91Ó°ÊÓ

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean number of hours spent working per week by college students who hold jobs is different from 20 hours b. To test whether or not a bank's ATM is out of service for an average of more than 10 hours per month c. To test if the mean length of experience of airport security guards is different from 3 years d. To test if the mean credit card debt of college seniors is less than \(\$ 1000\) e. To test if the mean time a customer has to wait on the phone to speak to a representative of a mailorder company about unsatisfactory service is more than 12 minutes

Step by step solution

01

Example a

Null hypothesis (H0): The mean number of hours spent working per week by college students who hold jobs is equal to 20 hours. Alternative hypothesis (H1): The mean number of hours spent working per week by college students who hold jobs is not equal to 20 hours. This is a case of a two-tailed test.

02

Example b

Null hypothesis (H0): The bank's ATM is out of service for an average of 10 hours per month. Alternative hypothesis (H1): The bank's ATM is out of service for more than an average of 10 hours per month. This is a case of a right-tailed test.

03

Example c

Null hypothesis (H0): The mean length of experience of airport security guards is equal to 3 years. Alternative hypothesis (H1): The mean length of experience of airport security guards is not equal to 3 years. This is a case of a two-tailed test.

04

Example d

Null hypothesis (H0): The mean credit card debt of college seniors is equal to $1000. Alternative hypothesis (H1): The mean credit card debt of college seniors is less than $1000. This is a case of a left-tailed test.

05

Example e

Null hypothesis (H0): The mean time a customer has to wait on the phone to speak to a representative of a mailorder company about unsatisfactory service is equal to 12 minutes. Alternative hypothesis (H1): The mean time a customer has to wait on the phone to speak to a representative of a mailorder company about unsatisfactory service is more than 12 minutes. This is a case of a right-tailed test.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), is a statement that assumes no effect or no difference in the population parameter being studied. It expresses the default or original state of nature which we presume true until proven otherwise. For example, if we wish to test if the average number of hours college students work per week is different from 20 hours, the null hypothesis would be that the mean number of hours is indeed 20. The null hypothesis acts as a starting point for statistical testing, and is generally formulated to include an equality sign, such as \( = \), \( \leq \), or \( \geq \). This serves as the backbone, providing a baseline for comparison against the alternative hypothesis.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis, denoted as \( H_1 \) or \( H_a \), presents a statement that indicates the presence of an effect or difference. It challenges the status quo suggested by the null hypothesis. For instance, continuing with the example of students' working hours, the alternative hypothesis would state that the mean number of hours is not equal to 20. The alternative hypothesis is what researchers typically aim to support with data. It encompasses potential deviations from the null, usually expressed with symbols such as \( eq \), \( < \), or \( > \). This hypothesis is critical, as it guides the direction of the statistical test.

Two-Tailed Test

A two-tailed test analyses whether a sample parameter is either significantly higher or lower than the hypothesized population parameter. In this type of test, deviations in both directions from the null hypothesis are considered. For example, to determine if the mean number of working hours per week for college students is different from 20, a two-tailed test is used because the concern is on both potential increase or decrease from 20 hours. This kind of test fits scenarios where the exact direction of the effect is unknown or irrelevant. It ensures that any substantial deviation, either positive or negative, can be detected.

Left-Tailed Test

In hypothesis testing, a left-tailed test is employed when the alternative hypothesis suggests that the parameter of interest is less than the value stated in the null hypothesis. Essentially, it explores if the actual parameter is significantly smaller. For example, if we want to test if the mean credit card debt of college seniors is less than \( \$1000 \), the alternative hypothesis will express this condition, and a left-tailed test will be appropriate. This focuses solely on detecting decreases from the hypothesized value, making it a targeted approach when a reduction is expected or needs verification.

Right-Tailed Test

A right-tailed test is applicable when the alternative hypothesis proposes that the parameter of interest is greater than the value mentioned in the null hypothesis. In this type of test, we aim to establish whether the sample parameter is significantly higher than the presumed value. Consider the case where we want to test if the average wait time for a customer is more than 12 minutes. The alternative hypothesis indicates this interest, leading to a right-tailed test. Such a test is efficient when primarily seeking to identify increases or upward trends from the baseline hypothesis. This focused inquiry helps validate claims of growth or elevation.