APSTA-GE.2001 Final Exam Fall, 2022

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Fall, 2022

APSTA-GE.2001

Final Exam

Instructions

1. Upload your response under the “Assignments” tab, just as you did your weekly assignments.

2. You must complete the exam on your own, but you may use your textbook, course notes, Stata or R, calculator, PowerPoint slides, and videos.

3. There is no dataset for you to download or use for this final exam. That is, you do not need Stata or R for this exam (still, you may use either if you want, e.g., as a calculator).

A headline on the front page of the NY Times announced the following: “Top Test Scores from Shanghai Stun Educators.” The headline was based on results from the standardized PISA test administered to 15-year olds who were selected as representatives of each of their respective countries.  In all, there were 65 countries world-wide that took part in this assessment of science, math, and reading achievement.  For the first time, China was included as one of these 65 countries.  

(4 pts) 1.  The article stated that “PISA scores are on a scale, with 500 as the average.  Two-thirds of students in participating countries score between 400 and 600.”  

What does this statement suggest about the likely overall shape and value of the standard deviation of the distribution of PISA scores and why?

(1) Shape ________________________________(2) Why? ___________________________________

___________________________________________________________________________________

(1) Standard Deviation _____ (2) Why? ___________________________________________________

(15 pts) 2.   The NY Times article published the scores on reading, math, and science for a sample of approximately half of the total number of countries that took the test.  Given the following stem-and-leaf plot of the science test results for the sample of 32 countries reported in the NY Times,  

 

(a) What is the minimum score on this test? _____________

(b) What is the mode score on this test?       _____________

(c) What is the median score on this test?     _____________

(d) How would you describe the shape of this distribution? ___________________

(e) Based on The NY Times headline, what score do you suppose Shanghai China received? ____________

(6 pts) 3. For this sample of data based on 32 countries, the means and standard deviations for the Science, Reading, and Math tests are:

If the United States scored 502 in science, 500 in reading, and 487 in mathematics, on which test did the United States do the worst relative to these 32 countries?  Show your work and explain your answer.

(3 pts) 4. Assuming that the scores on these tests are normally distributed, what is the United States’ percentile rank in math?  Interpret what this percentile rank value means.

(3 pts) 5.  Given the following scatterplot of science and math scores, describe the nature of the relationship in words that someone who did not take statistics would understand.

    

(10 pts) 6.  The following analysis addresses the extent to which a country’s Science achievement score on PISA may be predicted from its Math achievement score using a linear regression model.

 

(1 pts) (a) Write out the unstandardized regression equation for predicting a country’s science score from its math score.

(2 pts) (b) Interpret the b-weight (i.e., the slope, or estimated coefficient on math) within the context of this problem.

(2 pts) (c) Interpret the Y-intercept (constant, _cons) within the context of this problem.  Is it meaningful?

(1 pts) (d) Interpret the R-squared value from this output.

(2 pts) (e) What is the predicted science score for a country that scores 500 on math? (Show your work)

(1 pts) (f) What is the correlation between the actual science scores, Y, and the predicted science scores,?   r = ______

(1 pts) (g) What is the correlation between the math scores, X, and the predicted science scores, ?    r = ______

(13 pts) 7.  You wish to test whether the sample of 32 countries whose scores were reported in the NY Times represents a random sample of the 65 countries that were administered the PISA test.  If the sample is a random one, we would expect the mean science score for this sample to equal the population science mean of 500; and, otherwise not.  Assume all underlying assumptions of this test are met.  

(1 pts) (a) State the null and alternative hypotheses.  _______________________________________

(5 pts) (b) Unfortunately some ink spilled on your output and your advisor is in need of the results immediately.  Drawing on your statistical knowledge, fill in the values that are missing. Show your work where calculations are necessary to find these values.

 

Std. Err. = _________; t = _________ ;  df = __________   

 (2 pts) (c) Using the appropriate p-value, state your conclusion within the context of the problem.  Use a = .05.  

(2 pts) (d) Does the confidence interval lead to the same conclusion?  Why or why not?

(3 pts) (e) If the result is statistically significant, compute and interpret the effect size.

(16 pts) 8.  To determine whether countries, on average, score higher on Math than Reading on the PISA, the sample of 32 countries was compared in terms of their Math and Reading PISA scores.  Answer the following questions in connection with this test.  Assume all underlying assumptions of this test are met.  The reason for N=28 rather than N=32 in the table below is because there are four countries that had either their Reading or Math score missing in the report given in The NY Times.  Use a = .05.  

(1 pts) (a) State the null and alternative hypotheses.

(6 pts) (b) Use the excerpted output to compute the appropriate t-value and other results requested for this test.  Show your work in the space provided where calculations are needed.

Variable |     Obs      Mean      Std. Err.   Std. Dev.   [95% Conf. Interval]

    math |      28    519.5714    4.992436    26.41749    509.3278    529.8151

 reading |      28    507.9286    3.321777    17.57719    501.1128    514.7443

-------------------------------------------------------------------------------

    diff |      28    11.64286    3.221215    17.04507    5.033469    18.25225

                                 t = _________;     df = _________;       p-value = _________ 

(6 pts) (d) If the result is statistically significant, compute and interpret the effect size.

(3 pts) (e) Draw a conclusion and interpret your results in the context of the problem.

(15 pts) 9. To determine if European and U.K. countries as a group score differently in Science from those in Asia, Australia, New Zealand, and North America (as a group), the following partial results are obtained:

 

 

                 diff = mean(0) - mean(1)

(1 pts) (a) State the null and alternative hypotheses.  

(6 pts) (b) 1. State the null and alternative hypotheses in connection with Levene’s test.  2. Explain why Levene’s test is carried out in this example.   3. What do you conclude based on Levene’s test in this example (cite specific statistics to support your conclusion)?

(6 pts) (c)  Use the excerpted output above to compute the appropriate t-value and other results requested for this test.  Show your work in the space provided where calculations are needed.

                       t = ___________; df = ___________; p-value = ____________

(2 pts) (d) State your conclusions within the context of the problem with respect to the hypotheses stated in part (a).

(11 pts) 10.  We now wish to explore whether there are mean differences in Math for countries that are classified into one of the following three categories based on their Science scores -- low, middle, and high.  The name for this categorical science variable is catsci.  Use the following results to answer the following questions.  Show your work where calculations are needed.  

  

(1 pts) (a) State the null and alternative hypotheses.

(2 pts) (b) Why are df=2 for catsci (aka Between)?  

(2 pts) (c) Why are df=29 for Total?  

(2 pts) (d) What are the two values that form the ratio of F=12.10?   

(2 pts) (e) What are the two values that form the R-squared value of 0.4726?   

(2 pts) (f) Comment on the goodness-of-fit of this model. I.e.: How well does this model fit the data?   

(4 pts) 11. For each of the following variables, identify the most appropriate level of measurement as   either Nominal, Ordinal, Interval, or Ratio.

(a)   Student’s country of residence ____________

(b)   Student Math scores on the standardized PISA (Programme for International Student Assessment)

        exam as a measure of student achievement in mathematics ____________

(c)   Student time to finish the PISA test in Reading ____________

(d)   Student judgments of the quality of education in their country (excellent, very good, good, fair, poor)

       ____________