Assignment II PART A (Probable Answers)

 Part A:

1. Arithmetic Mean (Grouped Data):

Imagine you have exam scores for a class, but instead of individual scores, you only have the data grouped into intervals (e.g., 90-100, 80-89, etc.) with the number of students in each range (frequency). The arithmetic mean for grouped data considers these frequencies and the midpoints of each interval to calculate an average representative score for the class.

Example:

Exam Score Range	Frequency	Midpoint
90-100	5	95
80-89	8	84.5
70-79	3	74.5

Here, you'd calculate the mean using a formula that considers the frequencies and midpoints.

2. Four Data Classification Scales:

Data classification helps us understand the level of information our data conveys. Here's a breakdown with examples:

• Nominal: Categorical data with no inherent order. You can't rank or compare the categories.

  Example: Eye color (blue, brown, green) - There's no order from "best" to "worst" eye color.

• Ordinal: Categorical data with a rank or order, but the intervals between categories might not be consistent.

  Example: Customer satisfaction rating (very satisfied, satisfied, neutral, dissatisfied) - There's a clear order of satisfaction, but the difference between "very satisfied" and "satisfied" might not be the same as "satisfied" and "neutral."

• Interval: Numerical data with equal intervals between categories. The zero point doesn't necessarily represent an absence of the quantity being measured.

  Example: Temperature in Celsius (10°C, 15°C, 20°C) - The difference between 10°C and 15°C is the same as the difference between 15°C and 20°C (5°C). However, 0°C doesn't mean there's no heat.

• Ratio: Numerical data with a true zero point, meaning zero represents a complete absence of the quantity being measured.

  Example: Weight (50 kg, 70 kg, 100 kg) - 0 kg truly represents no weight. You can meaningfully compare and rank weights.

3. Histogram:

Imagine you have data on heights of students in a class. A histogram would visualize this data using bars on a graph. The horizontal axis represents height intervals (e.g., 150-155 cm, 156-160 cm, etc.), and the vertical axis represents the frequency (number of students) within each height range. The height of each bar corresponds to the number of students in that height range. (Draw an example histogram)

Example: A histogram for student heights might show several bars, with the tallest bar potentially representing the most common height range in the class.

4. Percentile:

Percentiles help you identify specific values within a sorted data set. Dividing the data set into 100 equal parts gives you 100 percentiles (from 1st to 99th). The pth percentile represents the value where p% of the data falls below it.

Example: The 25th percentile (Q1) is the first quartile, which means 25% of the data points are less than or equal to that value. The 50th percentile (Q2) is the median, where 50% of the data falls below and 50% falls above.

5. Bernoulli Distribution:

Imagine flipping a fair coin. The Bernoulli distribution models the probability of getting heads (success) or tails (failure) in a single coin toss. It assumes a constant probability (p) of success and (1-p) for failure.

Example: If p (success) = 0.6 for getting heads, then the Bernoulli distribution tells us the probability of getting tails (failure) is 1-0.6 = 0.4.

The Bernoulli distribution only needs one formula as it deals with a single binary experiment (success/failure) and doesn't involve calculating probabilities for multiple events like the Poisson distribution.

Here's the equation for Bernoulli's distribution:

P(X = x) = p^x * (1 - p)^(1-x)

where:

• P(X = x): Represents the probability of getting outcome "x". In a Bernoulli trial, "x" can only be 0 (failure) or 1 (success).
• p: Represents the probability of success in the experiment.
• (1 - p): Represents the probability of failure in the experiment (1 - p since success and failure are the only two possibilities and their probabilities must sum to 1).

This formula essentially calculates the probability of getting outcome "x" (either success or failure) by considering the probability of success (p) raised to the power of the number of successful outcomes (x) and the probability of failure (1-p) raised to the power of the number of failure outcomes (1-x) since there's only one trial.

6. Poisson Distribution:

Imagine counting the number of customer arrivals at a store in a given hour. The Poisson distribution models the probability of having a specific number of arrivals (events) within that fixed time interval (hour) given an average arrival rate (lambda).

Poisson Distribution:

The Poisson distribution models the probability of having a specific number (k) of events occurring in a fixed interval of time or space, given an average rate (lambda) of occurrence for those events.

• P(k events) = (e^-λ * λ^k) / k!

where:

• k is the number of events you're interested in (e.g., k = 3 customer arrivals).
• λ (lambda) is the average rate of events occurring in the interval (e.g., λ = 5 customers arriving per hour on average).
• e is the mathematical constant approximately equal to 2.71828.
• k! (factorial) is a mathematical operation where you multiply a number by all the positive integers less than or equal to itself (e.g., 3! = 3 * 2 * 1 = 6).

Another Example: If the average arrival rate is 5 customers per hour (lambda = 5), the Poisson distribution can calculate the probability of having 3 customers arrive, 10 customers arrive, or any other specific number of arrivals in that hour.

7. Random Variable (continued):

Continuing from the previous explanation, a random variable is a variable whose value depends on the outcome of a random experiment. It's essentially a numerical representation of the possible results from an experiment with some uncertainty.

Example:

• Rolling a die: The random variable here is the number rolled (1, 2, 3, 4, 5, or 6). The specific value depends on the random outcome of the die roll.
• Drawing a card from a deck: The random variable could be the suit (hearts, diamonds, clubs, or spades) or the card value (ace, king, queen, etc.). The specific outcome depends on the random draw from the deck.

8. Chi-Square Distribution (continued):

The chi-square distribution is a statistical tool used in hypothesis testing for categorical data. It helps assess the difference between the observed frequencies (how many times you actually see an outcome) and the expected frequencies (how many times you'd expect to see an outcome based on a hypothesis) in a chi-square test.

Chi-Square Test: Definition, Formula, and Civil Engineering Example

Definition:

The chi-square test (χ², pronounced "chi-squared") is a non-parametric statistical test used to assess whether two categorical variables are independent. It determines if the observed frequencies (counts) in different categories of a contingency table significantly differ from the expected frequencies based on a null hypothesis.

Formula:

The chi-square statistic is calculated using the following formula:

χ² = Σ ( (O_i - E_i)² / E_i )

where:

• χ²: Chi-square statistic
• Σ: Summation over all categories (i)
• O_i: Observed frequency in category i
• E_i: Expected frequency in category i

Interpretation:

• Higher chi-square values indicate a greater discrepancy between observed and expected frequencies, suggesting the null hypothesis (independence of variables) might be less likely.

Civil Engineering Example:

Scenario: An engineer is investigating the relationship between the type of foundation (slab, footing, pile) used in a building and the number of stories in the building (low-rise, mid-rise, high-rise). They collect data from 20 buildings:

Foundation Type	Low-Rise (1-3 Stories)	Mid-Rise (4-8 Stories)	High-Rise (9+ Stories)	Total
Slab	6	2	0	8
Footing	5	4	1	10
Pile	1	2	3	6
Total	12	8	4	20

Null Hypothesis (H₀): The type of foundation and the number of stories in a building are independent.

Steps:

1. Calculate Expected Frequencies: Based on the marginal totals (row and column sums) and assuming independence, calculate the expected frequency for each cell.

2. Compute Chi-Square: Apply the chi-square formula to each cell and sum the results.

Calculations (example shown for one cell):

• Expected frequency for Slab foundation in Low-Rise buildings (E_11):

  E_11 = (Total Low-Rise * Total Slab) / Total Buildings = (12 * 8) / 20 = 4.8

• Chi-square for Slab foundation in Low-Rise buildings:

  χ₁₁² = ( (O_11 - E_11)² / E_11 ) = ( (6 - 4.8)² / 4.8 ) ≈ 1.46

Complete the calculations for all cells and sum the chi-square values.

Decision:

• Use a chi-square distribution table to determine the critical value for a chosen significance level (e.g., α = 0.05) and the degrees of freedom (number of categories minus 1 for each variable minus 1, typically calculated as (rows - 1) * (columns - 1)).

• If the calculated chi-square value is greater than the critical value, reject the null hypothesis. This suggests a statistically significant association between foundation type and building number of stories.

Conclusion:

By analyzing the chi-square statistic, the engineer can evaluate the evidence against the null hypothesis. A significant result might suggest exploring the reasons behind the association, potentially influencing foundation design choices for different building heights in the future.

Note: This is a simplified example. Real-world scenarios might involve more categories and require additional statistical considerations.

Another Example:

Imagine surveying people about their preferred movie genre (comedy, action, drama, etc.). You can create a chi-square test to see if the observed genre preferences in your sample significantly differ from what you might expect based on population demographics. A low p-value from the chi-square test suggests the observed data deviates significantly from the expected data, potentially leading to rejecting the null hypothesis (which often states there's no difference between observed and expected frequencies).

I hope this provides a more comprehensive explanation with examples for each concept!