Statistics Exam Questions and Solutions: Data Description, Probability, and Distributions

Describing Data Numerically

Measures of Central Tendency and Spread

Numerical data can be summarized using measures such as the mean, median, and quartiles. These help describe the center and spread of the data.

• Mean (Average): The sum of all data values divided by the number of values.

• Median: The middle value when data are ordered.

• Quartiles (Q1, Q3): Values that divide the data into four equal parts.

• Box and Whisker Plot: A graphical representation showing the minimum, Q1, median, Q3, and maximum.

Example: For the data set {5, 5, 6, 6, 7, 6, 7}, the mean is 6, the median is 6, Q1 is 5, Q3 is 7.

Outliers and Shape of Distribution

Outliers are data points that fall far from the rest of the data. The shape of the distribution can be described as symmetric, positively skewed, or negatively skewed.

• Interquartile Range (IQR): $IQR = Q3 - Q1$

• Outlier Fences: Lower fence = $Q1 - 1.5 imes IQR$, Upper fence = $Q3 + 1.5 imes IQR$

• Skewness: If most data are on the left, the distribution is positively skewed.

Example: For the above data, IQR = 2, lower fence = 2, upper fence = 10. No outliers present.

Correlation

Association Between Variables

Correlation measures the strength and direction of a linear relationship between two variables.

• Correlation Coefficient (r): Ranges from -1 to 1. Values close to 1 or -1 indicate strong association.

• Interpretation: A value of $r = 0.775$ suggests a strong positive association.

Example: Number of visits to fast food restaurants (X) and monthly income (Y) are positively correlated.

Describing Data with Tables and Graphs

Frequency Distribution

Frequency tables summarize data by showing how often each value or range of values occurs.

Class	Frequency	Cumulative Frequency
10-14	9	9
15-19	4	13
20-24	11	24

• Cumulative Frequency: The running total of frequencies up to each class.

• Probability Calculation: Probability of a value at least 15 is $\frac{16}{40} = 0.4$.

Probability

Basic Probability Concepts

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).

• Marginal Probability: Probability of a single event occurring.

• Conditional Probability: Probability of one event given another has occurred.

• Joint Probability: Probability of two events occurring together.

Probability Tables

	MBA Degree	No Degree	Marginal Probabilities
Over 35	0.27	0.13	0.4
At most 35	0.28	0.12	0.6
Marginal probabilities	0.7	0.3

• Example: Probability that a randomly selected analyst has an MBA and is at most 35 is 0.28.

• Conditional Probability Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Binomial Distribution & Discrete Random Variables

Probability Distribution Function (PDF)

A probability distribution function assigns probabilities to each possible value of a discrete random variable.

• Requirements: All probabilities must be non-negative and sum to 1.

• Example: For $P(X=x) = k(x^3 + 4x^2 + 5x + 3)$ for $x=2,3,4,5$, find $k$ so that $\sum P(X=x) = 1$.

Expected Value and Variance

The expected value (mean) and variance measure the center and spread of a probability distribution.

• Expected Value: $E(X) = \sum x P(X=x)$

• Variance: $Var(X) = E[(X - \mu)^2] = \sum (x - \mu)^2 P(X=x)$

Example: For the distribution:

X	P(X=x)
0	0.64
1	0.32
2	0.04

• $E(X) = 1 \times 0.32 + 2 \times 0.04 = 0.32 + 0.08 = 0.4$

• $Var(X) = 0.32$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Tables have been reconstructed and formulas provided in LaTeX format for academic rigor.