Difficulty: Intermediate to Advanced | Prerequisites: Chapters 1 to 6 study notes
Chapters 8 through 10 are the heart of inferential statistics. You use sample data to make claims about the population: confidence intervals estimate a parameter within a range, and hypothesis tests decide whether the data support a specific claim. You need to know when to use a z-test (sigma known) versus a t-test (sigma unknown), and how to handle one-sample, two-sample independent, and paired designs. This is typically the most heavily weighted material on the final exam.
Confidence intervals give a range of plausible values for a population parameter. Hypothesis tests follow a four-step procedure: identify the parameter, state hypotheses, compute a test statistic and p-value, then make a decision. Type I error means rejecting a true null; Type II error means failing to reject a false null. Power = 1 minus the probability of a Type II error.
Confidence interval (CI)
A range of values, computed from sample data, that is likely to contain the true population parameter. A 95% CI means that if you repeated the sampling process many times, about 95% of the intervals would capture the true value.
Think of it as a net you cast around your estimate; wider nets catch the parameter more often.
Margin of error (ME)
The half-width of a confidence interval. It equals the critical value times the standard error.
Significance level (alpha)
The threshold for rejecting the null hypothesis. Common values are 0.05 and 0.01. It is also the probability of making a Type I error.
Null hypothesis (H0)
The default claim about the population parameter, typically a statement of no effect or no difference.
Alternative hypothesis (Ha)
The claim you are testing for, which contradicts the null. It can be one-sided (greater than or less than) or two-sided (not equal to).
P-value
The probability of observing results at least as extreme as the sample data, assuming the null hypothesis is true.
In simple terms, a small p-value means the data would be surprising if the null were true.
Type I error
Rejecting the null hypothesis when it is actually true. The probability of this is alpha.
Think of it as a false alarm.
Type II error
Failing to reject the null hypothesis when it is actually false. The probability of this is beta.
Think of it as a missed detection.
Power
1 minus beta. The probability of correctly rejecting a false null hypothesis. Higher power means you are more likely to detect a real effect.
Z-test
Used when the population standard deviation (sigma) is known.
T-test
Used when sigma is unknown and must be estimated by the sample standard deviation (s). Uses the t-distribution with degrees of freedom.
Paired test
Used when two measurements are taken on the same subjects (before/after, matched pairs). You work with the differences d-bar.
One-sample z (sigma known): x-bar +/- z_(alpha/2) * (sigma / sqrt(n)), df = N/A
One-sample t (sigma unknown): x-bar +/- t_(alpha/2, n-1) * (s / sqrt(n)), df = n minus 1
Two-sample independent t: (x-bar_1 minus x-bar_2) +/- t_(alpha/2, v) * sqrt(s1^2/n1 + s2^2/n2), df will be given to you
Paired (two-sample dependent): d-bar +/- t_(alpha/2, n-1) * (s_d / sqrt(n)), df = n minus 1
Sigma is known: use a z-test.
Sigma is unknown (you only have s): use a t-test. This is the more common scenario on exams.
Lower bound on mu: mu > x-bar minus z_alpha * (sigma / sqrt(n))
Upper bound on mu: mu < x-bar + z_alpha * (sigma / sqrt(n))
One-sample z: n = (z_(alpha/2) * sigma / ME)^2
One-sample t: n = (t'_(alpha/2, n-1) * s / ME)^2 (where m is the pilot sample size)
Identify the parameter(s) of interest and describe them in context.
State H0 and Ha.
Compute the test statistic and find the p-value. For z-tests, you calculate the p-value by hand using a z-table. For t-tests, the p-value or critical value is typically provided.
Decision: compare the p-value to alpha. If p-value < alpha, reject H0. State the conclusion in context.
One-sample z: z_ts = (x-bar minus mu_0) / (sigma / sqrt(n))
One-sample t: t_ts = (x-bar minus mu_0) / (s / sqrt(n)), df = n minus 1
Two-sample independent t: t'_ts = (x-bar_1 minus x-bar_2 minus delta_0) / sqrt(s1^2/n1 + s2^2/n2), df given
Paired t: t_ts = (d-bar minus delta_0) / (s_d / sqrt(n)), df = n minus 1
Upper-tailed (Ha: mu > mu_0): p = P(Z > z_ts) = 1 minus P(Z <= z_ts) for z; p = P(T > t_ts) for t
Lower-tailed (Ha: mu < mu_0): p = P(Z < z_ts) for z; p = P(T < t_ts) for t
Two-tailed (Ha: mu != mu_0): p = 2 * P(Z > |z_ts|) for z; p = 2 * P(T > |t_ts|) for t
Type I error probability = alpha (the significance level you chose)
Type II error probability = beta (depends on the true parameter value and the sample size)
Power = 1 minus beta
Increasing sample size increases power. Increasing alpha increases power but also increases the Type I error rate.
\text{One-sample z: } \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\text{One-sample t: } \bar{x} \pm t_{\alpha/2,\, n-1} \frac{s}{\sqrt{n}}\text{Two-sample t: } (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2,\, \nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\text{Paired: } \bar{d} \pm t_{\alpha/2,\, n-1} \frac{s_d}{\sqrt{n}}z_{ts} = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}t_{ts} = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}t'_{ts} = \frac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}t_{ts} = \frac{\bar{d} - \Delta_0}{s_d / \sqrt{n}}n = \left( \frac{z_{\alpha/2} \cdot \sigma}{ME} \right)^2Pharmaceutical trials use hypothesis tests to decide whether a new drug performs better than a placebo. Confidence intervals appear in election polling: "Candidate A leads with 52% +/- 3 points" is a confidence interval for the true proportion of voters.
Students often say a 95% confidence interval means there is a 95% probability the parameter is inside it. The parameter is fixed; the interval is random. Before the sample is drawn, there is a 95% chance the procedure will produce an interval that contains the parameter. After computation, the parameter either is or is not inside.
Students confuse failing to reject H0 with proving H0 is true. Failing to reject means the evidence was not strong enough. It does not confirm the null.
Students forget that a two-tailed p-value is doubled. If Ha is mu != mu_0, the p-value is 2 * P(Z > |z|), not just P(Z > |z|).
Students mix up Type I and Type II errors. Type I = false positive (rejecting a true null). Type II = false negative (keeping a false null).
The four-step hypothesis test procedure is the backbone of the exam. Practise writing it out fully, including the conclusion in context.
Know how to determine whether to use z or t. If sigma is given in the problem, use z. If only s is given, use t.
Expect a question asking you to compute the required sample size for a given margin of error.
Paired vs. independent two-sample designs: if the same subjects are measured twice (before/after), use the paired test. If two separate groups are compared, use the independent two-sample test.
True or False: A z-test is appropriate when sigma is unknown. (Answer: False, use a t-test.)
Fill in the blank: Power = 1 minus ______. (Answer: beta.)
True or False: A p-value of 0.03 means we reject H0 at the alpha = 0.05 level. (Answer: True.)
Fill in the blank: In a paired test, the degrees of freedom are ______. (Answer: n minus 1.)
True or False: Failing to reject H0 proves H0 is true. (Answer: False.)
Q: A STAT 300 midterm is normally distributed with a mean of 40 minutes and a standard deviation of 17 minutes. What is the probability that a student completes the exam in more than 50 minutes?
A: z = (50 minus 40) / 17 = 10/17 = approximately 0.59. P(Z > 0.59) = 1 minus P(Z < 0.59) = 1 minus 0.7224 = 0.2776.
Q: You are testing whether a new diet reduces average weight. H0: mu = 80 kg, Ha: mu < 80. Your sample of 25 gives x-bar = 77, s = 6. Compute the test statistic and state whether you would reject at alpha = 0.05.
A: t = (77 minus 80) / (6 / sqrt(25)) = minus 3 / 1.2 = minus 2.5, df = 24. For a one-tailed test at alpha = 0.05 with df = 24, the critical value is about minus 1.711. Since minus 2.5 < minus 1.711, reject H0. There is sufficient evidence that the diet reduces average weight.
Q: Is it a z-test or a t-test? A problem states: "sigma is known to be 3.104."
A: Since sigma is known, this is a z-test.
Q: A researcher measures blood pressure before and after a treatment on the same 15 patients. Which test design is appropriate: independent two-sample or paired?
A: Paired, because the same subjects are measured twice (before and after).
The hypothesis-testing framework extends directly into ANOVA (Chapter 11), where you test whether three or more group means are equal. It also underpins regression inference (Chapter 12), where you test whether the slope of a regression line is significantly different from zero. The concepts of Type I/II error and power apply everywhere inference is used.
STAT 101, Purdue, confidence interval, margin of error, hypothesis test, null hypothesis, alternative hypothesis, p-value, significance level, alpha, Type I error, Type II error, power, beta, z-test, t-test, one-sample, two-sample, independent samples, paired samples, matched pairs, degrees of freedom, test statistic, critical value, rejection region, normal distribution, t-distribution