For all problems, please show all your work. For any of the following calculations, feel free to mix calculations by hand with R calculations, but in most cases, you should try to do something by hand if it seems feasible, and only use R if the calculation is complex. In most cases you can only use the t.test function to verify your work.
1. Lebron: You think that the average person is taller than Lebron James, that stands at 6-fee 8-inches tall or about 2.03 meters. You measure 100 randomly selected people, and obtain a mean height of 5’9” (~ 1.75 m.) and standard deviation of 2.8 inches (~ 0.07 m). Please show your work for each question, and use meters as unit of measure. - you are only allowed to use the t.test function here to verify your work -.
a. Formalize the null hypothesis and alternative hypothesis using a one-tailed test.
b. Compute the corresponding t-statistic.
c. Can you reject the null hypothesis? (Assuming a significance level of 95%)
d. What is the 95% confidence interval of the test?
e. What is the p-value of the test?
2. Tetris: Your friend thinks that men and women have different skill levels in playing Tetris - for whatever reason -. To test this, you ask 50 men and 50 women play the game in a controlled setting. The mean score of the men is 1,124 with a standard deviation of 200 and the mean score for the women is 1,245, also with a standard deviation of 200. - you are only allowed to use the t.test function here to verify your work -.
a. Formalize the null hypothesis and alternative using a two-tailed test.
b. Compute the corresponding t-statistic.
c. Can you reject the null hypothesis? (Assuming a significance level of 95%)
d. What is the 95% confidence interval of the test?
e. What is the p-value of the test?
3. Alcohol:Your friend from the previous problem thinks that drinking the night before an exam
might help performance on the exam the next morning. To test this, you select 100 of your closest friends, and randomly get 50 of them drunk the night before the exam, which you denote the treatment group; the other 50 are going to be called the control group. The next day, the treatment group gets a mean of 78 with a standard deviation of 20 and the control group gets a 75 with a standard deviation of 5. - you are only allowed to use the t.test function here to verify your work -.
a. Is there evidence to question the hypothesis proposed by your friend? Test the hypothesis using a significance level of 90%, 95%, and, 99% (Hint: making a function that run the test based on the significance level can save you some time.)
4. Earnings and Education: For this example you need to use the CPSSW04 data - you can get this dataset in a similar way to the CPSSWEducation data from learning modules 5.2, simply replace CPSSWEducation with CPSSW04-. - in this problem you are required to use the t.test function -
a. Make a boxplot using ggplot2 with the average earnings of individuals with only a high school diplomma vs individuals that have a bachelor degree.
b. Use the t.test function to determine if individuals with a bachelor degree have a higher average earnings than individuals with a high school degree.
c. Use the t.test function to determine if individuals between 25 and 29 years earn the same as individuals between 30 and 34 years.
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