Statistical Analysis Chapter 1

• Statistical analysis notes
• Sangyung Lee, PHD
• Kyung Hee University

True and False Questions

• There are 15 True or False questions worth 2 points each, for a total of 30 points

Chapter 1

• Discrete data cannot be ratio data
• Even ratio data can be measured using an ordinal scale
• Inferential statistics deal with making inferences about a population
• A population can be considered a set of numbers
• The daily sleep duration of housewives in Korea cannot be considered a population
• Qualitative data refer to data that can be quantified

Chapter 1 Answers

• Discrete data can be ratio data (False)
• Even ratio data can be measured using an ordinal scale (True)
• Inferential statistics deal with making inferences about a population (True)
• A population can be considered a set of numbers (True)
• The daily sleep duration of housewives in Korea can be considered a population (False)
• Qualitative data cannot be quantified (False)

Population vs Sample

• In inferential statistics, population characteristics are called parameters
• Sample characteristics are called statistics
• Population parameters are: Mean (μ), Variance (σ²), and Standard Deviation (σ)
• Sample statistics are: Mean (X), Variance (S²), and Standard Deviation (S)

Data types

• Data is either Qualitative or Quantitative
• Quantitative data branch out into discrete or continuous data
• Qualitative data branch out into Nominal data or Ordinal data
• Quantitative data branch out into Interval data or Ratio data

Chapter 2

• A frequency distribution table consists of class intervals and frequencies
• Relative frequency is calculated by dividing the class frequency by the total frequency
• Cumulative frequency is the sum of all frequencies up to a specific class
• Frequency distribution may be represented using a graph
• The sum of relative frequencies in a frequency distribution table is always 0
• The first step in creating a frequency distribution table is determining the class width
• Class width is determined by dividing the range of data by a number of class intervals

Chapter 2 Answers

• A frequency distribution table consists of class intervals and frequencies (True)
• Relative frequency is calculated by dividing the class frequency by the total frequency (True)
• Cumulative frequency is the sum of all frequencies up to a specific class (True)
• Frequency distribution may be represented using a graph (True)
• The sum of relative frequencies in a frequency distribution table is always 1 (False)
• The first step in creating a frequency distribution table is determining the class interval (False)
• Class width is determined by dividing the range of data by a number of class intervals (True)

Chapter 3

• The mode is the value located at the center of a dataset arranged in ascending order
• Variance is always greater than or equal to zero
• The interquartile range can have a negative value
• To standardize ordinal data, the percentile should be calculated
• The mean of standardized values of any population is always 0
• Standardization refers to subtracting the mean from a data value and dividing it by the variance
• Degrees of freedom refer to the actual number of observations used in calculating a sample statistic
• The standard deviation has a different unit of measurement than the original data
• If the sample size is increased, the sample variance increases to the population variance

Chapter 3 Answers

• The mode is the median value located at the center of a ascending dataset (False)
• Variance is always greater than or equal to zero (True)
• Interquartile range cannot be a negative value (False)
• To standardize ordinal data, the percentile is calculated (True)
• The mean of standardized values of any population is always 0 (True)
• Standardization subtracts the mean from a data value and divides it by the standard deviation (False)
• Degrees of freedom is the actual number of observations used in calculating a sample statistic (True)
• The standard deviation has the same unit of measurement as the original data (False)
• When the sample size is adequately large, the sample variance is nearly equal to population variance (True)

Degrees of Freedom

• When calculating the sample mean, one degree of freedom is lost as remaining values are dependent on it
• When estimating variance from a sample, divide by n-1 instead of n to correct for bias
• Adjustment makes sample variance an unbiased estimator of population variance
• Case: When the population mean (μ) is known, then the following is true:
  • Formula: s² = 1/n * Σ(xᵢ - μ)²
  • Reason: Deviations are measured from the true mean, so no correction is needed
• Case: When population mean is unknown (typical), then the following is true:
  • Formula: s² = 1/(n-1) * Σ(xᵢ - x̄)²
  • Reason: The sample mean (x̄) replaces the unknown population mean, using one degree of freedom, thus a correction is required.

Chapter 4

• Bivariate data refers to data obtained by examining two variables
• A contingency table cannot be considered a frequency table that displays bivariate data
• The correlation coefficient is the covariance divided by the standard deviations of the two variables
• The sample correlation coefficient is a type of sample statistic
• The values of the sample correlation coefficient and the population correlation coefficient always match
• If the correlation coefficient is 0, no relationship exists between the two populations
• The correlation coefficient takes values between 0 and 1

Chapter 4 Answers

• Bivariate data refers to data obtained by simultaneously examining two variables (True)
• A contingency table can be considered a frequency table that displays bivariate data (False)
• The correlation coefficient is the covariance divided by the standard deviations of the two variables (True)
• The sample correlation coefficient is a type of sample statistic (True)
• The values of the sample correlation coefficient and the population correlation coefficient almost always match (True)
• If the correlation coefficient is 0, there is no relationship between the two populations (True)
• The correlation coefficient takes values between -1 and 1 (False)

Chapter 5

• Conditional probability can be expressed using joint probability and marginal probability
• "No correlation" and "statistical independence" are different concepts
• Bivariate data, where samples are drawn simultaneously from two populations, can be considered a compound event
• The joint probability of two events equals the product of their marginal probabilities, so the two events are considered independent
• A conditional probability can be expressed as the product of a joint probability and a marginal probability, then two events are considered independent

Chapter 5 Answers

• Conditional probability can be expressed using joint probability and marginal probability (True)
• Statistical independence equals 'no correlation' (False)
• In bivariate data where samples are simultaneously drawn from two populations, it can be considered a compound event (True)
• If two events' joint probability equals the product of marginal probabilities, the two events are independent (True)
• If conditional probability can be expressed as the product of a joint probability and marginal probability, the two events are considered independent (False)

Chapter 6

• A probability distribution consists of the sample space of a random variable and the associated probabilities
• Any random variable, when standardized, has a variance of 0
• The expected value of a random variable is conceptually the same as its mean
• The expected value of a random variable is calculated by multiplying each possible value by its corresponding probability and summing the results
• The standard deviation of a random variable is the square root of the average of the squared deviations from the mean
• The variance of a new random variable created by adding two random variables is equal to the sum of their individual variances

Chapter 6 answers

• The sample space of a variable and is related probabilies makes up a probability distribution (True)
• When standardized, any random variable has a variance of 1 (False)
• The expected value of a random variable is conceptually the same as its mean (True)
• The expected value of a random variable is calculated by multiplying each possible value by its corresponding probability and summing the results (True)
• A random variable’s standard deviation is the square root of the average of the squared deviations from the mean (True)
• A new random variable created by adding two random variables has a variance equal to the individual variances only when the two random variables are independent (False)

Chapter 7

• A binomial random variable can be expressed as the sum of Bernoulli random variables
• The variance of a binomial distribution is the mean multiplied by the failure probability
• Independent trials mean that the outcome of one trial does not affect the others
• The standard deviation of a binomial distribution is np(1-p)
• A discrete random variable takes only integer values
• A binomial random variable is a continuous random variable
• A Bernoulli trial is an independent trial with exactly two mutually exclusive outcomes
• If the probability of success is less than 0.5, the binomial distribution has a long tail to the left
• Even if the success probability is 0.5, the binomial distribution may not be symmetric
• The total of all binomial probabilities always sums to 1

Chapter 7 Answers

• A binomial random variable = sum of Bernoulli random variables (True)
• The binomial distribution variance = mean * the failure probability (True)
• Outcomes of independent trials means that one trial doesn't affect the others (True)
• A binomial distribution standard deviation = √np(1-p) (False)
• Discrete random values take on integer values (True)
• A binomial random variable is discrete (False)
• A Bernoulli trial is an independent trial with two mutually exclusive outcomes (True)
• If success prob < 0.5, binomial distribution has a long tail to the right (False)
• If the success probability is 0.5, the binomial distribution will be symmetric. (False)
• total of all binomial probabilities = 1 (True)

Problem Solving

• There are 10 Problem Solving questions for a total of 70 points
• The presenter suggested that you will need a calculator

Chapter 3 Formula

• Population mean μ = ΣΧ₁/Ν
• Sample mean x = ∑x₁/n
• Range = Maximum value – Minimum value
• Interquartile Range (IQR) = Q3 – Q1
• Population Deviation = Χ₁-μ
• Sample Deviation = x₁- x
• Population variance σ² = Σ(Χ₁-μ)²/Ν
• Population standard deviation σ = ν(Σ(Χ₁-μ)²/Ν)
• Sample variance s²= ∑(x₁- x)²/(n-1)
• Sample standard deviation s = v(∑(x-x)²/(n-1))
• Z score = (Χ₁-μ)/σ

C3-1

• Answer the questions based on the given data:
• Data: 6, 8, 10, 10, 10, 12, 14
• Calculate the mean and median
• Determine the range and interquartile range (IQR)
• Standardize the value 10 using Z-scores

C3-1 Answer

• Given Data: 6, 8, 10, 10, 10, 12, 14
• Mean = (6+8+10+10+10+12+14)/7 = 10
• Median = 10
• Range = max-min = 14-6 = 8
• IQR = Q3-Q1 = 12-8 = 4
• Z score of 10 = (Χ₁-μ)/σ = (10-10)/σ = 0

C3-2

• The following table outlines the steps for calculating the sample mean, variance, and standard deviation of variable x.
• You have to fill in the blanks

C3-2 Answer

• Calculations for the sample mean, variance, and standard deviation of variable x.

Chapter 4 Formula

• Population covariance σχγ = ∑((Χ₁-μχ)(Υ₁-μγ))/N
• Sample covariance sxy = ∑((x₁- ㄡ)(y₁- ӯ))/(n-1)
• Correlation σχγ/σχσγ = Sxy/SxSy

C4

• Need to calculate the sample covariance and correlation coefficient between statistics scores of five students
• Need to fill in the blanks

C4 Answer

Chapter 5 Formula

• Marginal Probability
  • P(A) = Probability that event A occurs
• Joint Probability
  • P(A∩B) = Probability that event A and B occur together
• Conditional Probability
  • P(A|B) = Probability that event A occurs given that event B has occurred
  • P(A|B) = P(A ∩ B) / P(B)
  • P(B|A) = P(A ∩ B) / P(A)
  • P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A)
• When events A and B are mutually independent:
  • P(AB) = P(A)
  • P(A ∩ B) = P(A|B) × P(B) = P(A) × P(B)

C5-1

• Determine if there a relationship between "opinion" or "gender" according to the table below
• Determine if there a relationship between "decision" or "ad-type" according to the table below

C5-1 Answer

• Opinion and gender can be related:
  • P(Agree ∩ Male) = 0.42
  • P(Agree) × P(Male) = 0.54 × 0.6 = 0.324
  • P(Agree ∩ Male) ≠ P(Agree) × P(Male)
• Answer: Related
• Decision and action are not related
  • P(Purchase ∩ Offline) = 0.324
  • P(Purchase) × P(Offline) = 0.54 × 0.6 = 0.324
  • P(Purchase n Offline) = P(Purchase) × P(Offline)
• Answer: Not related (Independent)

C5-2

• The test gives 90% right answers for those having the disease, and 90% for the other people who don't
• 10% of the world's population has the disease
• What is the probability of one having the disease, given a test comes back as positive?

C5-2 Answer

• P = Test Positive, PC = Test Negative
• D = Has disease, DC = No disease
• P(D) = 0.1
• P(P|D) = 0.9
• P(Pc| DC) = 0.9
• P(DNP) = P(D)*P(P|D) = 0.1×0.9 = 0.09
• therefore, if a test comes back positive, there is about 50/50 that you actually have the disease

Chapter 6 Formula

• X: Random variable; P(X): Assigned Probability of X; E(X): Expected Value of X; V(X):Variance of X; COV(X,Y): Covariance of X and Y
• Ε(X) = ∑(Χ×P(X))
• E(a) = a
• E(aX) = a × E(X)
• E(X+a) = E(X)+a
• E(X+Y) = E(X)+E(Y)
• V(X) = ∑((X-E(X))2×P(X))
• V(a) = 0
• V(aX) = a2×V(X)
• V(X+Y) = V(X)+V(Y)+2×COV(X,Y)

C6

• Given "Peter problem"
• With a 50% probability, he loses all his money.
• With a 30% probability, he breaks even.
• With a 20% probability, he wins $100.
  • What are the expected value and variance if his bet is 100?
  • What are the expected value and variance if his bet is 1000?
  • Under the assumption that the games are statically independent, what are the expected value and variance of his net gain if he bets 100 in each of 10 races?

C6-1

• need to compute the expected value and variance of the amount Peter can gain from a $100 bet?
• You have to fill in the blanks

C6-1 Answer

• What are the expected value and variance of the amount Peter can gain from a $100 bet?
• E(X)=(-100)×0.5+0×0.3+100×0.2=-30
• V(X)=(-100+30)²×0.5+(0+30)²×0.3+(100+30)²×0.2=6100

Q6-2

• Find the expected value and variance if he bets 1000
  • (Y=10X

Q6-2 Answer

• Y=10X
• (Y)=(10X)=10×E(X)=-300
• V(Y)=V(10X)=100×V(X)=610000

Q6-3

• Games are independent, what is the expected value and variance if he bets 10 times.

Q6-3 Answer

• Let Y=X1+X2+X3+...+X10
• E(Y)=E(X1+X2+X3+...+X10)=10×E(X)=-300
• V(Y)=V(X1+X2+X3+...+X10)=10×V(X)=61000

Chapter 7 formula

• The probability that a desired event (success) occurs X times in n independent Bernoulli trials with success probability p:
• X=X1+X2+…+X (where x=1 if success or x=0 if failure)
• P(X=k) = nCkpk(1-p)n-k
• nCk=n!/(k!x(n-k)!)
• n!=nx(n-1)×(n-2)×…×1
• Ε(X) = ∑(X×P(X)) = np
• V(X) = ∑((X-E(X))2×P(X)) = np(1-p)

C7-1

• Questions about if X~Binomial(n=25, p=0.3) using the binomial probability (with table provided)

C7-1 Answer

• If X~Binomial(n=25, p=0.3), P(X≤12)=0.983
• If X~Binomial(n=25, p=0.3), P(8≤X≤12)=P(X≤12)-P(X≤7)=0.983-0.512=0.471
• If X~Binomial(n=25, p=0.3), P(X≥12)=1-P(X≤11)=1-0.956=0.044
• If X~Binomial(n=25, p=0.3), P(X=12)=P(X≤12)-P(X≤11)=0.983-0.956=0.027

C7-2

• 25% of visiting customers make a purchase at the store; the average amount is $50
• The store expects about 20k visitors per year

C7-2 Answer

• If X~Binomial(n=20,000,p=0.25)
• Expected number of purchasing customers is: E(X)=np=20000×0.25=5000
• By the formula for the variance of a binomial distribution:
• V(X)=np(1-p)=20000×0.25×0.75=3750
• If each person spends on average $50:E(Y)=E(50X)=50×E(X)=50×5000=250000
• E(50X)= 50* Number of Customers
• V(Y)=V(50X)=502×V(X)=502×3750=9375000
• SQRT(V(Y))=SQRT(9375000)=3061.86, Standard Deviation