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Question 1 [17 marks]

A technology company has a very intensive hiring process, where each applicant is required

to complete four different tests. These tests are designed to assess various qualities,

skills and abilities that the company is looking for in prospective employees. The

technology company has collected some data on the applicants and their test scores.

Specifically, for a random selection of 500 applicants, they have recorded the following

for each applicant: their age in years (Age), their undergraduate degree (Degree) and

their score in each of the four tests (Test1, Test2, Test3 and Test4). The data is stored

in the file AssignmentData.RData in the dataframe Q1.df.

(a) [4 marks] Use some sample statistics to describe the scores for Test 1. Specifically,

based on the definitions given in the lectures, calculate the sample coefficient of

variation, the sample median and the sample range of the scores.

(b) [4 marks] Create a boxplot and a histogram of the scores for Test 1. Make sure

to give each plot a proper descriptive title and label the x-axis of the histogram

appropriately (do not just use the default title or labels). Based on these plots,

describe the distribution of scores for Test 1. Be specific in your description, making

sure to mention any interesting and/or important aspects of the distribution.

(c) [4 marks] The company wonders whether the distribution of scores for Test 1

is different between applicants who have an undergraduate degree in Computing

and applicants who have an undergraduate degree in Engineering. Create separate

histograms of the scores for Test 1 for these two groups of applicants (i.e., one

histogram for applicants with a Computing degree and one histogram for applicants

with an Engineering degree). Make sure to give each histogram a proper descriptive

title and a label for the x-axis. Based on these histograms, describe any differences

or similarities in the distribution of scores for Test 1 between these two groups of

applicants.

(d) [2 marks] The company wonders whether the spread in scores might be similar

for all tests. Determine whether or not the spread of the scores for Test 1 and Test

3 are similar. Do not conduct a hypothesis test, but make sure to provide a clear

justification for your answer based on the data.

(e) [3 marks] Test whether the mean score for Test 1 is greater than 7. Clearly state

your hypotheses and use a significance level of α = 10%. Do not use any functions

available in R or any R package that are designed to perform hypothesis tests.

Assignment S1 2021 Page 3 of 6 STAT7055

Question 2 [14 marks]

For this question you will be required to generate some sample data in R.

(a) [2 marks] First, in a single line of code, specify the seed for the random number

generator in R by using the set.seed function with your student number (without

the “u”) as the seed argument. For example, if your student number is u1234567

then you would use the line of code set.seed(1234567). Next, in a single line of

code, create a vector consisting of 100 observations that are randomly generated

from a normal distribution with mean µ = 65 and variance σ2 = 182.25 and call

this vector x (representing the variable X). Finally, in a single line of code, create

a vector consisting of 105 observations that are randomly generated from a uniform

distribution between a = 42 and b = 92 and call this vector y (representing the

variable Y ). These three lines of code must be executed in succession with no other

lines of code in between.

For the remaining parts of this question, assume that the values of µ, σ2, a and b are all

unknown and all that you have available is the sample data you generated in part (a).

(b) [4 marks] Calculate an 84% confidence interval for the population mean of the Y

values. Interpret the confidence interval. Do not use any functions available in R

or any R package that are designed to calculate confidence intervals.

(c) [4 marks] Test whether the population proportion of X values that are greater

than 75 is less than 0.351. Clearly state your hypotheses, making sure to define

any parameters, and use a significance level of α = 5%. Do not use any functions

available in R or any R package that are designed to perform hypothesis tests.

(d) [4 marks] If the X values and Y values are treated as independent samples, test

whether the population proportion of X values that are greater than 75 is greater

than the population proportion of Y values that are greater than 88. Clearly state

your hypotheses, making sure to define any parameters, and use a significance level

of α = 10%. Do not use any functions available in R or any R package that are

designed to perform hypothesis tests.

Assignment S1 2021 Page 4 of 6 STAT7055

Question 3 [23 marks]

Some data were collected on the university entrance scores for year 12 students from

four high schools over a period of four years. For each year, a sample of year 12 students

were randomly selected from each high school and their university entrance scores were

recorded. For a given high school, the same number of year 12 students were selected each

year. However, within a given year, the number of year 12 students selected may differ

between the four high schools. The data is stored in the file AssignmentData.RData in

the dataframe Q3.df. For a given year, the university entrance scores for all students

across all four high schools are given in the column named by the year (Year2005,

Year2006, Year2007 and Year2008) and the high school (1, 2, 3, or 4) to which each

student belonged is given in the column named HighSchool.

For this question, you will be analysing the university entrance scores from

2005.

(a) [2 marks] Calculate the sample mean university entrance score for each high school.

(b) [2 marks] Calculate the sample variance of university entrance scores for each high

school.

(c) [3 marks] Test whether the population variance of university entrance scores is

the same for high school 1 and high school 2. Clearly state your hypotheses and

use a significance level of α = 5%. Do not use any functions available in R or any

R package that are designed to perform hypothesis tests.

(d) [3 marks] Test whether the population mean university entrance score for high

school 2 is greater than that for high school 1 by more than 5. Clearly state your

hypotheses and use a significance level of α = 5%. Do not use any functions

available in R or any R package that are designed to perform hypothesis tests.

You will now conduct a one-way ANOVA on the university entrance scores from 2005

with high school as the factor.

(e) [8 marks] Discuss whether or not the assumptions for a one-way ANOVA hold

for this data. You do not need to conduct any hypothesis tests, but make sure to

provide clear justifications for your answer.

(f) [2 marks] Calculate the sum of squares for treatment for the one-way ANOVA.

Do not use any functions available in R or any R package that are designed to

perform an ANOVA.

(g) [3 marks] Test whether the population mean university entrance score is the same

for all four high schools. Clearly state your hypotheses and use a significance level

of α = 5%. Do not use any functions available in R or any R package that are

designed to perform hypothesis tests or an ANOVA.

Assignment S1 2021 Page 5 of 6 STAT7055

Question 4 [16 marks]

A think tank has been developing aptitude tests which they are hoping could eventually

be used as a replacement for IQ tests. They have conducted a long-term study where

they selected a random sample of 200 people and, for each person, recorded their scores

in an age-appropriate aptitude test every five years from age 5 to age 25 (Age5, Age10,

Age15, Age20 and Age25). The data is stored in the file AssignmentData.RData in the

dataframe Q4.df. The think tank is interested in whether the score in the aptitude test

taken at age 5 could be used to predict the score in later year aptitude tests.

For this question, you will be analysing the aptitude test scores at ages 5

(Age5) and 10 (Age10).

(a) [3 marks] Create a scatter plot of the aptitude test scores at age 10 against the

aptitude test scores at age 5. Make sure to give your plot an appropriate title and

appropriate labels for the x and y axes. Describe the relationship between these

two variables.

(b) [3 marks] Test whether the correlation between the aptitude test scores at age

10 and the aptitude test scores at age 5 is greater than zero. Clearly state your

hypotheses and use a significance level of α = 5%. Do not use any functions

available in R or any R package that are designed to perform hypothesis tests.

(c) [2 marks] Fit a simple linear regression model with aptitude test scores at age

10 as the dependent variable and aptitude test scores at age 5 as the independent

variable. Write down the estimated regression model.

(d) [5 marks] Discuss whether or not the assumptions for a simple linear regression

model hold for this data, making sure to provide clear justifications for your answer.

(e) [3 marks] Test whether the intercept is less than 5. Clearly state your hypotheses

and use a significance level of α = 10%. Do not use any functions available in R

or any R package that are designed to perform hypothesis tests.

END OF ASSIGNMENT

Assignment S1 2021 Page 6 of 6 STAT7055

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