ISA 225

Planted 02021-08-17

Notes for Principles of Business Analytics - ISA 225

Prepping resources:

• Video
  • MIT Introduction to Statistics
  • Introduction to Business Statistics
  • Business Statistics (STAT101)
  • Professor Mitchell - Business Statistics
  • Professor Mitchell - Statistics
• W3 Schools Statistics

Review of Statistical Inference

• Population:
  • N Size
  • μ Mean
  • 𝜎 Standard Deviation
  • P Ratio (proportion)
• Sample:
  • n Size
  • p̂ Ratio (proportion) (sample portion size / sample size)
  • x̄ Mean
  • Portion Size

Checks

• Checks:
  • Independent:
    • ✅ Randomly selected
    • ✅ 10% condition: n < 10% of P
  • Normality:
    • ✅ Success/Failure:
      • ✅ np >= 10
      • ✅ n(1-p) >= 10
    • ✅ Expect at least 10 success and 10 failures

Common Z-Values

Determining Sample Size

Notes

The t-distribution describes the statistical properties of sample means that are estimated from small samples; the standard normal distribution is used for large samples.

Hypothesis test for population proportion and mean

A hypothesis is a claim about a population parameter (proportion, mean)

Steps to compute a hypothesis test:

1. State hypothesis
2. Calculate test statistics
3. Find p-value
4. Make conclusions based on p-value

The null hypothesis H_o, is the starting assumption (nothing has changed).

The alternative hypothesis, or H_a is a claim the population parameter value differs from the null hypothesis. It can take these different forms depending on what you want to test (H_a):

Left-tailed hypothesis test: \(H_a: populationParameter \lt claimedValue\)

Right-tailed hypothesis test: \(H_a: populationParameter \gt claimedValue\)

Two-tailed hypothesis test: \(H_a: populationParameter \neq claimedValue\)

Step 2: Calculate the Test Statistics

• Test statistics about population proportion P (One-prop Z-test)
  • \[Z-score = {p̂ - P \over \sqrt{p(1-p) \over n} }\]
• Test statistics about population mean μ (One-sample test)
  1. When the population 𝜎 Standard Deviation is known: (One-sample Z-test)
     • \[Z-score = {sampleMean - claimedValue \over{populationMean \over \sqrt{sampleSize}} }\]
     • \[Z-score = {x̄ - μ \over{𝜎 \over \sqrt{n}} }\]
  2. When the population 𝜎 Standard Deviation is unknown (One-same t test / student t-test)
     • \[t = {sampleMean - claimedValue \over{sampleStdDev \over \sqrt{sampleSize}} }\]
     • \[t = {x̄ - μ_0 \over{s \over \sqrt{n}} }\]

Step 4: Make conclusion based on p-value

Compare p-value with significance level α (always given before test). The smaller α, the more accurate the test is.

1. Type I errors, the null hypothesis is true, but we reject it (false negative)
2. Type II errors, the null hypothesis is false, but we fail to reject it (false positive)

If p-value < α, then reject null hypothesis, we have enough evidence to support H_a.

If p-value > α, then do not reject null hypothesis, we do not have enough evidence to support H_a.

Comparing Two Population Parameters

Two Sample t-test (comparing two population means)

1. State hypothesis
2. Check assumptions and calculate test statistics
3. Find p-value based on test statistics
4. Make conclusion based on p-value

Since population standard deviations are unknown, we use the standard errors instead:

Confidence Interval for Difference between Two Population Means

Two sample Z-interval (when \(𝜎_1\) and \(𝜎_2\) are known)

Two sample Z-interval (when \(𝜎_1\) and \(𝜎_2\) are unknown)

The \(t^*_{df, a/2}\) here depends on the confidence level 100(1-α)% and the calculated df.

Interpretation of C.I. is similar to one-sample test

Chi-Square Tests

• One variable?
  • Goodness of Fit Test
  • H₀: model fits data
  • H_a: model does not fit data
• Two variables?
  • Test for independence
  • H₀: variables are independent
  • H_a: variables are not independent

Goodness-of-Fit Tests (one variable)

A χ2 goodness of fit test is applied when you have one categorical variable from a single population.

1. State the hypothesis:
   • H₀: model fits. (hypothesized model fits the sample we collected)
   • H_a: model doesn’t fit. (hypothesized model doesn’t fit the sample we collected)
2. Assumptions and Test Statistics:
   • Assumptions:
   • Counted Data Condition – The data must be counts for the categories of a single categorical variable.
   • Independence Assumption – The counts should be independent of each other.
   • Randomization Condition – The counted individuals should be a random sample of the population.
   • Sample Size Assumption – We expect at least 5 individuals per cell. - Test statistics:
   • \[{\chi^2 = \sum_{allCells} {(Obs - Exp)^2\over{Exp} } }\]
3. Find p-value based on the test statistics
   • Df= (#cells -1), use the χ2 table, fix the line of df, then with the test statistics to find the corresponding p-value, which is the right-tail probability of the test statistics.
   • (or by technology) p-value= P(χ2 > test statistics)
4. Make Conclusions based the p-value
   • If p-value < α, reject the H₀, which means the hypothesized model doesn’t fit the sampled data.
   • If p-value > α, fail to reject H₀, we do not have significant evidence to say the model doesn’t fit the sampled data.

Chi-Square test for Independence (two variables)

1. State the hypothesis:
   • H₀: variables are independent.
   • H_a: variables are not independent.
2. Assumptions and Test Statistics:
   • Assumptions:
   • Counted Data Condition – The data must be counts for the categories of a single categorical variable.
   • Randomization Condition – The counted individuals should be a random sample of the population.
   • Sample Size Assumption – We expect at least 5 individuals per cell. - Test statistics:
   • \[{\chi^2 = \sum_{allCells} {(Obs - Exp)^2\over{Exp} } }\]
   • Assuming H0 is true, which means that the variables are independent.
     • \[{Exp_{ij} = {totalRow_i \times totalCol_i}\over{tableTotal} }\]
3. Find p-value based on the test statistics
   • Df= (# of rows -1)×(# of cols-1), use the χ2 table, fix the line of df, then with the test statistics to find the corresponding p-value, which is the right-tail probability of the test statistics.
   • p-value= P(χ2 > test statistics)
4. Make Conclusions based the p-value
   • If p-value < α, reject the H₀, which means the two variables are not independent.
   • If p-value > α, fail to reject H₀, we do not have significant evidence to say the two variables are not independent.

Simple regression (linear)

Sample regression line:

• ŷ the predicted value of response variable (y), when x is given as a specific value.
• b₀ the sample y-intercept
• b₁ the sample slope
• r the correlation coefficient
  • value from -1 to 1
  • closer to 0, the weaker relationship they have
• r² the proportion of the observed variation in y that can be accounted for by x, or modeling by x.
  • shows how well the model fits the data
  • value from 0 to 1
  • closer is to 1, the stronger the regression relationship.
• \({e = y - \hat{y}}\) the residual, difference between predicted (ŷ), and observed (y) values
• Ɛ the population mean residual
• μ_y the population mean of y at a given value of x
• 𝛽₀ the population mean value of Y when X = 0
• 𝛽₁ the population mean value of Y for each unit increase in X

Step 1: State the hypothesis

• \[H_o: \beta_0 = 0\]
• \[H_a: \beta_1 \ne 0\]

Step 2: Test statistics

• df=n-2
• Se is called “Root Mean Squared Error”
• \[{t-test = {b_1-\beta_1\over{SE(b_1)}}}\]
• Confidence interval = \({b_1 \pm t^*_{df,{\alpha\over{2}}} \times SE(b_1) }\)

Regression Assumptions

1. Linearity Assumption: scatterplot looks like a linear relationship
2. Independence Assumption: randomly selected
3. Equal Variance Assumption: scatterplot equally spread out, no clumping and spread around the line in residual plot is reasonably consistent at line 0
4. Normal Population Assumption: the residuals satisfy the Nearly Normal Condition and

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