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QUESTION 1 To answer Question 1, use data uploaded on MyTimes. This is a second...

May 28, 2024

QUESTION 1 To answer Question 1, use data uploaded on MyTimes. This is a secondary data set from a car rental company and it contains the following variables: • Price of the cars (in RM) • Age of the cars (in years) • Mileages travelled by the cars • Type of the cars • Maintenance of the cars (in RM) • Model of the cars PART A (34 marks) 1. For all six variables given above, identify the type of each variable (qualitative or quantitative), and state the level of measurement. (6 marks) 2. Refer to the variable ‘Mileages travelled by the cars.’ Construct a frequency distribution table (show all the relevant working clearly). Draw an appropriate graph for this variable and explain. (8 marks) 3. By using Excel, display the summary statistics for both price of the cars and maintenance of the cars. Include quartile 1, quartile 3 and inter quartile range for both variables. (6 marks) 4. Write a short report based on your analysis in question 3 above. Your report must include interpretation of relevant measures of location, measures of dispersion and the shape of the distribution for each variable. Appropriate measures of location and dispersion for both variables must be added at the end of the report. (14 marks) PART B (34 marks) 1. Use correlation and regression analysis to investigate the relationship between maintenance and mileage of the car. (i) Identify the dependent and the independent variable. Use Excel to draw a scatterplot. Add a trend line. Explain your graph. (6 marks) (ii) Use Excel to generate a summary output for regression analysis. From the output, determine the regression equation. (4 marks) (iii) Write a short report based on your results in part (ii). Your report must include the strength of the relationship, the outcome of the model fit and the significance of the model. (5 marks) 2. Use correlation and regression analysis to investigate the relationship between maintenance of the car, age of the car and mileage travelled by the car. (i) Identify the dependent and the independent variables. (3 marks) (ii) Use Excel to generate a summary output for regression analysis. From the output, determine the regression equation. (4 marks) (iii) Write a short report based on your results in part (ii). Your report must include the outcome of the model fit, significance of the model and the significance of the individual variables. (6 marks) 3. Based on your results in question 1 and 2 above discuss which is a better model – model from Q1(ii) or Q2(ii), and why. Also state other relevant factor/s that will affect the stated dependent variable. Must include references to support these answers. (6 marks)

Solution by Steps

step 1

The variable "Price of the cars" is quantitative and measured at the ratio level because it has a true zero point and meaningful ratios

step 2

The variable "Age of the cars" is quantitative and measured at the ratio level because it has a true zero point and meaningful ratios

step 3

The variable "Mileages travelled by the cars" is quantitative and measured at the ratio level because it has a true zero point and meaningful ratios

step 4

The variable "Type of the cars" is qualitative and measured at the nominal level because it categorizes cars without a specific order

step 5

The variable "Maintenance of the cars" is quantitative and measured at the ratio level because it has a true zero point and meaningful ratios

step 6

The variable "Model of the cars" is qualitative and measured at the nominal level because it categorizes cars without a specific order

Answer

Price: Quantitative, Ratio; Age: Quantitative, Ratio; Mileage: Quantitative, Ratio; Type: Qualitative, Nominal; Maintenance: Quantitative, Ratio; Model: Qualitative, Nominal

# 2. Frequency distribution table and graph for "Mileages travelled by the cars"

step 1

Collect the mileage data from the table and determine the range

step 2

Divide the range into equal intervals (bins)

step 3

Count the number of observations in each interval to create the frequency distribution table

step 4

Use the frequency distribution table to draw a histogram

Answer

Frequency distribution table and histogram for mileage

# 3. Summary statistics for price and maintenance

step 1

Use Excel to calculate the mean, median, mode, standard deviation, quartile 1 (Q1), quartile 3 (Q3), and interquartile range (IQR) for both price and maintenance

step 2

The formulas in Excel are: - Mean: =AVERAGE(range) - Median: =MEDIAN(range) - Mode: =MODE.SNGL(range) - Standard Deviation: =STDEV.S(range) - Q1: =QUARTILE.INC(range, 1) - Q3: =QUARTILE.INC(range, 3) - IQR: =Q3 - Q1

Answer

Summary statistics for price and maintenance

# 4. Short report based on summary statistics

step 1

Interpret the mean, median, and mode to understand the central tendency of the data

step 2

Interpret the standard deviation and IQR to understand the dispersion of the data

step 3

Discuss the shape of the distribution based on skewness and kurtosis

Answer

Report on price and maintenance statistics

Part B # 1. Correlation and regression analysis between maintenance and mileage

step 1

Identify the dependent variable (maintenance) and the independent variable (mileage)

step 2

Use Excel to create a scatterplot and add a trend line

step 3

Use Excel to generate the regression output and determine the regression equation

step 4

Write a report on the strength of the relationship, model fit, and significance

Answer

Scatterplot, regression equation, and report on maintenance and mileage

# 2. Correlation and regression analysis between maintenance, age, and mileage

step 1

Identify the dependent variable (maintenance) and the independent variables (age and mileage)

step 2

Use Excel to generate the regression output and determine the regression equation

step 3

Write a report on the model fit, significance of the model, and significance of individual variables

Answer

Regression equation and report on maintenance, age, and mileage

# 3. Compare models from Q1(ii) and Q2(ii)

step 1

Compare the R-squared values to determine which model explains more variance

step 2

Consider the significance of the models and individual variables

step 3

Discuss other relevant factors that might affect maintenance

Answer

Comparison of models and discussion on other factors

Key Concept

Understanding variable types and levels of measurement is crucial for proper statistical analysis.

Explanation

Identifying whether a variable is qualitative or quantitative and its level of measurement (nominal, ordinal, interval, ratio) helps in choosing the appropriate statistical methods for analysis.

Solution by Steps

step 1

To draw an appropriate chart, we first need to decide the type of chart that best represents the data. Given the data is categorical (age groups) and numerical (number of people with disabilities), a bar chart or a stacked bar chart would be appropriate

step 2

Create a bar chart with age groups on the x-axis and the number of people on the y-axis. Separate bars for each type of disability (vision, hearing, physical) and gender (male, female) can be used

step 3

Plot the data from the table into the bar chart. For each age group, there will be six bars representing the number of males and females with vision, hearing, and physical disabilities

step 4

Analyze the chart to write a short report. Observe the trends and patterns, such as which age group has the highest number of disabilities, and any noticeable differences between genders

Answer

The bar chart shows that the age group 45-50 years has the highest number of people with disabilities, particularly physical disabilities. Males generally have higher numbers in all categories compared to females.

Key Concept

Bar chart representation of categorical and numerical data

Explanation

A bar chart helps visualize the distribution of disabilities across different age groups and genders, making it easier to identify trends and patterns.

Part (b)

step 1

Categorize the data by gender. Sum the number of males and females for each type of disability across all age groups

step 2

Create a new variable for the age groups: 18 years and less, 19 to 44 years, and above 45 years. Sum the number of people in these new age groups for each gender

step 3

Construct a new table with the new age groups and the number of people by gender

Answer

The new table will show the total number of males and females in each of the three new age groups.

Key Concept

Re-categorization of data

Explanation

Re-categorizing data helps in simplifying the analysis and making it more relevant to the specific questions being asked.

Part (c)(i)

step 1

Calculate the total number of females

step 2

Calculate the number of females aged 45 years and above

step 3

Use the formula for probability:

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

step 4

Substitute the values into the formula to find the probability

Answer

The probability that a randomly chosen female is 45 years and above is calculated as

\frac{\text{Number of females aged 45 and above}}{\text{Total number of females}}

Key Concept

Probability calculation

Explanation

Probability is the measure of the likelihood that an event will occur, calculated by dividing the number of favorable outcomes by the total number of outcomes.

Part (c)(ii)

step 1

Calculate the total number of people

step 2

Calculate the number of males with physical disabilities

step 3

Use the formula for probability:

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

step 4

Substitute the values into the formula to find the probability

Answer

The probability that a randomly chosen person is a male with physical disability is calculated as

\frac{\text{Number of males with physical disabilities}}{\text{Total number of people}}

Key Concept

Probability calculation

Explanation

Probability is the measure of the likelihood that an event will occur, calculated by dividing the number of favorable outcomes by the total number of outcomes.

Part (c)(iii)

step 1

Calculate the total number of people

step 2

Calculate the number of females with vision disabilities

step 3

Calculate the number of people aged 18 years or below

step 4

Use the formula for probability:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

step 5

Substitute the values into the formula to find the probability

Answer

The probability that a randomly chosen person is a female with vision disability or a person aged 18 years or below is calculated using the formula for the union of two events.

Key Concept

Union of probabilities

Explanation

The probability of the union of two events is the sum of the probabilities of each event minus the probability of their intersection.

Part (c)(iv)

step 1

Define the events: A = male with hearing disability, B = age group above 45 years

step 2

Calculate

P(A)

P(B)

, and

P(A \cap B)

step 3

Use the formula for independence:

P(A \cap B) = P(A) \cdot P(B)

step 4

Compare

P(A \cap B)

with

P(A) \cdot P(B)

step 5

P(A \cap B) = P(A) \cdot P(B)

, the events are independent; otherwise, they are not

Answer

The events male with hearing disability and age group above 45 years are statistically independent if

P(A \cap B) = P(A) \cdot P(B)

Key Concept

Statistical independence

Explanation

Two events are statistically independent if the occurrence of one does not affect the probability of the occurrence of the other.