Wk 2 - BIOSTATS

What is a Z score?

A unitless measure showing how far a value is from the mean, in standard deviations (SD). Formula: Z = \frac{(x - \text{mean})}{SD} x = IQ that is given

What does a Z score allow you to do?

Identify outliers and compare values across different scales or units.

If IQ = 132, Mean = 100, SD = 15, what is Z?

Z = (132 - 100) / 15 = 2.13 → more than 2 SD above the mean → likely an outlier.

What percentage of data lies within 1 SD and 2 SD of the mean in a normal distribution?

68 \% within \pm 1 SD; 95 \% within \pm 2 SD.

When is a data point considered an outlier using Z score?

When Z > +2 or Z < -2.

What does a Z score of 5 mean?

Extremely rare (about 1 in 3 million chance) — almost certainly an outlier.

What is the null hypothesis (H_0)?

It proposes that there is no real difference between the populations being compared.

Example of a null hypothesis for comparing salaries?

“There is no difference in mean graduate salaries between Flinders University and UniSA.”

What is the alternative hypothesis (H_1)?

States there is a difference between the groups.

What does the p value represent?

The probability of obtaining a difference this large (or larger) if the null hypothesis were true.

What does p < 0.05 mean?

There’s less than a 5 \% chance the result is due to random variation → statistically significant → reject H_0.

What does p \ge 0.05 mean?

The difference is likely due to chance → fail to reject H_0 . Not statistically significant.

Give an example of interpreting p values:

p = 0.4 → no significant difference (fail to rejectH_0)

p = 0.02 → significant difference (rejectH_0)

Why do we compare mean values of samples?

To see whether groups come from the same population (no difference) or different populations (true difference).

What causes means to differ between groups?

Random sampling, true population differences, or measurement bias.

Example of comparing means in biology?

Comparing mean blood pressure between control and treatment groups.

What is sampling variability?

The natural variation in sample means when repeatedly sampling from a population.

What is the Standard Error of the Mean (SEM)?

A measure of how precisely the sample mean estimates the population mean.

Formula: SEM = \frac{SD}{\sqrt{n}}

How does sample size affect SEM?

Larger n → smaller SEM → more precise mean.

What is a 95 \% Confidence Interval (CI)?

The range within which the true population mean is expected to lie 95 \% of the time.

Formula: \text{Mean} \pm 1.96 \times SEM

Example interpretation of CI:

Flinders mean salary = \$64,500 (95 \% CI = \$63,100– \$65,900) → the true mean is likely within that range.

What does it mean if two 95 \% CIs overlap?

The difference is probably not significant; confirm with a t-test.

What if 95 \% CIs do not overlap?

The difference is likely significant (but still confirm with a t-test).

How do you use graphs to compare CIs?

Plot mean \pm 95 \% CI; visually check overlap; confirm significance with a t-test.

Example of overlapping CIs interpretation:

Flinders 63.1–65.9k vs UniSA 61.3–64.0k → overlap → not significantly different (p = 0.12).

What is the main purpose of a t-test?

To determine whether two sample means differ more than expected by chance.

When do you use an unpaired t-test?

When comparing independent groups (e.g., Flinders vs UniSA graduates).

When do you use a paired t-test?

When comparing the same individuals before and after treatment (e.g., pre/post diet weight).

What is a two-tailed t-test?

Tests for any difference (higher or lower) between two means — default test type.

What is a one-tailed t-test?

Tests for a specific directional difference (e.g., weight increases).

What is an equal variance (homoscedastic) t-test?

Used when both samples have similar spread (SD).

What is an unequal variance (Welch’s) t-test?

Used when sample spreads are different.

What does p < 0.05 mean in a t-test?

Reject H_0 → groups differ significantly.

What does p \ge 0.05 mean in a t-test?

Fail to reject H_0 → no significant difference detected.

What assumption does a t-test make about data distribution?

That sample means are approximately normally distributed.

When might this assumption fail?

When the data are highly skewed or sample size is very small.

What test can be used if data are not normal?

A non-parametric test, such as the Mann–Whitney U test.

What do error bars represent?

Variability or uncertainty in the estimate of the mean.

What are the main types of error bars?

• 95 \% CI → range of true mean (preferred)

• SEM → precision of sample mean

• SD → spread of data (not precision)

Which error bars are best for comparing group means?

95 \% Confidence Intervals.

Flinders vs UniSA salary comparison — which test?

Unpaired, two-tailed, equal variance t-test → p = 0.12 → Fail to reject H_0 (no significant difference).

Boys vs Girls height — which test?

Unpaired, two-tailed, equal variance t-test → p < 0.0001 → Reject H_0 (boys taller).

Keto diet weight loss — which test and result?

Paired, two-tailed, equal variance t-test → p = 0.21 → Fail to reject H_0 (no significant weight loss).

Appropriate null hypothesis for comparing COVID death age between men and women?

The mean age of COVID fatalities is the same for men and women.

Why is it invalid to perform a t-test on 3 men and 5 women to compare height?

Because the sample size is too small — not enough data for reliable statistical inference.

Significance threshold for p value?

p < 0.05 statistically significant.

What’s the difference between SD and SEM?

SD = variability of data; SEM = precision of mean estimate.

Why is “no difference” not the same as “not significant”?

Failing to reject H_0 means no detected difference, not proof that groups are identical.

When can you use a one-tailed test?

Only when you have a clear, justified directional hypothesis.

What’s a Type II error?

Failing to detect a real difference because of low sample size or power.