Lesson 104 — z-test and Student's t-test

$H_0: \mu = 20$, $H_1: \mu \neq 20$. Data: $n = 16$, $\bar X = 22$, $s = 5$. t-statistic and conclusion with $\alpha = 0.05$.

$H_0: \mu = 50$, $H_1: \mu < 50$. Data: $n = 36$, $\bar X = 48$, $s = 8$. Calculate $T$ and the p-value. Conclusion with $\alpha = 0.05$.

Blood pressure of 10 patients before and after diet. Differences: 2, 3, 1, 4, 2, 3, 1, 2, 3, 1 mmHg. $H_0: \mu_D = 0$, $H_1: \mu_D > 0$. At the 5% level, did the diet reduce pressure?

Group A: $n_1 = 30$, $\bar X_1 = 85$, $s_1 = 4$. Group B: $n_2 = 25$, $\bar X_2 = 80$, $s_2 = 3$. Welch t two-tailed at the 5% level.

You want to compare average salaries between two departments of a company, with possibly different variances. What is the best strategy?

Population average IQ: $\mu_0 = 100$, $\sigma = 15$ (known). A class of $n = 64$ has $\bar X = 102$. At the 5% level, does the class have an average IQ different from the population?

Reaction time (ms) of 10 drivers before and after a cup of coffee. Differences: 2, -1, 3, 1, 2, 0, 1, 2, -1, 2. At the 5% level, did coffee alter reaction time?

Group 1: $n_1 = 15$, $\bar X_1 = 75$, $s_1 = 10$. Group 2: $n_2 = 20$, $\bar X_2 = 70$, $s_2 = 9$. Use pooled t two-tailed at the 5% level (assuming equal variances).

A doctor measures the blood pressure of 20 patients before and after a treatment. Which test is most appropriate?

Study hours of $n = 20$ university students: $\bar X = 4.5$ h/day, $s = 1.2$ h. $H_0: \mu = 5$. Two-tailed at the 5% level.

Coin flipped 100 times: 52 heads. At the 5% level, is the coin fair? (Use z-test for proportion with $\sigma_{\hat p} = \sqrt{p_0(1-p_0)/n}$.)

Classes A and B: $n_1 = 20$, $\bar X_1 = 45$, $s_1 = 7$; $n_2 = 30$, $\bar X_2 = 40$, $s_2 = 10$. Welch t at the 5% level.

What is the relationship between the z-test and the t-test for large samples?

Typing speed (wpm) of 10 students before and after course: before [40, 35, 50, 45, 38, 42, 47, 36, 44, 41], after [45, 38, 57, 47, 42, 48, 48, 39, 49, 45]. At the 5% level, did the course improve speed?

A production line must fill bottles with 200 mL. Sample of $n = 25$: $\bar X = 198$ mL, $s = 12$ mL. At the 5% level, is the process unregulated?

A school applied a Math tutoring program. SAEB score of 10 students before: [60, 55, 70, 65, 58, 62, 67, 56, 64, 61]. After: [75, 63, 82, 85, 63, 72, 85, 63, 76, 74]. Use the appropriate test at the 5% level.

A school network tested a new teaching methodology. Control school ($n_1 = 50$): $\bar X_1 = 505$ points, $s_1 = 20$. Pilot school ($n_2 = 60$): $\bar X_2 = 520$, $s_2 = 25$. At the 1% level, did the methodology improve results? Also calculate effect size.

Industrial process temperature: $\mu_0 = 30$°C, $\sigma = 8$°C. Calculate the two-tailed z-test for $\bar X = 31$°C with $n = 100$ and then with $n = 10000$. Compare p-values and discuss the difference between statistical and practical significance.

Show algebraically that the paired test has a smaller standard error than the Welch t (for the same $n$ per group), when the correlation $\rho$ between pairs is positive. How much smaller is the standard error for $\rho = 0.7$ and $s_1 = s_2$?

Derive the Welch-Satterthwaite formula for the effective degrees of freedom $\nu$ of the Welch test. Verify that for $n_1 = n_2$ and $s_1 = s_2$, the formula reduces to $\nu = 2(n-1)$, coinciding with the pooled t.