Study Guide for Exam 3

This will be a closed-book exam. You will be provided with this formula sheet. You will need a calculator. A calculator that performs arithmetic operations, reciprocals, square roots, powers and logarithms (base 10) is sufficient. Graphing calculators are permitted.

To do well on the exam, you should be able to do the following:

Chapter 5

Section 5.6: Normal as Approximation to Binomial

  1. Use Triola & Triola’s rule of thumb to determine when the normal distribution approximation to the binomial distribution is appropriate.
  2. Determine the appropriate mean and standard deviation for the normal distribution approximating a binomial distribution with \(n\) trials and success probability \(p\).
  3. Explain why, in terms of the probability histogram for a binomial distribution, the continuity correction is desirable when computing binomial probabilities using the normal density histogram. You will not need to use the continuity correction on homework and exams.
  4. For a binomial random variable \(X\) with \(n\) trials and success probability \(p\), use the normal distribution approximation to the binomial distribution to compute binomial probabilities such as \(P(X \leq x)\), \(P(X < x)\), \(P(X \geq x)\), \(P(X > x)\), and \(P(X = x)\).

Chapter 6

Section 6.2: Estimating a Population Proportion

  1. State the large-sample margin of error for the sample proportion \(\widehat{p}\) in terms of the population proportion \(p\).
  2. State the large-sample confidence interval for the population proportion \(p\)
  3. Determine the large-sample confidence interval to use depending on when a pilot estimate of \(p\) is known or unknown.
  4. Given a sample size \(n\), sample proportion \(\widehat{p}\), and confidence level \(c\), construct a \(100c\%\) confidence interval for the population proportion \(p\).
  5. Given a confidence level \(c\) and a desired precision \(E\), determine the sample size necessary to attain that level of precision at that confidence level.
  6. Explain, qualitatively, how the precision of the sample proportion varies with the confidence level \(c\), sample size \(n\), and population proportion \(p\).

Section 6.3: Estimating a Population Mean: \(\sigma\) Known

  1. Explain how a point estimator is related to a population characteristic.
  2. Give examples of point estimators for population parameters, like the population mean and standard deviation.
  3. Compare and contrast an interval estimator and a point estimator.
  4. Explain what a margin of error \(E\) corresponds to in an interval estimator.
  5. Give the \(100c\%\) confidence interval / interval estimator for the population mean \(\mu\) when the population standard deviation \(\sigma\) is known.
  6. State under what assumptions the confidence interval / interval estimator from the previous item is appropriate.
  7. Relate the confidence level \(c\) to the size \(\alpha\) of a hypothesis test for the population mean.
  8. State what quantity a confidence interval / interval estimator is for. That is, what are we confident about / what are we constructing an interval estimator for?
  9. Explain what the \(100c\%\) confidence level indicates about a confidence interval / interval estimator in a way that a layperson would understand.
  10. Give the sample size \(n\) that is necessary to achieve a margin of error \(E\) for the population mean \(\mu\) when the population standard deviation \(\sigma\) is known.
  11. Describe (qualitatively) the interaction between the the sample size \(n\), population standard deviation \(\sigma\), confidence level \(c\), and margin of error \(E\) in the confidence interval / interval estimator for the population mean \(\mu\).
  12. Use a \(100c\%\) confidence interval for the population mean \(\mu\) when the population standard deviation \(\sigma\) is known to perform a size \(\alpha = 1 - c\) two-tailed hypothesis test for \(\mu\) when the population standard deviation \(\sigma\) is known.

Section 6.4: Estimating a Population Mean: \(\sigma\) Unknown

  1. Give the \(100c\%\) confidence interval / interval estimator for the population mean \(\mu\) when the population standard deviation \(\sigma\) is unknown.
  2. State under what assumptions the confidence interval / interval estimator from the previous item is appropriate.
  3. Use a \(100c\%\) confidence interval for the population mean \(\mu\) when the population standard deviation \(\sigma\) is unknown to perform a size \(\alpha = 1 - c\) two-tailed hypothesis test for \(\mu\) when the population standard deviation \(\sigma\) is unknown.

Chapter 7

Section 7.1: Overview

  1. Answer the question: does a statistical hypothesis make a statement about a sample or a population?
  2. State the “Rare Event Rule for Inferential Statistics,” as given in Triola and Triola.

Section 7.2: Basics of Hypothesis Testing

  1. Given a claim, identify the null hypothesis and the research / alternative hypothesis, and express both as equalities / inequalities involving the population parameter(s).
  2. Given a research hypothesis, identify the corresponding null hypothesis by determining the research hypothesis’s complement, and vice versa.
  3. Define the significance level of a hypothesis test.
  4. Given a test statistic, the sampling distribution of the test statistic, the significance level of the test, and the null and alternative hypotheses, identify the critical region for a hypothesis test.
  5. Given a claim and sample data, calculate the value of a relevant test statistic.
  6. Determine whether a one-sided or two-sided hypothesis test is appropriate for a given pair of null and research hypotheses. The textbook calls these one-tailed (left-tailed and right-tailed) and two-tailed hypothesis tests.
  7. Specify the types of decision errors that can occur when performing a hypothesis test, and give their standard names.
  8. Relate the probability of a Type I error to the size of the rejection region / significance level \(\alpha\) of a hypothesis test.
  9. Define the power of a hypothesis test when a particular alternative hypothesis is assumed true.
  10. Relate the probability of a Type II error to the power of the hypothesis test.
  11. Explain the analogy between the default balance between Type I and Type II errors in hypothesis testing and the presumption of innocence in the US criminal justice system.
  12. Determine which of the two error types we control in our construction of the rejection region of a hypothesis test.
  13. Given a test statistic and the sampling distribution of the test statistic when the null hypothesis is true, determine the \(P\)-value corresponding to the observed value of the test statistic.
  14. Give the correct interpretation of a \(P\)-value as the probability of observing a test statistic at least as extreme as the observed test statistic when the null hypothesis is true.
  15. Explain what small and large \(P\)-values indicate.
  16. Given a \(P\)-value for an observed test statistic and the desired size of the rejection region / significance level, determine whether the null hypothesis should be rejected.
  17. State what is commonly meant by a \(P\)-value indicating ‘statistical significance.’
  18. State what, at the end of the day, is statistically significant.
  19. Compare and contrast practical significance and statistical significance.
  20. Relate the conclusion of a hypothesis test back to the original claim, and state the conclusion in plain English that a non-statistician could understand.

Section 7.3: Testing a Claim About a Proportion

  1. Recognize the notation \(p\) for the population proportion and \(\widehat{p}\) for the sample proportion.
  2. State the appropriate null and alternative hypotheses when given a claim about a population proportion.
  3. State the \(Z\) test statistic for a population proportion.
  4. Test a claim about a population proportion \(p\) using the appropriate \(Z\) test statistic for the population proportion using:
  5. State when a hypothesis test for the population proportion \(p\) using the \(Z\) test statistic is appropriate.
  6. State the margin of error for a population proportion \(p\) when we use the normal distribution approximation to the binomial distribution.
  7. Construct a confidence interval for the population proportion \(p\) using the normal distribution approximation to the binomial distribution.

Section 7.4: Testing a Claim About a Mean: \(\sigma\) Known

  1. Specify the requirements necessary for the \(Z\)-based hypothesis test of a population mean to be valid.
  2. Given a claim and sample data, determine whether the \(Z\)-based hypothesis test is appropriate to test the claim.
  3. Given a claim and sample data, perform a \(Z\)-based hypothesis test for the claim about the population mean.

Section 7.5: Testing a Claim About a Mean: \(\sigma\) Unknown

  1. Compare and contrast the density histograms for the standard normal distribution and the \(T\)-distribution with df degrees of freedom.
  2. Compute a \(T\)-statistic when given a sample mean, sample standard deviation, and sample size for a simple random sample.
  3. Determine the sampling distribution of the \(T\)-statistic when the null hypothesis about the population mean is true for a random sample of size \(n\) from the population.
  4. Determine how the sampling distribution of the \(T\)-statistic behaves as the sample size \(n\) increases.
  5. Specify the requirements necessary for the \(T\)-based hypothesis test of a population mean to be valid.
  6. Given a claim and sample data, determine whether the \(T\)-based hypothesis test is appropriate to test the claim.
  7. Use the T-table from Appendix A–3 of Triola and Triola to determine the critical values / rejection regions for the \(T\)-statistic when testing a null hypothesis using a rejection region of size / significance level \(\alpha\).
  8. Given a claim and summary statistics about a data set, perform a \(T\)-based hypothesis test for the claim about the population mean.