A consumer research group is interested in testing an...

A consumer research group is interested in testing an automobile manufacturer's claim that a new economy model will travel at least 25 miles per gallon of gasoline $\left(H_0 ; \mu \geq 25\right)$. a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based on the value of $\bar{x}$ for the test to determine whether the manufacturer's claim should be rejected? Assume that $\sigma$ is 3 miles per gallon. b. What is the probability of committing a Type II error if the actual mileage is 23 miles per gallon? c. What is the probability of committing a Type II error if the actual mileage is 24 miles per gallon? d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon?

A consumer research group is interested in testing an automobile manufacturer's claim that a new economy model will travel at least 25 miles per gallon of gasoline $\left(H_0 ; \mu \geq 25\right)$.
a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based on the value of $\bar{x}$ for the test to determine whether the manufacturer's claim should be rejected? Assume that $\sigma$ is 3 miles per gallon.
b. What is the probability of committing a Type II error if the actual mileage is 23 miles per gallon?
c. What is the probability of committing a Type II error if the actual mileage is 24 miles per gallon?
d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon?

Key Concepts

Type I Error

A Type I error occurs when the null hypothesis is rejected even though it is true. The probability of committing a Type I error is defined by the significance level, ?. This error reflects finding a false positive result in the context of hypothesis testing.

Type II Error

A Type II error happens when the null hypothesis is not rejected even though it is false. The probability of a Type II error (commonly denoted by ?) is influenced by factors such as the sample size, the actual effect size, and the chosen significance level. It reflects the test's failure to detect a true effect when one exists.

Rejection Region (Decision Rule)

The rejection region is the range of values for which the null hypothesis will be rejected. It is determined by the significance level and the sampling distribution under the null hypothesis. If the computed test statistic or the related parameter (such as the sample mean) falls within this region, the null hypothesis is rejected in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) represents a baseline or claim that is assumed to be true, while the alternative hypothesis (H?) represents the contradicting claim that researchers want to test. Correct formulation of these hypotheses is critical because it sets the stage for the type of error that might be committed (Type I or Type II) and influences the overall test design.

Significance Level

The significance level, often denoted by ?, is the probability threshold for rejecting the null hypothesis when it is actually true. It defines the criterion for the rejection region; a result is considered statistically significant if its probability under the null hypothesis is less than ?. This parameter is crucial for controlling the rate of Type I error.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis (what the researcher wants to test), and then using sample data to determine whether to reject the null hypothesis in favor of the alternative.

Z-Test Statistic

The Z-test is a hypothesis test used when the population standard deviation is known and the sampling distribution of the sample mean is normal (or approximately normal). It involves computing a test statistic by subtracting the hypothesized population mean from the sample mean and dividing by the standard error, which helps determine how many standard deviations the sample mean is from the hypothesized mean.

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