HSCI 190 Module 4: T Tests, Chi
Square, and Statistical Analysis
Techniques 2026|2027 Edition!!!
, HSCI 190 Module 4
Learning outcomes:
1. Compare z distributions, t distributions and chi square distributions to identify most
appropriate test for a given scenario
2. Describe assumptions of independent t tests, dependent t tests, non parametric t tests and
categorical analyses to apply the correct statistical test
3. Utilize statistical software to perform t test and chi square calculations
4. Describe statistical methods and findings clearly and effectively
mcne
SECTION 1
Recall: calculating probability using z scores
- Need to know population SD (how much variability in population), which is rare
Creation of student’s t distribution
- Student’s t distribution (variation of z distribution) created to deal with data whose
population SD is unknown
- Created in early 1900’s by the Guinness beer company
Student’s t distribution
- Similarities to z distribution
- Mean = 0
- Symmetric around mean
- Total area = 1
- Difference
- T distribution changes shape as sample size changes
- Large number of samples (< 30) - looks similar to z distribution
- Small number of samples (> 30) - t distribution has heavier tails and lower peak
than z distribution (e.g. flatter and wider); greater variance
- As df increases, t distribution tends towards standard normal distribution
, Calculating t scores
T score Z score
Formula
Degrees of - Defined by degrees of freedom - fairly stable for any sample size
freedom (df) over 30
- df: number of observations - 1;
used to provide more reliable
estimate
- df always noted with the t symbol
(e.g. tn-1)
Sample mean - numerator: x̄ - μ - numerator = X - μ
- specifies X to be x̄ (sample mean)
- μ (population mean)
Denominator - denominator: s / √n - denominator: σ (population SD)
- s (sample SD)
- n (number of observations)
- can be used when population SD
is unknown
T distribution charts
- Can use standard normal chart to find area under the curve like with z scores
- Shape of t distribution changes with degrees of freedom, so the area under the curve also
changes with different sample sizes
Sample calculation:
- Random sample of 12 smokers was obtained from the general population and their
cholesterol levels were measured. The sample mean cholesterol level is 217 mg/100ml,
with a sample SD of 46 mg/100ml. What is the t score associated with the highest 5% of
cholesterol levels?
- Df = 11 (on left side of chart)
- Probability = 0.05 (on top of the chart)
- T value = 1.796 (where they meet)
Square, and Statistical Analysis
Techniques 2026|2027 Edition!!!
, HSCI 190 Module 4
Learning outcomes:
1. Compare z distributions, t distributions and chi square distributions to identify most
appropriate test for a given scenario
2. Describe assumptions of independent t tests, dependent t tests, non parametric t tests and
categorical analyses to apply the correct statistical test
3. Utilize statistical software to perform t test and chi square calculations
4. Describe statistical methods and findings clearly and effectively
mcne
SECTION 1
Recall: calculating probability using z scores
- Need to know population SD (how much variability in population), which is rare
Creation of student’s t distribution
- Student’s t distribution (variation of z distribution) created to deal with data whose
population SD is unknown
- Created in early 1900’s by the Guinness beer company
Student’s t distribution
- Similarities to z distribution
- Mean = 0
- Symmetric around mean
- Total area = 1
- Difference
- T distribution changes shape as sample size changes
- Large number of samples (< 30) - looks similar to z distribution
- Small number of samples (> 30) - t distribution has heavier tails and lower peak
than z distribution (e.g. flatter and wider); greater variance
- As df increases, t distribution tends towards standard normal distribution
, Calculating t scores
T score Z score
Formula
Degrees of - Defined by degrees of freedom - fairly stable for any sample size
freedom (df) over 30
- df: number of observations - 1;
used to provide more reliable
estimate
- df always noted with the t symbol
(e.g. tn-1)
Sample mean - numerator: x̄ - μ - numerator = X - μ
- specifies X to be x̄ (sample mean)
- μ (population mean)
Denominator - denominator: s / √n - denominator: σ (population SD)
- s (sample SD)
- n (number of observations)
- can be used when population SD
is unknown
T distribution charts
- Can use standard normal chart to find area under the curve like with z scores
- Shape of t distribution changes with degrees of freedom, so the area under the curve also
changes with different sample sizes
Sample calculation:
- Random sample of 12 smokers was obtained from the general population and their
cholesterol levels were measured. The sample mean cholesterol level is 217 mg/100ml,
with a sample SD of 46 mg/100ml. What is the t score associated with the highest 5% of
cholesterol levels?
- Df = 11 (on left side of chart)
- Probability = 0.05 (on top of the chart)
- T value = 1.796 (where they meet)
