Statistics Problems i need answers to with showed work

7

STA 2023 Prof. Sampson

Project 2, Chapters 4-6

Instructions:

1. Project text will be typed, double-spaced, in 12-point Times New Roman font.

2. Tables and charts will be single spaced.

3. Project answers will be labeled exactly as the questions are presented in the project document.

4. Projects answers will be written in complete sentences with correct grammar, syntax, punctuation, and spelling.

5. Every answer must be justified.

6. Calculations will be typed in equation editor or a similar feature of the word processing program used. If you need help navigating equation editor, ask.

7. Group work is encouraged (check your calculations), but all projects must be uniquely stated. Write your own interpretations.

8. Hint: Use this document. Type your answers in place of the questions. Leave headers and labels in place. When you are finished, delete the instructions and make sure that no tables split across multiple pages.

Chapter 4: Distribution of Random Variables (50 points)

Binomial distributions: One of our in-class activities included several probability experiments in which you flipped coins and recorded the results. If we define a random variable , then these experiments are binomial in nature.

1. For the experiment with four tosses: (16 points)

   1. What was your result?

   2. Identify and the sample space for .

   3. Calculate and .

   4. Calculate . Was this event unlikely?

   5. Calculate using the binomial distribution.

   6. Calculate using the normal approximation to the binomial distribution.

   7. Explain whether the normal approximation is appropriate.

   8. The following graph represents the probability distribution for this experiment. Discuss which events are unlikely. In a true research scenario, was this number of trials adequate?

1. For the experiment with eleven tosses: (16 points)

   1. What was your result?

   2. Identify and the sample space for .

   3. Calculate and .

   4. Calculate . Was this event unlikely?

   5. Calculate using the binomial distribution.

   6. Calculate using the normal approximation to the binomial distribution.

   7. Explain whether the normal approximation is appropriate.

   8. The following graph represents the probability distribution for this experiment. Discuss which events are unlikely. In a true research scenario, was this number of trials adequate?

1. For the experiment with thirty tosses: (18 points)

   1. What was your result?

   2. Identify and the sample space for .

   3. Calculate and .

   4. Calculate . Was this event unlikely?

   5. Calculate using the binomial distribution.

   6. Calculate using the normal approximation to the binomial distribution.

   7. Explain whether the normal approximation is appropriate.

   8. How did your answer to parts e, f, and g change as the number of trials increased?

   9. The following graph represents the probability distribution for this experiment. Discuss which events are unlikely. In a true research scenario, was this number of trials adequate?

Chapter 5: Foundation for inference (50 pts)

The following contingency tables show the comparison between party and gender in two different United States Congresses.

1. Construct the following for the 114th Congress: (16 pts)

   1. Calculate and for women in Congress.

   2. Calculate and for men in Congress.

   3. Calculate and for Republican women in Congress.

   4. Calculate and for Republican men in Congress.

   5. Calculate and for Democrat women in Congress.

   6. Calculate and for Democrat men in Congress.

   7. Calculate and for other party women in Congress.

   8. Calculate and for other party men in Congress.

2. Construct the following for the 115th Congress: (16 pts)

   1. Calculate and for women in Congress.

   2. Calculate and for men in Congress.

   3. Calculate and for Republican women in Congress.

   4. Calculate and for Republican men in Congress.

   5. Calculate and for Democrat women in Congress.

   6. Calculate and for Democrat men in Congress.

   7. Calculate and for other party women in Congress.

   8. Calculate and for other party men in Congress.

3. These data are population data for an individual Congress, but they are cluster samples for the whole history of Congress. Let us assume that the data from the 114th Congress provides our parameters, and we want to see if there have been meaningful changes in the 115th Congress. (18 pts)

   1. Verify conditions to ensure that the standard error estimate is reasonable and the point estimate is unbiased.

   2. The null and alternative hypotheses ( and ) for the claim that the proportion of women in Congress has not really changed.

   3. Calculate the test statistic .

   4. Find the -value. Compare it to , and state whether to reject or fail to reject .

   5. Interpret your results in context.

   6. Identify Type 1 and Type 2 errors.

Chapter 6: Inference for Categorical Data (50 pts)

In 2010, Gallup polling found that 31% of Americans identify as Democrats, 29% as Republicans, and 40% as independent or other (wikipedia.org…don’t judge). Many Independents lean to one of the major parties, and when this fact is taken into account, Democrats and Republicans each claim 45% of Americans, and Other claims 10%.

1. For the 115th Congress, perform an Independence test for party affiliation in Congress against gender: (14 pts)

   1. We wish to compare Congress to the American public. Is this a simple random sample? Is it still appropriate to use a test?

   2. Verify the necessary conditions.

   3. Write the hypothesis in plain language.

   4. Calculate the test statistic and degrees of freedom.

   5. Find the -value. Compare it to , and state whether to reject or fail to reject .

   6. Interpret your results in context.

   7. Identify Type 1 and Type 2 errors.

2. For the 115th Congress, perform a Goodness of Fit test for party affiliation in Congress against the Gallup poll claimed distributions without leanings: (18 pts)

   1. Verify the necessary conditions.

   2. Write the hypothesis in plain language.

   3. Calculate the test statistic and degrees of freedom.

   4. Find the -value. Compare it to , and state whether to reject or fail to reject .

   5. Interpret your results in context.

   6. Identify Type 1 and Type 2 errors.

3. For the 115th Congress, perform a Goodness of Fit test for party affiliation in Congress against the Gallup poll claimed distributions with leanings: (18 pts)

   1. Verify the necessary conditions.

   2. Write the hypothesis in plain language.

   3. Calculate the test statistic and degrees of freedom.

   4. Find the -value. Compare it to , and state whether to reject or fail to reject .

   5. Interpret your results in context.

   6. Identify Type 1 and Type 2 errors.

4. For the 115th Congress, perform an Independence test for party affiliation in Congress against gender: (14 pts)

   1. Verify the necessary conditions.

   2. Write the hypothesis in plain language.

   3. Calculate the test statistic and degrees of freedom.

   4. Find the -value. Compare it to , and state whether to reject or fail to reject .

   5. Interpret your results in context.

   6. Identify Type 1 and Type 2 errors.