Interpreting a Decision In Exercises 43–48, determine whether the claim represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that
         
b. fails to reject the null hypothesis?


Marketing A fitness equipment company claims that its competitor’s home gym does not have a customer satisfaction rate of 99%.

Verified step by step guidance

1. Step 1: Understand the problem. The fitness equipment company is making a claim about the customer satisfaction rate of its competitor's home gym. Specifically, the claim is that the satisfaction rate is not 99%. This is a two-tailed hypothesis test because the claim involves 'not equal to' (≠).
2. Step 2: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents the default assumption, which is that the satisfaction rate is equal to 99%. Mathematically, H₀: p = 0.99. The alternative hypothesis represents the claim being tested, which is that the satisfaction rate is not 99%. Mathematically, H₁: p ≠ 0.99.
3. Step 3: Interpret the decision to 'fail to reject the null hypothesis.' Failing to reject the null hypothesis means that there is not enough statistical evidence to support the alternative hypothesis. In this context, it means there is insufficient evidence to conclude that the competitor's home gym satisfaction rate is different from 99%.
4. Step 4: Understand the implications of failing to reject H₀. This does not prove that the null hypothesis is true; it simply means that the data does not provide strong enough evidence to reject it. The satisfaction rate could still be 99%, or it could be different, but the test did not detect a significant difference.
5. Step 5: Consider the context of the test. The result of failing to reject H₀ might be due to factors such as a small sample size, low statistical power, or variability in the data. These factors should be considered when interpreting the outcome of the hypothesis test.

Key Concepts

Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference, serving as a default position in hypothesis testing. In this context, it would assert that the competitor's home gym does indeed have a customer satisfaction rate of 99%. The null hypothesis is tested against an alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis

The alternative hypothesis (H1) represents a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, the alternative hypothesis would claim that the competitor's home gym does not have a customer satisfaction rate of 99%. This hypothesis is what the fitness equipment company is trying to support with evidence.

Failing to Reject the Null Hypothesis

Failing to reject the null hypothesis means that the evidence collected from the sample data is not strong enough to support the alternative hypothesis. In practical terms, this suggests that there is insufficient evidence to conclude that the competitor's home gym has a customer satisfaction rate different from 99%. It does not prove that the null hypothesis is true, only that there is not enough evidence to dispute it.