Exam #1
- What are the two types of statistics? (5pts.)
The two types of statistics are descriptive and inferential. In descriptive statistics, the data of a population is summarized graphically or numerically through derivation of statistics related to central tendency and dispersion around the same. The shape and nature of distribution is also used in descriptive statistics.
In inferential statistics, the parameters of a population are inferred according to sample statistics. This involves the estimation of statistics of a group allowing for approximation of information data relating to the entire population. In this type of statistics, modeling are used to generate mathematical equations which illustrate how two variable interrelate.
- What is the relationship between a population and a sample? (5 pts.)
Population refers to all the possible outcomes or measurements that may be obtained for use in a study. On the other hand, sample is a section of the population that represents the entire population from which it was obtained. Population contains a sample and it is always bigger in number as compared to the sample. In picking a sample, it is assumed that it has similar attributes as those of an entire population.
- What are the two types of variables we work with in statistics? (5 pts.)
The two types of variables being used in statistics are independent and dependent variables. Independent variable are normally the input which may be controlled or manipulated so as to alter the outcome which is dependent variable. During a study or experiment, the values of independent variables are known. On the other hand, dependent variables represent the outcome of an experiment or study and they are determined by the independent variables.
- What is the difference between a continuous variable and a discrete variable? Give an example of each. (5 pts.)
Continuous variables refer to those, which can have values between two values whereas discrete variables can take definite values, normally whole numbers. Continuous variables may be presented in fractions and decimals.
5 3 8 12
10 6 7 14
7 3 4 12
- Calculate the following:
- Population Mean
Population Mean, μ=i=1Nxi
5+3+8+12+10+6+7+14+7+3+4+1212=7.5833
- Sample Mean (use bottom row as the sample)
Sample Mean, μ=i=1Nxi
7+3+4+124=6.5
- Population Variance (Pretend only middle row is the Population while using Population Mean you constructed in Part A and make the number of observations for the denominator that which corresponds to the middle row)
Population Variance, σ2=1Ni=1Nxi-μ2
Replacing i=1Nxi-μ2 in the equation;
Population Variance, σ2=50.542364=12.6356
- Population Standard Deviation (Pretend only middle row is Population)
Population Standard Deviation, σ=σ2=12.6356=3.5547
- Sample Variance (Use bottom row as the sample)
Population Variance, σ2=1N-1i=1Nxi-μ2
Replacing i=1Nxi-μ2 in the equation;
Population Variance, σ2=494-1=16.3333
- Sample Standard Deviation (use bottom row as the sample)
Population Standard Deviation, σ=σ2=16.3333=4.0414
- Median (use population)
3 3 4 5 6 7 7 8 10 12 12 14
Median=7+72=142=7
- Range (use population)
Range=14-3=11
- Use the concept of mutual exclusivity to explain why the Special and General Rules of Addition are different (10 pts.). What are the rules of addition specifically used to calculate (10 pts.)? If bowling and going to the movies are mutually exclusive, with P(bowling)=0.5 and P(movies)=0.15, what is the probability of going bowling or to the movies (10 pts.)? (30 pts. total)
Mutual exclusivity refers to an attribute of probability and statistics where the existence of one event rules out the presence of another one. In other words, the two events cannot coexist. In the specific Rules of Addition, if two events A and B are mutually exclusive,
P(A or B)=P(A)+P(B), conversely, in general Addition Rule which related to Non-Mutually Exclusive events, P(A or B)=P(A)+ P(B) – P (A and B) to eliminate the overlap.
Probability of going bowling or to the movies=Pbowling+Pmovies=0.5+0.15=0.65
- Suppose that you are in charge of discovering which Asset Back Securities (ABSs) primarily consist of liar's loans (loans that were fraudulently originated). After doing so research, your subjective probability estimate is that 200/500 ABSs primarily contain liar's loans. If this is correct, calculate the probability of having the first two randomly selected ABSs that are examined primarily contain liar's loans. (10 pts.)
Probability 1st two ABSs has liar's loans=P(1st has liar's loans)×P(2nd has liar's loan)
Probability 1st two ABSs has liar's loans=200500×199499=0.4008≅0.4
Exam #2
- What are the three primary characteristics of a normal probability distribution? What is the standard normal probability distribution? What is the formula for converting a normal probability distribution into the standard normal probability distribution? In your own words, explain what a z score is.
Characteristics of normal distribution:
- The shape of normal curve is similar on either side of the mean, µ, which is at the middle
- The total area covered by the curve equals to one
- The shape of the curve is determined by standard deviation and mean only
Standard normal probability distribution:
Standard normal probability distribution is a normal distribution having a standard deviation of 1 and mean of 0
Converting normal probability distribution into the standard normal probability distribution:
Considering x as a continuous random variable with normal distribution and having mean and standard deviation represented by µ and ???? respectively, normal curve has the following equation:
y=1σ2πe-x-μ2/2σ2
When the mean, µ=0 and standard deviation ????=1, the normal distribution becomes standard normal distribution represented by the equation;
y=e-x2/22π
Z-score refers to a measure indicating the standard deviations a value occupies as measured from mean. X values can be transformed into z-score using the relationship;
z=Value, x-Mean, μStandard Deviation, σ
- Suppose that a doctor’s office wants to know how many flu cases to expect this month. From past experience, they know that the population mean number of cases is 10 with a population standard deviation of 2. What is the probability that there will be between 7 and 12 flu cases this month?
Corresponding values of z-scores;
z1=7-102=-1.5 and z2=12-102=1
P7<x<12=P-1.5<z<1
=Pz<1-P(z<-1.5)
0.8413-0.0668=0.7745
- Suppose that the distribution of total returns on the S&P 500 is normal with a mean of 10% and a standard deviation of 1%. What is the probability of purchasing a stock that yields less than 7.9%?
Mean, µ=0.1
Standard Deviation, ????=0.01
Value, x less than 0.079
z-score corresponding to 7.9%; z=0.079-0.10.01=-2.1
Px<0.079=Pz<-2.1
Therefore, the probability of purchasing a stock the yields less than 7.9% is 0.0179 or 1.79%
- Suppose that the distribution of total returns on the S&P 500 is normal with a mean of 10% and a standard deviation of 1%. What is the probability of purchasing a stock that yields between 11.5% and 12.5%?
Corresponding values of z-scores;
z1=0.115-0.10.01=1.5 and z2=0.125-0.10.01=2.5
P11.5%<x<12.5%=P1.5<z<2.5
=Pz<2.5-P(z<1.5)
0.9938-0.9332=0.0606
Therefore, the probability of purchasing a stock the yields between 11.5% and 12.55 is 0.0606 or 6.06%.
- In your own words, what is the significance of the Sampling Distribution of the Sample Mean and the Central Limit Theorem?
The Central Limit Theorem explains the features of the group of the means that has been formed from the means of a vast number of (N)-sized random samples, all of which are obtained from a specific parent population.
A Central Limit Theorem’s outcome is that if measurements of a given quantity are averaged, the average’s distribution inclines to a normal one. Additionally, if, in reality, a calculated variable is a collection of various uncorrelated variables, which are all contaminated with an arbitrary error of whichever distribution, the measurements are liable to contamination with an arbitrary error, which is usually distributed with increase in the number of the variables.
Work Cited
Anderson, David, Dennis Sweeney, and Thomas Williams. Essentials of statistics for business
and economics. Mason, OH: Cengage Learning, 2010.
