the sample of measurements. Based on the sample results, the device determines whether there is convincing evidence that the individual's actual mean systolic pressure is greater than \(130 .\) If so, it recommends that the person seek medical attention. (a) State appropriate null and alternative hypotheses in this setting. Be sure to define your parameter. (b) Describe a Type I and a Type II error, and explain the consequences of each.

You manufacture and sell a liquid product whose electrical conductivity is supposed to be \(5 .\) You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be \(5 .\) You will therefore test $$ \begin{array}{l} H_{0}: \mu=5 \\ H_{a}: \mu \neq 5 \end{array} $$ If the true conductivity is \(5.1,\) the liquid is not suitable for its intended use. You learn that the power of your test at the \(5 \%\) significance level against the alternative \(\mu=5.1\) is 0.23. (a) Explain in simple language what "power \(=0.23 "\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? (c) If you decide to use \(\alpha=0.10\) in place of \(\alpha=0.05\), with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(\mu=5.2\), with no other changes, will the power increase or decrease? Justify your answer.

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus. The appropriate hypotheses for the significance test are (a) \(H_{0}: \mu=18 ; H_{a}: \mu \neq 18\). (b) \(H_{0}: \mu=18 ; H_{a}: \mu>18\). (c) \(H_{0}: \mu<18 ; H_{a}: \mu=18\) (d) \(H_{0}: \mu=18 ; H_{a}: \mu<18\). (e) \(H_{0}: \bar{x}=18 ; H_{a}: \bar{x}<18\).
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