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Chapter 8: Problem 43

A plan for an executive travelers' club has been developed by an airline on the premise that \(5 \%\) of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level \(.01\) the null hypothesis that the company's premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the company's premise will be judged correct when in fact \(10 \%\) of all current customers qualify?

Short Answer

Expert verified
Reject the null hypothesis; probability of Type II error is very low.

Step by step solution

01

Understand the Problem

The problem gives us information about testing a hypothesis, based on a company's premise about customer qualification for a club membership. We need to test whether the sample result of 40 out of 500 customers supports this premise. The company's premise states that 5% would qualify, but the sample suggests a different percentage.
02

Define Hypotheses

We will define the null hypothesis as: \[ H_0: p = 0.05 \] where \( p \) is the proportion of customers who would qualify according to the company's premise. The alternative hypothesis is: \[ H_a: p eq 0.05 \] We want to test this at a 0.01 significance level.
03

Calculate Test Statistic

Our test statistic will be calculated using the formula for a proportion:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]where \( \hat{p} = \frac{40}{500} = 0.08 \) is the sample proportion, \( p_0 = 0.05 \) is the claimed proportion, and \( n = 500 \). We calculate\[ z = \frac{0.08 - 0.05}{\sqrt{\frac{0.05 \times 0.95}{500}}} \approx 3.4641 \].
04

Determine Critical Value

Using the standard normal distribution table, we check the critical value for a 0.01 level of significance for a two-sided test. This critical value is approximately 2.576. Since we're doing a two-tailed test, our rejection regions are any \( z \) less than -2.576 or greater than 2.576.
05

Decide and Conclude

The calculated test statistic \( z \approx 3.4641 \) is greater than the critical value 2.576, indicating that we are in the rejection region. Thus, we reject the null hypothesis at the 0.01 significance level.
06

Probability of Type II Error

The probability of judging the company's premise correct when in fact 10% of customers qualify (Type II Error) is calculated by determining the \( z \) for 10%. The new \( \hat{p} \) is 0.1. Using the previous formula,\[ z = \frac{0.1 - 0.05}{\sqrt{\frac{0.05 \times 0.95}{500}}} \approx 7.071 \].This \( z \) falls well outside the critical region, indicating very low probability that the null hypothesis holds when 10% qualify.

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Key Concepts

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

Null Hypothesis
Understanding the null hypothesis is crucial in hypothesis testing. In this context, the null hypothesis (\( H_0 \) ) posits that the real proportion of customers qualifying for the executive travelers' club is exactly 5%.
This serves as the default or starting assumption.
  • In this problem, the null hypothesis is mathematically expressed as \( H_0: p = 0.05 \), where \( p \) is the proportion of qualifying customers.
  • The null hypothesis usually aims to assert that there is no significant difference from the assumed premise or situation.
It acts as a standard against which the actual sample results are compared. If evidence is insufficient or inconclusive, the null hypothesis may not be rejected.
Alternative Hypothesis
The alternative hypothesis represents what we are trying to detect or measure as different from the null hypothesis.
It challenges the null hypothesis by suggesting that there is a significant difference.
In this exercise:
  • The alternative hypothesis (\( H_a \)) is formulated as \( H_a: p eq 0.05 \), implying that the true proportion of qualifying customers is not equal to 5%.
  • The test is considered two-tailed because the alternative hypothesis is not directional (it does not specify whether the proportion is less than or greater than 5%, just that it is different).
When analyzing data, if the evidence suggests that the real-world distribution deviates significantly from what the null hypothesis claims, we consider the alternative hypothesis.
Type II Error
A Type II Error occurs when the test fails to reject the null hypothesis, despite it being incorrect.
Simply put, it means failing to detect a true effect or difference when one truly exists.
In our problem:
  • A Type II Error would occur if we concluded that the company's premise of 5% qualifying stands, when actually 10% of customers qualify.
  • It鈥檚 important to understand that a Type II Error is not the same as accepting the null hypothesis; it's more like being unable to prove the null hypothesis wrong.
The probability of making a Type II Error can be influenced by various factors, including sample size and the chosen significance level.
Significance Level
The significance level, often denoted by \( \alpha \), is a threshold to decide whether a result is statistically significant or not.
It is a pre-determined probability of rejecting the null hypothesis when it is actually true, known as a Type I Error.
In this scenario:
  • The significance level used for the hypothesis test is 0.01, which indicates a 1% risk of concluding that a difference exists when there is none.
  • Choosing a low significance level, like 0.01, means that more substantial evidence is required to reject the null hypothesis, offering a more stringent and conservative test.
Ultimately, the significance level helps in managing the trade-off between Type I and Type II Errors, guiding decision-making in hypothesis testing.

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