Unit 5- Activity 1: Comprehensive Study Notes on Categorical Variables

Conceptual Framework: This unit focuses on analyzing the connection between two categorical variables using contingency tables. These tables allow for the calculation of probabilities to determine how categories within one variable relate to categories within another.
Mathematical Definitions: * Marginal Probability: The probability of a single categorical outcome based on the total row or column count relative to the grand total. It focuses on one variable at a time (e.g., the total percentage of males in a group regardless of other factors). * Conditional Probability: The probability of an outcome given that another specific condition is already met. This narrows the denominator to a specific row total or column total (e.g., the percentage of females, specifically within the group of highly motivated individuals).

Gender	Low Motivation	Medium Motivation	High Motivation	Total
Female	$32$	$52$	$15$	$99$
Male	$28$	$55$	$26$	$109$
Total	$60$	$107$	$41$	$208$

Analysis and Multiple Choice Questions: * Q1: Percentage of Males (Marginal Probability): * Formula: $\frac{\text{Total Males}}{\text{Grand Total}}$ * Calculation: $\frac{109}{208} \approx 0.52403$ * Answer: d) $52.4\%$ * Q2: Percentage of Males reporting High (Conditional Probability): * Formula: $\frac{\text{Male and High Motivation}}{\text{Total Male}}$ * Calculation: $\frac{26}{109} \approx 0.2385$ , which rounds to $23.9\%$ * Answer: a) $23.9\%$ * Q3: Percentage of High Motivated who are Female (Conditional Probability): * Formula: $\frac{\text{Female and High Motivation}}{\text{Total High Motivation}}$ * Calculation: $\frac{15}{41} \approx 0.36585$ * Answer: b) $36.6\%$ * Q4: Least common motivation level for both genders: * Comparison of totals: Low ( $60$ ), Medium ( $107$ ), High ( $41$ ). * Answer: c) High
Short Answer Analysis: * Q5: Total Reporting High Motivation: $41$ * Q6: Percentage of Overall Medium Motivation (Marginal Probability): * Formula: $\frac{\text{Total Medium Motivation}}{\text{Grand Total}}$ * Calculation: $\frac{107}{208} \times 100 \approx 51.44\%$ * Q7: Comparison of Low Motivation Rates: * Female Low Rate: $\frac{32}{99} \times 100 \approx 32.3\%$ * Male Low Rate: $\frac{28}{109} \times 100 \approx 25.7\%$ * Comparison: Females have the higher rate of low motivation by approximately $32.3\% - 25.7\% = 6.6\%$ .

School Type	Low Quality	Medium Quality	High Quality	Total
Private	$9$	$30$	$25$	$64$
Public	$10$	$155$	$36$	$201$
Total	$19$	$185$	$61$	$265$

Calculated Proportions: * a) Percentage of Medium Quality Teachers in Private Schools: * Calculation: $\frac{30}{64} \times 100 = 46.875\%$ * b) Percentage of High Quality Teachers in Public Schools: * Calculation: $\frac{36}{201} \times 100 \approx 17.91\%$

Group	PC Sale	Mac Sale	Total
Males	$65$	$45$	$110$
Females	$35$	$85$	$120$
Total	$100$	$130$	$230$

Short Answer Analysis: * a) Percentage of All Sales that are MAC: * Calculation: $\frac{130}{230} \times 100 \approx 56.52\%$ * c) Percentage of PC Customers who are Male: * Calculation: $\frac{65}{100} \times 100 = 65\%$ * d) Percentage of Sales from Female Customers (Marginal Probability): * Calculation: $\frac{120}{230} \times 100 \approx 52.17\%$ * e) Percentage of MAC Customers who are Female: * Calculation: $\frac{85}{130} \times 100 \approx 65.38\%$

Age Group	Use Instagram: Yes	Use Instagram: No	Total
18—29	$225$	$127$	$352$
30—49	$211$	$317$	$528$
50—64	$114$	$430$	$544$
65+	$52$	$477$	$529$
Total	$602$	$1351$	$1953$

Multiple Choice Questions Analysis: * Q1: Percentage of all respondents who DO NOT use Instagram: * Formula: $\frac{\text{Total No}}{\text{Grand Total}}$ * Calculation: $\frac{1351}{1953} \approx 0.69175$ * Answer: d) $69.2\%$ * Q2: Percentage of all respondents in the 18–29 age group: * Formula: $\frac{\text{Total 18—29}}{\text{Grand Total}}$ * Calculation: $\frac{352}{1953} \approx 0.18023$ * Answer: b) $18.0\%$ * Q3: Percentage of those aged 30–49 who use Instagram: * Formula: $\frac{\text{30—49 Yes}}{\text{Total 30—49}}$ * Calculation: $\frac{211}{528} \approx 0.39962$ * Answer: a) $40\%$ * Q4: Percentage of those aged 65+ who do not use Instagram: * Formula: $\frac{\text{65+ No}}{\text{Total 65+}}$ * Calculation: $\frac{477}{529} \approx 0.9017$ * Answer: d) $90.2\%$

Survey Narrative: In a statistics project, students were surveyed about commuting by car and receiving parking tickets. * Total who commute by car = $25$ . From this group, $19$ received a ticket. * Total who do not commute by car = $28$ . From this group, $7$ received a ticket.
Variable Identification: * Explanatory Variable: Commute to school by car (Yes or No). This is the variable that predicts or influences the other. * Response Variable: Whether the student had ever received a parking ticket (Yes or No). This is the outcome variable being measured.
Completed Contingency Table:

Group	Received Ticket: Yes	Received Ticket: No	Total
Commute by Car: Yes	$19$	$6$ (Calculated: $25 - 19$ )	$25$
Commute by Car: No	$7$	$21$ (Calculated: $28 - 7$ )	$28$
Total	$26$	$27$	$53$