Biostatistics Midterm 1

What does a confidence interval tell us?

A confidence interval gives a range where we believe the population mean falls.

How do I know if two group means are significantly different using confidence intervals?

If the confidence intervals overlap → not significant.
If the confidence intervals do not overlap → significant difference.

What is the formula for a t confidence interval?

s = sample standard deviation
n = sample size
t= t-value from table

How do I find degrees of freedom for a t confidence interval?

Degrees of freedom = n−1

When do we use t instead of z?

Use t when the population standard deviation is unknown and you are using the sample standard deviation.
T confidence intervals are usually wider than Z confidence intervals because t-values are larger.

What are the common Z values?

90% = 1.645
95% = 1.96
99% = 2.576
These are used when the population standard deviation is known.

What does the p-value help us decide?

The p-value tells us whether to reject or fail to reject the null hypothesis.
If p < 0.05 → reject null.
If p ≥ 0.05 → fail to reject null.

What is the null hypothesis?

The null hypothesis usually says there is no difference between groups.

What is the alternative hypothesis?

The alternative hypothesis says there is a difference between groups

When do I use an independent samples t-test?

Use it when comparing the means of two independent groups.

When do I use a paired samples t-test?

Use it when the two measurements are connected or from the same person.

How do I know if data is independent or paired?

Independent = two separate groups.
Paired = same people measured twice or matched pairs.

When do I use Student’s t-test or Welch’s t-test?

Student’s t-test = equal variances.
Welch’s t-test = unequal variances.
Mann-Whitney U compares medians, not means

What does the homogeneity test check?

It checks if the two groups have equal variances.
If p ≥ 0.05 → equal variances → use Student’s t-test.
If p < 0.05 → unequal variances → use Welch’s t-test

What is the difference between descriptive and statistical difference?

Descriptive difference = the means look different.
Statistical difference = the test says the difference is significant using the p-value.

How do I write a conclusion if p < 0.05?

Since p < 0.05, we reject the null hypothesis. There is a significant difference in means between the two groups.

How do I write a conclusion if p ≥ 0.05?

Since p ≥ 0.05, we fail to reject the null hypothesis. There is not a significant difference in means between the two groups.

What is a two-tailed test?

A two-tailed test asks if two means are different, without saying which one is higher or lower. This is usually what is used for independent samples tests.

What is a left-tailed test?

A left-tailed test checks if group 1 is less than group 2.
Example: μ1<μ2

What is a right-tailed test?

A right-tailed test checks if group 1 is greater than group 2.
Example: μ1>μ2

When do I use ANOVA?

Use ANOVA when comparing the means of three or more groups.
If there are only two groups, use a t-test.

What variables do I need for one-way ANOVA?

Dependent variable = numerical variable you can average.
Independent variable/factor = categorical variable with 3 or more groups.

What are the ANOVA hypotheses?

Null: all group means are equal.
Alternative: at least two group means are different.

How do I interpret ANOVA p-value?

If p < 0.05 → significant ANOVA → at least two groups are different.
If p ≥ 0.05 → no significant difference between group means.

What does a significant ANOVA not tell us?

It tells us at least two groups are different, but it does not tell us exactly which groups are different. For that, use post hoc tests.

When do I use Tukey?

Use Tukey after a significant ANOVA to see which specific groups are different from each other.

What test statistic does ANOVA use?

ANOVA uses an F test statistic.
F = mean square between groups ÷ mean square within groups.

How do I find ANOVA degrees of freedom?

Between groups df = k−1
Within groups df = N−k
k = number of groups
N = total sample size.

What is a Type I error?

A Type I error is saying there is a difference when there really is not. It is a false positive. Alpha = 0.05 means a 5% chance of Type I error.

Why use ANOVA instead of many t-tests?

Running many t-tests increases the chance of a Type I error. ANOVA lets us test three or more groups with one main test.

What is sampling variation?

Different random samples from the same population can give different sample means. The sample means usually stay close to the population mean. In the GMAT example, the population mean was 520.78, and different samples had different means.

What is an estimator?

An estimator is a statistic from a sample used to estimate a population parameter. Example: sample mean estimates the population mean.

What is an estimate?

An estimate is the actual value of the estimator from one sample. Example: if your sample mean is 504.6, then 504.6 is the estimate.

What is the difference between a population parameter and a sample statistic?

Population parameters usually use Greek letters. Sample statistics usually use Roman letters.
Examples:
Population mean = μ
Sample mean = x̄
Population standard deviation = σ
Sample standard deviation = s

What is a sampling distribution?

A sampling distribution is the probability distribution of all possible values a statistic can have from random samples of the same size.

What is sampling error?

Sampling error is the difference between the sample statistic and the true population parameter. It happens because samples vary.

What is bias?

Bias is a systematic difference between the expected value of the estimator and the true population parameter. Sampling error is random, but bias is systematic.

What is an unbiased estimator?

An unbiased estimator does not overstate or understate the true parameter on average. The sample mean is an unbiased estimator of the population mean.

What does efficiency mean?

Efficiency refers to the variance of the estimator’s sampling distribution. A more efficient estimator has a smaller variance.

What is a consistent estimator?

A consistent estimator gets closer to the true population parameter as the sample size increases.

What does the Central Limit Theorem help us do?

It helps predict where sample means are expected to fall. It also supports using confidence intervals for means.

What is standard error?

Standard error is the standard deviation of the sampling distribution of the sample mean. It tells how much sample means vary.

Formula:
SE = σ / √n
or sometimes written with sample standard deviation:
SE = s / √n

What happens to standard error when n increases?

Standard error gets smaller as sample size increases. Bigger sample size = more precise estimate.

What is the confidence interval formula when population standard deviation is known?

x̄ ± z × σ / √n

Where:
x̄ = sample mean
z = z-score
σ = population standard deviation
n = sample size

What is a confidence level?

The confidence level is the probability that the confidence interval contains the true population mean. 95% confidence is the most common.

What happens when the confidence level increases?

The confidence interval gets wider. Higher confidence means less precision.

What does 95% confidence mean?

If we made 100 confidence intervals, about 95 would contain the true population mean, and about 5 would not.