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). The null hypothesis states that p ≤ 0.20. This means we are testing whether the population proportion p is less than or equal to 0.20.
2. Step 2: Analyze confidence interval (a). The interval given is 0.21 < p < 0.23. Since the entire interval is above 0.20, it does not include the value specified in the null hypothesis. Therefore, this confidence interval suggests rejecting H0.
3. Step 3: Analyze confidence interval (b). The interval given is 0.19 < p < 0.23. This interval includes the value 0.20, which is part of the null hypothesis. Therefore, this confidence interval does not provide enough evidence to reject H0.
4. Step 4: Analyze confidence interval (c). The interval given is 0.175 < p < 0.205. This interval includes the value 0.20, which is part of the null hypothesis. Therefore, this confidence interval does not provide enough evidence to reject H0.
5. Step 5: Summarize the reasoning. Confidence intervals that do not include the value specified in the null hypothesis (0.20) provide evidence to reject H0. In this case, only interval (a) leads to rejecting H0, while intervals (b) and (c) do not.

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 statistical testing. In this context, H0 states that the population proportion (p) is less than or equal to 0.20. To determine whether to reject H0, we compare the confidence intervals of the sample proportions to this threshold.

Confidence Interval

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

Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. In hypothesis testing, if the confidence interval for a sample proportion does not overlap with the null hypothesis value, it indicates that the result is statistically significant, leading to the potential rejection of H0. This concept is crucial for interpreting the results of the confidence intervals presented.