Solved Problems: Applying Confidence Intervals to Hypothesis Testing Scenarios

📚 Quick Study Guide

• 📊 Confidence Interval (CI): A range of values likely to contain the true population parameter. Calculated as: $\text{Sample Statistic} \pm (Critical Value \times \text{Standard Error})$.
• 🧪 Hypothesis Testing: A method to test a claim about a population using sample data. We typically set up a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).
• 𝛼 Significance Level ($\alpha$): The probability of rejecting the null hypothesis when it is true (Type I error). Common values are 0.05 and 0.01.
• ⚖️ Relationship: A confidence interval can be used to perform a hypothesis test. If the hypothesized value falls *outside* the CI, we reject the null hypothesis.
• 💡 One-tailed vs. Two-tailed: One-tailed tests consider direction (e.g., greater than), while two-tailed tests consider any difference. Confidence intervals are usually linked to two-tailed tests.
• 🔢 Z-test vs. T-test: Use a Z-test when the population standard deviation is known or the sample size is large (n > 30). Use a T-test when the population standard deviation is unknown and the sample size is small.
• 📝 P-value: The probability of observing a test statistic as extreme as, or more extreme than, the statistic obtained from a sample, assuming that the null hypothesis is true. If the p-value is less than $\alpha$, we reject the null hypothesis.

Practice Quiz

1. Question 1: A 95% confidence interval for the mean is (10, 20). Which of the following null hypotheses would be rejected at the 5% significance level?
   1. A) $H_0: \mu = 12$
   2. B) $H_0: \mu = 15$
   3. C) $H_0: \mu = 19$
   4. D) $H_0: \mu = 21$
2. Question 2: If a 99% confidence interval for the difference in means between two populations does *not* contain zero, what can we conclude about the null hypothesis that the means are equal, at a significance level of 1%?
   1. A) Fail to reject the null hypothesis.
   2. B) Reject the null hypothesis.
   3. C) Accept the null hypothesis.
   4. D) The test is inconclusive.
3. Question 3: A researcher calculates a 90% confidence interval for a population proportion to be (0.45, 0.55). If they wanted to test the hypothesis $H_0: p = 0.5$ vs. $H_a: p \neq 0.5$ at a significance level of 10%, what would be the conclusion?
   1. A) Reject the null hypothesis.
   2. B) Fail to reject the null hypothesis.
   3. C) Accept the null hypothesis.
   4. D) The sample size is too small to conclude.
4. Question 4: Which of the following statements is true regarding the relationship between confidence intervals and hypothesis testing?
   1. A) A confidence interval can only be used for one-tailed tests.
   2. B) If the hypothesized value falls within the confidence interval, we reject the null hypothesis.
   3. C) If the hypothesized value falls outside the confidence interval, we reject the null hypothesis.
   4. D) Confidence intervals are not related to hypothesis testing.
5. Question 5: A 95% confidence interval for the slope of a regression line is (-0.5, 0.2). At the 5% significance level, can we conclude that there is a significant linear relationship between the variables?
   1. A) Yes, there is a significant positive linear relationship.
   2. B) Yes, there is a significant negative linear relationship.
   3. C) No, there is no significant linear relationship.
   4. D) The test is inconclusive.
6. Question 6: What happens to the width of a confidence interval as the sample size increases, assuming all other factors remain constant?
   1. A) The width increases.
   2. B) The width decreases.
   3. C) The width remains the same.
   4. D) The width becomes zero.
7. Question 7: A researcher constructs a 95% confidence interval for the population mean. Which of the following is the correct interpretation of the confidence interval?
   1. A) There is a 95% probability that the sample mean lies within the interval.
   2. B) There is a 95% probability that the population mean lies within the interval.
   3. C) 95% of all sample means will lie within the interval.
   4. D) The interval contains 95% of the data values.