SOLVING PROBLEMS: Show as m ork as possible for partial credit. You may round the answers to 2 decimal places. Please box o the final answer, where applicable. Use the attached standard normal table, t-table, and F-distributi a le as needed.
1) You found a report from CDC that says, "The age of a patient hospitalized from COVID-19 is normally distributed with a mean of 60.6 years and a population standard deviation of 3.2 years". Based on a wide range of other reports you have your suspicions towards this report and you think the mean age has to be higher than that.
a. A sample of 200 hospitals around Los Angeles is taken and the mean age of a patients hospitalized from COVID-19 for this sample was found to be 63.7 years. If your significance level is .01, what can your conclude about this hypothesis? Use the six step process including a diagram. Also calculate the p-value? (5pts)
b. You don't trust the population standard deviation from the above CDC report so, you look for data from Los Angeles county hospitals to test this hypothesis. You notice one hospital had released information on patients' age presented in the following table:
Number of patients Age 20 70 30 75 10 90 5 25
If the level of significance is .10, what can you conclude about your hypothesis? Use the six step process including a diagram. Does your decision change at .0005 level of significance? (6 pts)
c. For the above sample of patients (from b), calculate the coefficient of skewness. What can you say about the shape of this sample? Hint: Use the patient's age to sort the sample. (3 pts)
2) Taylor Swift is looking for an opener for her new post pandemic world tour. The final contest boils down to two YouTubers: you and some-annoying-nobody. She asks both YouTubers to sing a cover for some of her songs and measures the numbers of views (in millions) to select the best candidate. She plans on selecting whoever can deliver more views on their YouTube covers. The following table lists the number of views on each song for both:
You Some-Annoying-Nobody 8 7 7 10 9 2 1.5 1.2 5.5 1.5 4.2 1.9 3.3 3 5.5 2.5 6
Test the hypothesis that you should be selected for her world tour. Use .10 level of significance. Assume population standard deviations are unknown but assumed equal. Assume this scenario satisfies all necessary independence and distribution conditions. (7 pts)
3) Once the pandemic lockdowns is over and people start getting back to their work/school, they realize the difference in air quality due to pollution. You are included as part of a group of scientists trying to study the impact of age of cars on their CO2 emission per mile levels and on mileage levels. The motive is to try to educate people about this relationship so they would consider switching to better options. The below table lists the data collected on a sample of cars, their emission levels per mile, and their mileage levels. (24 pts)
a. Calculate the correlation coefficient between age of cars and CO2 emission per mile levels.
b. Test the hypothesis that the correlation between age of cars and emission levels per mile is non-zero at 1% level of significance. Use the six step process including a diagram.
c. Represent the relationship between age of cars and emission levels per mile as the general form of linear regression equation: 9 = a + bx. Specifically, solve for the values of a and b, and interpret them. What is the predicted CO2 emission per mile levels for a 1.5 year old car?
d. Test the hypothesis that the slope of the above equation is greater than zero at the 5% level of significance. Use the six step process including a diagram.
e. Consider the relationship between age of cars and mileage levels as the general form of linear regression equation. Test the hypothesis that the slope of this regression equation is less than zero at the 1% level of significance. Use the six step process including a diagram.
f. Find the standard error of estimate and coefficient of determination for both relationships (d and e). Interpret both estimates. For convenience, you can use excel here.
Age of Car (years) CO2 emission per mile (in grams) Mileage (miles per gallon) 2.5 523 55 4 611 51.4 1 150 60 10 1007 40.2 15 6321 31.6 9 841 38.4 8 666 38 3 212 53 8 884 39 7 921 38 20 10,000 20.1 6 721 45 3.5 552 49.1 5.5 632 45 13 2431 22 6.5 814 47 9.5 1001 42.8
4) Research continues in the development of cleaner driving mechanism. About a year later, engineers at t University design three alternatives for fuel in cars. As a statistician, you are being ., ., tasked to check if there is a difference in the mean emission levels among these three alternative fuels at 0.05 level of significance. The following table presents the data on a sample of cars from the experiment:
Fuell Fuel2 Fuel3 120 166 210 150 167 195 190 169 201 100 173 203 155 204
Build an ANOVA Table to solve the problem. For maximum credit, show work on how to calculate SST, SSE, degrees of freedom (both). (10 pts)
5) In an abundance of caution, authorities all over the world are recommending that people wear masks when they go out in public. A study shows that wearing a mask can reduce the likelihood of infection by 60%. Assume mask effectiveness follows a binomial distribution. A group of 40 people are gathered in a bar to celebrate a promotion. (10 pts)
a. What is the probability that 18 people in the bar will not get infected?
b. What is the probability that no-one will be infected?
6) Netflix hires you to study the number of hours people spent binging in front of the screen during the pandemic lockdown. Consider the following scenario: (10 pts)
a. They sampled 500 individuals and found the mean to be 25 hours. Assuming we know the population standard deviation is 4.1 hours, what is the confidence interval at the 80% level of confidence?
b. They sampled 500 individuals and found the mean to be 25 hours. Assuming we know the population standard deviation is 4.1 hours, what is the confidence interval at the 90% level of confidence?
c. They sampled 90 individuals and found the mean to be 21 hours. Assuming we don't know the population standard deviation, but we estimate the sample standard deviation to be 5 hours, what is the confidence interval at the 95% level of confidence?
d. Assuming we know the population standard deviation is 4.1 hours, what is the required sample size if the error should be less than 20 minutes with a 95% level of confidence?
7) A recent survey shows the average number of miles driven per capita in the state of California is 10,476 miles with a standard deviation of 1,263. Use a diagram to solve the following: (12 pts)
a. You are trying to save on car insurance by driving less. You talk to your insurance agent, and found out that people who fall in bottom 20% of driver miles usually see their rates reduced. What is the maximum number of miles you can drive to qualify for this rate reduction?
b. What is the minimum number of miles you have to drive to fall in the top 13%?
c. We take a sample of 64 drivers studying/working at Loyola Marymount University, and find that the average miles driven per capita for this sample is 7,895 miles. What are the odds that you get a sample like this?
8) In a coffee shop near LMU, 60% of the employees are female, while 40% male. 85% of the females attended college, 70% of the males attended college. An employee is selected at random: (6 pts) a. What is the probability that the employee selected is a female who did not attend college? b. What is the probability that the employee selected is a male who did not attend college?
9) A cell phone keypad has 10 different digits (0, 1... 9) that can be used to form a 4 digit password. (7 pts)
a. What is the total number of possible combinations of passwords that you can form with these 10 digits?
b. An app in the cellphone asks you to use alphabets in addition to digits to form a 6 character password. What is the possible number of combinations of passwords that you can form for this app? Assume its standard 26 English alphabet characters (A, B ....Z).