91Ó°ÊÓ

A highway construction zone has a posted speed limit of 40 miles per hour. Workers working at the site claim that the mean speed of vehicles passing through this construction zone is at least 50 miles per hour. A random sample of 36 vehicles passing through this zone produced a mean speed of 48 miles per hour. The population standard deviation is known to be 4 miles per hour. a. Do you think the sample information is consistent with the workers' claim? Use \(\alpha=.025\). b. What is the Type I error in this case? Explain. What is the probability of making this error? c. Will your conclusion of part a change if the probability of making a Type I error is zero? d. Find the \(p\) -value for the test of part a. What is your decision if \(\alpha=.025\) ?

Step by step solution

01

Formulating Hypotheses

The hypotheses for this problem would be set up as follows: Null hypothesis \( H_0 \): The mean speed is 50 miles/hour \( \mu = 50 \) Alternative hypothesis \( H_1 \): The mean speed is less than 50 miles/hour \( \mu < 50 \)

02

Calculate the Z-Score

We can calculate the z-score using the following formula: \( Z = \frac{\(\bar{x} - \mu\)}{ \frac{\sigma}{\sqrt{n}} } \) . Where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. Plugging our values we get: \( Z = \frac{48 - 50}{ \frac{4}{\sqrt{36}} } \) which simplifies to \( Z = - 3 \)

03

Find Critical Value and Make a Decision

Given that \( \alpha = .025 \), we can look up the critical Z-value for a one-tailed test in a standard Z-table which is approximately 1.96. Our calculated Z-score is -3 which falls in the region of rejecting the null hypothesis as it is less than -1.96.

04

Identify Type I Error and Its Probability

A Type I error happens when you reject the null hypothesis even though it is actually true. In this case, it would mean concluding that the mean speed is less than 50 mph when it is actually 50 mph. The probability of making a Type I error is \(\alpha\), which we have set to 0.025. This means there is a 2.5% chance of committing this error.

05

Answer Part C

If the probability of making a Type I error is zero, it means we are certain that our decision to reject the null hypothesis is correct. However, our conclusion won't change because it is based on the evidence from our sample data.

06

Calculate p-value

The p-value is the smallest level of significance at which we are able to reject the null hypothesis. For a given z-score, we can look up its probability in a Z-table. In this case, since the Z score is -3, the p-value (which signifies the probability of our data given the null hypothesis is true) is 0.00135. As 0.00135 is less than our significance level of 0.025, we can reject the null hypothesis.

07

Final Decision

If \( \alpha = .025 \), then our decision is to reject the null hypothesis. This is because our p-value of 0.00135 is less than our significance level of 0.025.

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Key Concepts

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

Type I Error

When conducting hypothesis testing, a Type I error occurs when the null hypothesis is rejected, even though it is actually true. To put it in simpler terms: you are making a mistake by thinking a change or effect exists when it doesn't. In the context of the exercise,

• A Type I error would mean concluding that the mean speed of vehicles is less than 50 mph, while it actually remains at 50 mph.
• The probability of making a Type I error is represented by the significance level, denoted by \( \alpha \).

For this problem, the significance level \( \alpha \) is 0.025, which means there is a 2.5% chance of rejecting the true null hypothesis. Thus, Type I errors are important as they indicate the likelihood of making false conclusions based on sample data.

Z-Score Calculation

The Z-score is a measure that describes how far away a data point is from the mean, in terms of standard deviations. It helps us understand where the sample mean lies in relation to the expected population mean. In hypothesis testing, it is crucial for determining whether to reject or not reject the null hypothesis. To compute the Z-score, we use the formula:

\[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Here:

• \( \bar{x} \) is the sample mean (48 mph in the exercise).
• \( \mu \) is the population mean according to the null hypothesis (50 mph).
• \( \sigma \) is the population standard deviation (4 mph).
• \( n \) is the sample size (36 vehicles).

Using these values, the Z-score is calculated as -3, indicating that the sample mean is significantly lower than the hypothesized population mean. A negative Z-score suggests that the observed mean is less than the expected mean, and its magnitude shows how extreme this observation is.

Significance Level

The significance level, denoted by \( \alpha \), represents the threshold used to determine when to reject the null hypothesis. It is a pre-set probability level that defines how willing one is to make a Type I error.

• In practice, common values for \( \alpha \) are 0.01, 0.05, and 0.10, but for this exercise, we use \( \alpha = 0.025 \).
• This signifies that if the probability of observing our sample data is less than 2.5% (assuming the null hypothesis is true), we will reject the null hypothesis.

Choosing a smaller \( \alpha \) makes tests more stringent, reducing the chance of making a Type I error, but possibly increasing the chance of a Type II error (failing to detect a real effect). In our exercise, with a significance level of 0.025, the decision rule becomes more conservative, only rejecting the null hypothesis if the test statistic provides strong evidence against it.

p-value

The p-value is a critical component in hypothesis testing as it provides the probability of observing the sample data, or something more extreme, assuming that the null hypothesis is true. It helps us assess the strength of the evidence against the null hypothesis.

• If the p-value is less than or equal to the significance level \( \alpha \), we reject the null hypothesis.
• In the exercise, the calculated p-value is 0.00135.
• Since 0.00135 is less than 0.025, this indicates strong evidence against the null hypothesis, leading us to reject it.

This means that the probability of observing the sample mean of 48 mph, or something more extreme, under the assumption that the true mean speed is 50 mph, is very low. A small p-value strengthens the case for rejecting the null hypothesis, suggesting that the speed limit might indeed be exceeded.