Quantitative Analysis I

The Outline...

The foundational class for critically analyzing and describing data for a specific objective.

 (Reference:  Statistical Techniques in Business and Economics, 14th Edition.  Lind, Marchal, Wathen.  2010)

1. Statistics

   1. Types of Statistics

      1. Descriptive Statistics:  organizing, summarizing and presenting data  in an informative way.

      2. Inferential Statistics:  methods used to estimate a property of a population on a basis of a sample.

         1. Population

         2. Sample

   2. Types of Variables

• • 1. Qualitative

    2. Quantitative

       1. Discrete

       2. Continuous

1. 3. Levels of Measurement

• • 1. Nominal:  only be classified and counted with no natural order to outcomes

    2. Ordinal:  represented  by sets of labels or names (high, medium, low) that have relative order

    3. Interval:  equal differences are represented by equal differences in the measurements 

    4. Ratio: same as interval, but difference between two numbers is meaningful.

2. Describing Data- pictorial

• 1. Constructing a Frequency Table

  2. Relative Class Frequencies:  mutually exclusive, percentage of the whole

  3. Graphic Presentation of Qualitative Data

     1. Bar Chart

     2. Pie Chart

3. Describing Data- numerical measures

• 1. Population Mean

  2. Parameter- characteristic of a population

  3. Sample Mean

  4. Median- midpoint of values

  5. Statistic- a characteristic of a sample

  6. Mode- most frequently occurring observation

  7. Relative positions of the mean, median, and mode

     1. symmetric distribution (normal)

     2. Skewed distribution is NOT symmetrical

     3. Positive Skewness:, the mean is the largest of the three, median, then mode is smallest

     4. Negative Skewness:  the mean is the smallest, median, then the mode is the largest

  8. Measures of Dispersion

• • 1. Range

    2. Mean Deviation

    3. Variance

    4. Standard Deviation

    5. Other Measures

       1. Quartiles

       2. Deciles

       3. Percentiles

• 9. Interpretation and Uses of the Standard Deviation

• • 1. Chebyshev's Theorem

       1. 75% of data lie between +- 2 Standard Deviation

       2. 89% of data lie between +- 3 Standard Deviation

       3. 96% of data lie between +- 5 Standard Deviation

    2. Empirical Rule

• • • 1. 68% of data lie between +- 1 Standard Deviation

      2. 95% of data lie between +- 2 Standard Deviation

      3. 99.7% of data lie between +- Standard Deviation

4. Continuous Probability Distributions

• 1. Types of Distributions

     1. Random is discrete (0 &1)

        1. Binomial, hypergeometric, Poisson

        2. Probability distribution is a table that links each possible value that a random variable can assume with its probability of occurrence

     2. Continuous probability distributions

• • • 1. random variable is continuous

      2. types:

         1. uniform

         2. normal

         3. Student's t

         4. chi-square

         5. F distributions

      3. Infinite number of values

      4. Probability distribution is the graph of an equation

• 2. Standard Normal Probability Distribution

• • 1. normal distribution

    2. AKA the z-distribution

    3. z-value (also called z-score) is the signed distance between a selected value, designated X, and the population mean µ, divided by the population standard deviation, σ.  

    4. sample size >30

• 3. t-Distribution

• • 1. symmetrical

    2. sample size <30

    3. STD DEV differ according to sample size.

    4. defined by the degrees of freedom (df = n-1)

• 4. f-Distribution

• • 1. must be positive

    2. positively skewed

    3. usually used to compare two variances

• 5. Chi-Square (χ2)  Distribution

• • 1. positively skewed

    2. defined by df (df = n-1)

    3. usually used to compare expected versus observed value

5. Estimation and Confidence Intervals

• 1. Point Estimate

     1. single value (point) derived from a sample and used to estimate a population value.

     2. estimate of the population parameter, from that estimate, we build our confidence interval

     3. Sample mean might be the best estimate of the population parameter

  2. Confidence Interval Estimates

• • 1. a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability.

    2. The specified probability is called the level of confidence.

    3. C.I. = point estimate ± margin of error.

• 3. Factors affecting confidence interval estimates:

• • 1. The sample size, n.

    2. The variability in the population, σ, usually estimated by s.

    3. Desired level of confidence.

6. One-Sample Test of Hypothesis

• • 1. The Hypothesis: A statement made about a population of interested developed for the purpose of testing

    2. Testing:  A procedure based on sample evidence and probagility theory to determine whether the hypothesis is a reasonable statement.

    3. NULL HYPOTHESIS (H0) - A statement about the value of a population parameter developed for the purpose of testing; the statement assumed to be true unless sufficient evidence from sample data show that it is false.

       1. ALTERNATE HYPOTHESIS (Ha or H1) - A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.

    4. Rules about the two

• • • 1. H0: null hypothesis

      2. Ha alternate hypothesis.

      3. H0 and H1 are mutually exclusive and collectively exhaustive.

      4. H0 is always presumed to be true

      5. Ha (H1) has the burden of proof

      6. “Reject H0” or “Fail to reject H0”

      7. Never “Accept H0

         1. Population is always changing and we never know what it is.

         2. Philosophical: we can only take the current data today; can’t really say for sure that something works for all of the population.

      8. Equality is always part of H0 (e.g. “=” , “≥” , “≤”)

• 1. Errors

     1. Type I Error:  Defined as the probability of rejecting the null hypothesis when it is actually true

        1. 1. This is denoted by the Greek letter “a”

           2.  Also known as the significance level of a test, or “alpha level”

           3. 10%- predicting that the hypothesis is correct 90% of the time (surveys & behavioral health)

           4. 5%- predicting that the hypothesis is correct 95% of the time (management studies)

           5. 1%- predicting that the hypothesis is correct 99% of the time.  (clinical studies)

     2. Type II Error:  Defiend as the probability of failing to reject the null hypothesis when it is actually false.

• • • 1. This is denoted by the Greek letter “β”

7. Hypothesis Testing Steps

• 1. Step 1: State the null (H0) and alternate (Ha) hypotheses

  2. Step 2: Select level of significance (α) or probability of rejecting the null hypothesis

  3. Step 3: Determine which statistical test to use (e.g., one sample t-test, ANOVA, etc.)

  4. Step 4: Formulate decision criteria

     1. Calculate critical values for the selected α using appropriate distribution

     2. State decision rule for when to reject H0

  5. Step 5: Evaluate sample, make decision and interpret results

• • 1. Evaluate sample statistic against the critical values

    2. Reject H0 and accept Ha OR Fail to reject Ho

    3. Interpret results

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Decision Tree to Determine the

Appropriate Test Statistic in

Step 3 Below.

(Click on the picture for an expanded view)

Other Helps for Statistics

1.  What statistical analysis tool should I use? 

Hypotheis Testing Steps

Step 1: State the null (H0) and alternate (Ha) hypotheses

Step 2: Select level of significance (α) or probability of rejecting the null hypothesis

Step 3: Determine which statistical test to use (e.g., one sample t-test, ANOVA, etc.)

Step 4: Formulate decision criteria

• 1. Calculate critical values for the selected α using appropriate distribution

  2. State decision rule for when to reject H0

Step 5: Evaluate sample, make decision and interpret results

• 1. Evaluate sample statistic against the critical values

  2. Reject H0 and accept Ha OR Fail to reject Ho

  3. Interpret results