Question

The mean length of long-distance telephone calls placed with a particular phone company was known to be \(7.3\) min under an old rate structure. In an attempt to be more competitive with other long-distance carriers, the phone company lowered long-distance rates, thinking that its customers would be encouraged to make longer calls and thus that there would not be a big loss in revenue. Let \(\mu\) denote the true mean length of long-distance calls after the rate reduction. What hypotheses should the phone company test to determine whether the mean length of long-distance calls increased with the lower rates?

Short answer

The phone company should test the null hypothesis \(H_0: \mu = 7.3\) and the alternative hypothesis \(H_1: \mu > 7.3\).

Step-by-step solution

Defining Null hypothesis

The null hypothesis would state that the mean length of calls (\(\mu\)) under the new rate structure is the same as under the old rate structure i.e., at \(7.3\) minutes. This can be mathematically expressed as \(H_0: \mu = 7.3\) minutes.

Formulating Alternative hypothesis

The alternative hypothesis is what the company wants to test, i.e., whether the mean length of calls has increased due to the reduced rate. So, it can be mathematically represented as \(H_1: \mu > 7.3\) minutes.

Explanation of Hypotheses

So, in this context, the null hypothesis \(H_0: \mu = 7.3\) states that the rate decrease has not affected the mean length of the calls, whereas the alternative hypothesis \(H_1: \mu > 7.3\) suggests that the mean length of calls increased with the lower rates.

Key concepts

Null Hypothesis

Understanding the null hypothesis is a fundamental aspect of hypothesis testing in statistics. It is the default assumption that there is no effect or no difference between two measured phenomena. For instance, if a telephone company is analyzing the impact of a change in their rate structure on the duration of calls, the null hypothesis would assert that this change has not altered the average call length.

In formal terms, the null hypothesis is declared as a specific value that the parameter of interest (such as the mean length of long-distance telephone calls, denoted as \(\mu\)) is equal to. If the known average call length under the old rate structure was 7.3 minutes, the null hypothesis (\(H_0\)) would be: \[H_0: \mu = 7.3 \text{ minutes}.\] The essence of the null hypothesis lies in its neutrality and the assumption of 'no change' or 'no effect,' which serves as a baseline for statistical testing.

Alternative Hypothesis

Contrasts against the null hypothesis, the alternative hypothesis (\(H_1\)) represents a researcher's conjecture that there is indeed an effect, or a difference that exists. In regards to the mean length of telephone calls after a rate reduction, the alternative hypothesis would state that the mean duration of calls has increased—that is, it is greater than the mean under the old rate structure.

Mathematically, this hypothesis could be formulated as \[H_1: \mu > 7.3 \text{ minutes}.\] The creation of an alternative hypothesis is a critical step as it embodies the idea you're testing and is often the hypothesis that the researcher wants to prove. In a scenario where a phone company has reduced rates with the expectation that call durations will extend, demonstrating an increase in the average length of calls will support their marketing strategy.

Mean Length of Calls

The mean length of calls, denoted mathematically by the symbol \(\mu\), is a measure of central tendency that calculates the average duration of telephone calls made within a certain period. This statistic is highly valuable to telephone companies in understanding customer behavior and optimizing rate plans.

When a company changes its rate structure, investigating the mean length of calls before and after the change is central to determining the impact of their business decision. In the case of a decrease in rates, the company hypothesizes that this will lead to customers engaging in longer calls, potentially compensating for the reduced rate by an increase in usage. The mean length of calls serves as a critical metric in conducting a hypothesis test to validate or refute this assumption.

Rate Structure Analysis

Rate structure analysis involves examining how changes in billing for services affect customer usage patterns. In the context of a telephone company, this could imply altering per-minute charges, implementing flat fees, or introducing discounted rates during off-peak hours. The analysis aims to determine if such changes are economically beneficial or detrimental to the company.

By conducting a rate structure analysis, a company assesses the business impact of changing its pricing model. For our particular case, the company aims to find out if lowering long-distance rates leads to a significant change in the mean length of calls. This type of analysis integrates statistical hypothesis testing to evaluate if the observed differences in call lengths are likely due to chance or if they are statistically significant and, hence, possibly attributed to the rate reduction.