26: Statistics - Classical Tests & the LRT Framework

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🎞️ Սլայդեր — 10 stat

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1) Tests for Means

01 ✏️ Coffee Shop Cups

A coffee shop claims their large cups contain 350 mL on average. You measure 25 cups and find $ar{X} = 345$ mL with $S = 10$ mL.

1. State $H_0$ and $H_1$ (two-sided).
1. Compute the $t$-statistic and p-value. Do you reject at $lpha = 0.05$?
1. Build a 95% CI for $\mu$. Does it contain 350?

02 ✏️ Paired vs Unpaired

Eight runners record their 5K times before and after a training program:

Runner	1	2	3	4	5	6	7	8
Before	23.1	25.4	22.8	24.0	26.2	23.5	24.8	25.0
After	22.5	24.1	22.0	23.5	25.0	23.0	24.2	24.0

1. Run a paired $t$-test. State the hypotheses, compute $t$, find the p-value.
1. Now (incorrectly) run an unpaired two-sample $t$-test on the same data. Compare the p-value.
1. Why is the paired test more powerful here? When would unpaired be appropriate?

03 ✏️ Proportion Test: Is the Coin Fair?

You flip a coin 200 times and get 112 heads.

1. Test $H_0: p = 0.5$ using the proportion $z$-test. (Use $p_0$ in the SE, not $\hat{p}$.)
1. Compute the p-value. Reject at $lpha = 0.05$?
1. Build a 95% Wilson CI. Does it contain 0.5?

2) Chi-Squared Tests

04 ✏️ Are the Dice Fair?

You roll a die 120 times and observe:

Face	1	2	3	4	5	6
Count	25	17	22	15	23	18

1. State $H_0$ (the die is fair) and compute expected counts.
1. Compute $\chi^2 = \sum rac{(O_i - E_i)^2}{E_i}$ and state the degrees of freedom.
1. Find the p-value. At $lpha = 0.05$, is the die fair?

05 ✏️ Independence: Does Study Method Matter?

A university surveys 200 students about study method and exam outcome:

	Pass	Fail
Self-study	40	30
Group study	55	15
Tutor	35	25

1. Under the null hypothesis of independence, compute all expected cell counts.
1. Compute the $\chi^2$ statistic and degrees of freedom.
1. Test at $lpha = 0.05$. What do you conclude?
1. Which cell contributes the most to $\chi^2$? Interpret what this means in context.

3) Nonparametric Tests

06 🐍 Mann-Whitney U in Action

Two teaching methods are compared. Test scores:

Method A: [78, 85, 90, 72, 88, 65, 92]
Method B: [95, 82, 88, 91, 76, 84, 89]
1. Why might a nonparametric test be preferred here? (Think about sample size and normality.)
1. Run a Mann-Whitney U test using . Report the p-value.
1. Compare with the Welch $t$-test result. Do they agree?

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--- title: "26: Statistics - Classical Tests & the LRT Framework" format: html: css: homework-styles.css --- <script src="homework-scripts.js"></script> # 📚 Նյութը ::: {.callout-tip collapse="true"} ## ⚠️ Note YouTube links in this section were auto-extracted. If you spot a mistake, please let me know! ::: ## Դասախոսություն - [🎞️ Սլայդեր — 10 stat](Lectures/stat/10_stat.pdf) # 🏡 Տնային --- ## 1) Tests for Means ### 01 ✏️ Coffee Shop Cups {data-difficulty="1"} A coffee shop claims their large cups contain 350 mL on average. You measure 25 cups and find $ar{X} = 345$ mL with $S = 10$ mL. - a) State $H_0$ and $H_1$ (two-sided). - b) Compute the $t$-statistic and p-value. Do you reject at $lpha = 0.05$? - c) Build a 95% CI for $\mu$. Does it contain 350? ### 02 ✏️ Paired vs Unpaired {data-difficulty="2"} Eight runners record their 5K times before and after a training program: | Runner | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |--------|---|---|---|---|---|---|---|---| | Before | 23.1 | 25.4 | 22.8 | 24.0 | 26.2 | 23.5 | 24.8 | 25.0 | | After | 22.5 | 24.1 | 22.0 | 23.5 | 25.0 | 23.0 | 24.2 | 24.0 | - a) Run a **paired** $t$-test. State the hypotheses, compute $t$, find the p-value. - b) Now (incorrectly) run an **unpaired** two-sample $t$-test on the same data. Compare the p-value. - c) Why is the paired test more powerful here? When would unpaired be appropriate? ### 03 ✏️ Proportion Test: Is the Coin Fair? {data-difficulty="1"} You flip a coin 200 times and get 112 heads. - a) Test $H_0: p = 0.5$ using the proportion $z$-test. *(Use $p_0$ in the SE, not $\hat{p}$.)* - b) Compute the p-value. Reject at $lpha = 0.05$? - c) Build a 95% Wilson CI. Does it contain 0.5? --- ## 2) Chi-Squared Tests ### 04 ✏️ Are the Dice Fair? {data-difficulty="1"} You roll a die 120 times and observe: | Face | 1 | 2 | 3 | 4 | 5 | 6 | |------|---|---|---|---|---|---| | Count | 25 | 17 | 22 | 15 | 23 | 18 | - a) State $H_0$ (the die is fair) and compute expected counts. - b) Compute $\chi^2 = \sum rac{(O_i - E_i)^2}{E_i}$ and state the degrees of freedom. - c) Find the p-value. At $lpha = 0.05$, is the die fair? ### 05 ✏️ Independence: Does Study Method Matter? {data-difficulty="2"} A university surveys 200 students about study method and exam outcome: | | Pass | Fail | |---|---|---| | Self-study | 40 | 30 | | Group study | 55 | 15 | | Tutor | 35 | 25 | - a) Under the null hypothesis of independence, compute all expected cell counts. - b) Compute the $\chi^2$ statistic and degrees of freedom. - c) Test at $lpha = 0.05$. What do you conclude? - d) Which cell contributes the most to $\chi^2$? Interpret what this means in context. --- ## 3) Nonparametric Tests ### 06 🐍 Mann-Whitney U in Action {data-difficulty="2"} Two teaching methods are compared. Test scores: - Method A: [78, 85, 90, 72, 88, 65, 92] - Method B: [95, 82, 88, 91, 76, 84, 89] - a) Why might a nonparametric test be preferred here? (Think about sample size and normality.) - b) Run a Mann-Whitney U test using . Report the p-value. - c) Compare with the Welch $t$-test result. Do they agree? # 🎲 38 (01) TODO - ▶️[ToDo]() - 🔗[Random link]() - 🇦🇲🎶[ToDo]() - 🌐🎶[ToDo]() - 🤌[Կարգին]() <a href="http://s01.flagcounter.com/more/1oO"><img src="https://s01.flagcounter.com/count2/1oO/bg_FFFFFF/txt_000000/border_CCCCCC/columns_2/maxflags_10/viewers_0/labels_0/pageviews_1/flags_0/percent_0/" alt="Flag Counter"></a>