Hypothesis Testing: Types of Hypothesis Tests

Question 1

T-Test

Answer 1

• tests for a Student’s t-distribution
• In a normally distributed population where standard deviation is unknown and sample size is comparatively small
  
  • Pared t-test compare two samples

Question 2

One Sample T Hypothesis Test (Student’s T Test)

Answer 2

• Used to resolve hypothesis tests around comparing process means.
  
  • The underlying chart makes use of the T distribution
• Student’s T distribution leverages the T distribution and is used for finding confidence intervals for the population mean when the same size is less than 30 and the population standard deviation is unknown
  
  • What is a T Distribution?
    
    • The T distribution and T tests are used to determine the likelihood of various conditions occurring for small sample sizes (< = 30)

Question 3

What is the T Distribution Table?

Answer 3

• The t distribution table values are critical values of the t distribution.
• The column header are the t distribution probabilities (alpha)
• The row names are the degrees of freedom (df)
• Student t tables gives the probability that the absolute t value with a given degree of freedom lies above the tabulate value

Question 4

Student’s T Distribution Example

Answer 4

Student’s T distribution can be used for finding confidence intervals for the population mean when the sample size is less than 30 and the population standard deviation is unknown.

Confidence of a mean 1 Example

• At a soda bottling factor, the normal filling specification (goal) is 16 fluid ounces. A sample of 20 bottles is tested with the following results: x = 16.13 ox and s = 0.24 fl oz. What interval would allow you to say, with 95% confidence, that the interval contains the actual mean of the filling process?

Question 5

What is a One Sample T Hypothesis Test?

Answer 5

• The One Sample T Hypothesis Test (Student’s T Test) allows to compare the (small) population mean to some hypothesized value or one sample mean if they are significantly different.
  
  • For example, if we know the average weight of chickens in a farm is 3lbs, and compare the average weight of sample black hens to the population mean

Question 6

When would you use a One Sample T Hypothesis Test?

Answer 6

• One sample t test is a type of parametric test because the assumption is samples are randomly distributed
• It tests whether the sample mean is significantly different than a population mean when the standard deviation of the population is unknown
• T test is used when the population standard deviation is unknown and the sample size is below 30, otherwise use Z-test (for known variance)

Question 7

Assumptions of One Sample T Hypothesis Test

Answer 7

• Data is continuous and quantitative at the scale level (ratio or interval data)
• The sample should be randomly selected from the population
• Samples are independent to each other
• Data should follow normal probability distribution
• Assumes it don’t have extreme outliers in the dependent variable

Question 8

One Sample T Hypothesis Test Formula

Answer 8

• Where
  
  • x̅ is observed sample mean
  • μ₀ is population mean
  • s is sample standard deviation
  • n is the number of the observations in the sample

Question 9

Steps to Calculate One Sample T Hypothesis Test

Answer 9

1. State the claim for the test and determine the null hypothesis and alternative hypothesis
2. Determine the level of significance
3. Calculate the degrees of freedom (df = N - 1)
4. Find the critical value
5. Calculate the test statistics
6. Make a decision, the null hypothesis will be rejected if the test statistic is in the rejection region
7. Finally, interpret the decision in the context of the original claim

Question 10

Chi Square Distribution

Answer 10

Best Method to test a population variance against a known or assumed value of the population variance.

Continuous distribution with degrees of freedom

Uses goodness of fit of a distribution of data, whether data series are independent, and for estimating confidences surrounding variance and standard deviation for a random variable from a normal distribution

Question 11

ANOVA Analysis of Variation

Answer 11

• Parametric statistical technique used to compare the data sets
• ANOVA is best applied where more than 2 populations or samples are meant to be compared
• It is used to text statistical significance of the relationship between a dependent variable (Y) and a single or multiple independent variables (X’s)

Question 12

Types of ANOVA

Answer 12

• One-Way
  
  • Measures single factor from multiple scores
  • Uses only one technician / one measurer
• Two-Way (without replicates)
  
  • Measures 2 factors
  • Uses only one technician (unless technicians are one of the factors)
• Two-Way (with replicates)
  
  • Measures 2 factors, but has multiple repetitions of each combination
  • Uses only one technician (unless the technicians are one of the factors)

Question 13

ANOVA Sum of Squares Correction Factor

Answer 13

• Grand total of all runs (G) = ΣX
• N= Total number of runs
• Correction factor (CF)= (ΣX)² /N = (G)² /N

Question 14

Terms used in ANOVA

Answer 14

• Degrees of Freedom (df): The number of independent conclusions that can be drawn from the data
• SS_Factor: It measures the variation of each group mean to the overall mean across all groups
• SS_Error: It measures the variation of each observation within each factor level to the mean of the level
• Main effect: A main effect is the effect where the performance of one variable considered in isolation by neglecting other variables in the study
• Interaction: An interaction effect occurs where the effect of one variable is different across levels of one or more other variables
• Mean Square Error (MSE): The mean square error (MSE) is divide the sum of squares of the residual error by the degrees of freedom
• F-test Statistic: The null hypothesis that the category means are equal in the population is tested by F statistic based on the ratio of mean square related to X and mean square related to error
• P-Value: It is the smallest level of significance that would lead to rejection of the null hypothesis (H0). If α = 0.05 and the p-value ≤ 0.05, then reject the null hypothesis, similarly if p-value > 0.05, then fail to reject the null hypothesis.

Question 15

Analysis of Variance (ANOVA) has three types:

Answer 15

1. One-Way Analysis
2. Two-Way Analysis
3. K-Way Analysis: L-Way ANOVA can be two-way ANOVA or three-way ANOVA or multiple ANOVA

Question 16

Assumptions of One-Way ANOVA

Answer 16

• One-way ANOVA (one-way analysis of variance) is a statistical method to compare means of two or more populations
• Assumptions
  
  • The sample data drawn from k populations are unbiased and representative
  • The data of k populations are continuous
  • The data of k populations are normally distributed
  • The variation within each factor or factor treatment combination is the same, and hence it is also called homogeneity of variance
  • Finally, the variances of k populations are equal

Question 17

Steps for Computing One-Way ANOAV

Answer 17

1. Establish the hypotheses. H₀: µ1= µ2= µ3 and H₁: At least one of the group means is different from the others.
2. In ANOVA, the total variance is subdivide into two independent variance; the variance due to the treatment and variance due to random error
3. Calculate SS_between
4. Calculate SS_Within
5. Calculate SS_Total
6. SS_T = SS_b + SS_w
7. Calculate the ANOVA table with degrees of freedom (df), calculate for the group, error and total sum of squares.
8. SS_b= sum of squares between treatments
9. SS_w= sum of squares due to error
10. MS_b= mean square for treatments
11. MS_W= mean square for error
12. SS_T= total sum of squares
13. T= number of treatment levels
14. n= number of runs at a particular level
15. N= total number of runs
16. F= the calculate F statistic with k-1 and N-k are the degrees of freedom
17. Determine the critical value. F critical value from the F distribution table.

• Finally, Draw the statistical conclusion. If Fcalc< Fcritical fail to reject the null hypothesis and if Fcalc > Fcritical, reject the null hypothesis.