Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.

Verified step by step guidance

1. Step 1: Understand the null hypothesis (H0). In this case, H0 states that the population mean μ is less than or equal to 54 (H0: μ ≤ 54). The goal is to determine whether each confidence interval provides evidence to reject H0.
2. Step 2: Analyze confidence interval (a): The interval is 53.5 < μ < 56.5. Since this interval includes values greater than 54, it suggests that the population mean could be greater than 54. This provides evidence to reject H0.
3. Step 3: Analyze confidence interval (b): The interval is 51.5 < μ < 54.5. Since this interval includes 54 but does not extend beyond it, there is insufficient evidence to reject H0. The population mean could still be less than or equal to 54.
4. Step 4: Analyze confidence interval (c): The interval is 54.5 < μ < 55.5. Since this interval is entirely above 54, it provides strong evidence to reject H0. The population mean is likely greater than 54.
5. Step 5: Summarize the findings: Confidence intervals (a) and (c) provide evidence to reject H0, while confidence interval (b) does not provide sufficient evidence to reject H0.

Key Concepts

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing. In this case, H0 states that the population mean (μ) is less than or equal to 54. The goal of hypothesis testing is to determine whether there is enough evidence to reject this null hypothesis in favor of an alternative hypothesis.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a certain level of confidence (commonly 95%). Each interval provides an estimate of where the true population mean (μ) may lie. If a confidence interval does not include the value specified in the null hypothesis, it suggests that the null hypothesis may be rejected.

Decision Rule for Hypothesis Testing

The decision rule in hypothesis testing involves comparing the confidence intervals to the null hypothesis. If the entire confidence interval lies above the value specified in H0 (in this case, 54), we reject H0. Conversely, if the interval includes or is below this value, we fail to reject H0. This rule helps in making informed decisions based on statistical evidence.