Midterm review

# Midterm review

**Thursday, February 11**

]

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# The midterm is on Tuesday

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> Details </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> When </td>
   <td style="text-align:left;"> Tuesday, February 16, in class, 80 minutes </td>
  </tr>
  <tr>
   <td style="text-align:left;"> What </td>
   <td style="text-align:left;"> Chapters 1 to 9, which are Topics 1 to 8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> How </td>
   <td style="text-align:left;"> Free response: show your work and explain your thinking </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Bring </td>
   <td style="text-align:left;"> A calculator, and any print materials, including your own notes </td>
  </tr>
  <tr>
   <td style="text-align:left;"> We attach </td>
   <td style="text-align:left;"> The z, t and F tables, and a formula sheet in both sets of symbols </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Not allowed </td>
   <td style="text-align:left;"> Any other device </td>
  </tr>
</tbody>
</table>

.box-8.small[Today is practice, in pairs. Nothing is scored]

???

General format only. No exam content in this deck, since it's public. The practice uses Mindspark students the exam won't use.

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# Plan for today

.box-1.medium.sp-after-half[Describe: Topics 1 to 4]

.box-6.medium.sp-after-half[Probability and sampling: Topics 5 and 6]

.box-3.medium.sp-after-half[Tests and intervals: Topics 7 and 8]

.box-inv-8.small[Each round: work alone for a few minutes, then with a neighbor. Then we compare]

???

Three rounds of about 25 minutes, with the answers after each. Walk the room during the pair work: that's where you'll see the slips worth naming. Calculators and the tables out.

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# One question runs through all of it

.box-inv-1.small[A fact about the population: μ, σ, σ<sub>M</sub>]

.box-inv-1.small[Usually unknown]
]

.box-inv-6.small[From our sample: M, σ̂, σ̂<sub>M</sub>, g]

.box-inv-6.small[Changes from sample to sample]
]

.box-8.small[Every test asks: is our estimate too far from what H₀ says the parameter is?]

???

The Topic 1 question, one more time. If students can say for each symbol on the formula sheet whether it's a parameter or an estimate, most of the midterm's reasoning follows.

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# Describe

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# Practice 1: Six students

.box-inv-8.small.sp-after-half[Six Govindpuri students' baseline math scores: **8, 5, 9, 12, 14, 12**]

.box-8.small.sp-after-half[**b.** Find SS, then σ̂, our estimate of the SD of all Govindpuri students]
]

.box-8.small.sp-after-half[**d.** Which scale is math score? Which is the center a student goes to?]
]

.box-inv-8.small[Work alone for 3 minutes, then compare with a neighbor]

???

About 8 minutes. Common slips: dividing SS by n instead of n − 1, and forgetting the square root at the end.

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# Practice 1: Answers a and b

???

n − 1 on the bottom because these six are a sample, and σ̂ estimates the SD of all Govindpuri students. The six students' own SD, with n on the bottom, would describe only them, and tends to run low as an estimate.

The median needs the scores in order first: 5, 8, 9, 12, 12, 14.

---

# Practice 1: Answers c and d

.box-8.medium.sp-after-half[z = (14 − 10) / 3.29 = 1.22]

.box-inv-8.small.sp-after-half[The 14 sits 1.22 standard deviations above the mean of these six]

.box-inv-8.small.sp-after-half[**Math score:** ratio. A score of 0 means no answers right]

.box-inv-8.small[**Center:** nominal. It's a name, with no order]

???

Ratio, as in Topic 1. Some students will say interval, and that's a fair argument to hear: a 0 on this test doesn't mean no math at all. On the exam, an answer with a reason is what counts.

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# Parameter or estimate? Hands up

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:center;">   </th>
   <th style="text-align:left;"> The number </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> A </td>
   <td style="text-align:left;"> The mean of all 619 Mindspark students at baseline, treated as our population </td>
  </tr>
  <tr>
   <td style="text-align:center;"> B </td>
   <td style="text-align:left;"> The mean of the six Govindpuri students, 10 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> C </td>
   <td style="text-align:left;"> σ̂ = 3.29 from the six </td>
  </tr>
  <tr>
   <td style="text-align:center;"> D </td>
   <td style="text-align:left;"> The SD of the sampling distribution of M, for samples of 16 from the population </td>
  </tr>
  <tr>
   <td style="text-align:center;"> E </td>
   <td style="text-align:left;"> The mean math score of every fifth grader in Delhi </td>
  </tr>
</tbody>
</table>

.box-8.small[Hands up for parameter, then for estimate]

???

About 3 minutes. Answers: A parameter, μ. B estimate, M. C estimate, σ̂. D parameter, σ<sub>M</sub>: it's set by σ and n, before any sample. E parameter, and one nobody knows. The Mindspark students aren't a random sample of Delhi, so M from them isn't a good estimate of it.

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# Probability and sampling

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# Practice 2: One score or a mean?

.box-inv-8.small.sp-after-half[Treat all 619 students as a population, with μ = 11.12 and σ = 3.90. Treat their scores as normal]

.box-8.small.sp-after-half[**b.** What score is the 25th percentile?]
]

.pull-right[
.box-8.small.sp-after-half[**c.** We draw samples of 16. What are the mean and SD of the sample means?]

.box-8.small.sp-after-half[**d.** In what share of those samples is M above 12.5?]
]

.box-inv-8.small[Sketch a curve and shade it before you open the table]

???

About 8 minutes, with the unit normal table. Common slips: reading the body when the question wants the tail, and using σ instead of σ<sub>M</sub> in part d.

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# Practice 2: Answers a and b

.box-inv-8.small.sp-after-half[The tail beyond z = 0.99 is **0.1611**, about 16% of students]
]

.pull-right[
.box-inv-8.small.sp-after-half[**b.** A tail of 0.2500 is closest at z = 0.67. Below the mean, that's z = −0.67]

.box-inv-8.small.sp-after-half[X = 11.12 + (−0.67) × 3.90 = **8.51**]
]

.box-8.small[Going right: X to z to area. Going left: area to z to X]

???

For part b, the table lists only positive z. The 25th percentile is below the mean, so find the tail closest to 0.2500, then make z negative. 0.67 gives 0.2514 and 0.68 gives 0.2483, so 0.67 is closer.

The normal curve is a model. In the real data, 11.6% of the 619 scored above 15, because scores are whole numbers and the shape isn't a perfect bell. Use the model on the exam unless a question gives you the raw scores.

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# Practice 2: Answers c and d

.box-inv-8.small.sp-after-half[**c.** The mean of the means is 11.12. σ<sub>M</sub> = 3.90 / √16 = **0.975**]

.box-inv-8.small[**d.** z = (12.5 − 11.12) / 0.975 = 1.42, and the tail beyond it is **0.0778**]

???

The picture is the point: a mean of 16 students is much less spread out than one student, so a mean above 12.5 is rarer than a single score above it. That's σ<sub>M</sub> = σ / √n, and it's why tests work at all.

The central limit theorem lets us use the normal curve for means of 16 even if single scores weren't quite normal.

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# Tests and intervals

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# Practice 3: All of Govindpuri, σ known

.box-inv-8.small.sp-after-half[Govindpuri's 82 students: M = 11.60. All 619: μ = 11.12 and σ = 3.90]

.box-8.small.sp-after-half[If Govindpuri's students were a random draw from the 619, would a mean this high be too unexpected? Run all four steps, with α = 0.05 and two tails]

.box-inv-8.small[Work alone, then compare with a neighbor. Then hands: reject or maintain?]

???

About 6 minutes. Common slip: using σ = 3.90 on the bottom instead of σ<sub>M</sub>.

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# Practice 3: Answer

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;"> Step </th>
   <th style="text-align:left;"> Our numbers </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1. State H₀ </td>
   <td style="text-align:left;"> H₀: μ = 11.12. H₁: μ ≠ 11.12 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2. Set the criteria </td>
   <td style="text-align:left;"> α = 0.05, two tails, so z beyond ±1.96 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3. Observe </td>
   <td style="text-align:left;"> σ<sub>M</sub> = 3.90 / √82 = 0.431, so z = (11.60 − 11.12) / 0.431 = 1.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4. Decide </td>
   <td style="text-align:left;"> <strong>Maintain H₀</strong>: z = 1.11 is inside ±1.96, p = 0.265 </td>
  </tr>
</tbody>
</table>

.box-8.small.sp-after-half[Govindpuri's mean isn't surprisingly far from 11.12]

.box-inv-8.small[Maintain, not accept: the test can't show that Govindpuri's μ is exactly 11.12]

???

Topic 7's ninth graders gave a reject with the same recipe. This one maintains, and the wording matters as much as the arithmetic.

For anyone who asks: Govindpuri's students are part of the 619, which makes the true standard error a little smaller. It doesn't change the decision.

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# Practice 4: Ninth graders, σ unknown

.box-inv-8.small.sp-after-half[Chhatarpur's 32 ninth graders: M = 12.62 and SS = 299.5 at baseline]

.pull-left[
.box-8.small.sp-after-half[**a.** Test H₀: μ = 11.12 with α = 0.05 and two tails. Find σ̂, σ̂<sub>M</sub> and t]

.box-8.small.sp-after-half[**b.** The table skips df = 31. Which row do you use?]
]

.box-8.small.sp-after-half[**d.** Find a 95% confidence interval for μ. Does it hold 11.12?]
]

.box-inv-8.small[Work alone for 4 minutes, then compare with a neighbor]

???

About 10 minutes. This is the longest problem today, and the most midterm-like. Common slips: dividing SS by n, and using n instead of n − 1 for df.

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# Practice 4: Answers a and b

.box-inv-8.small.sp-after-half[σ̂<sub>M</sub> = 3.11 / √32 = **0.549**]

.box-inv-8.small[t = (12.62 − 11.12) / 0.549 = **2.74**]
]

.pull-right[
.box-inv-8.small.sp-after-half[**b.** Use df = 30, the next lower row. Its cutoff is ±2.042]

.box-inv-8.small[A lower df has a wider cutoff, so it's the cautious choice]
]

.box-8.small[2.74 is beyond ±2.042: **reject H₀**. t(31) = 2.74, p = 0.010]

???

Report the real df, 31, even though the cutoff came from the row for 30. The exact cutoff for 31 is 2.040, so the decision is the same.

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# Practice 4: Answers c and d

.box-inv-8.small[About half an SD. Large next to the effects of real education programs in Topic 7's table]
]

.box-inv-8.small[It doesn't hold 11.12, which matches the reject]
]

.box-8.small[The test says real. g says how big. The interval says how sure. None of them says why: these are older students]

???

g divides by σ̂, not σ̂<sub>M</sub>. That's the most common mix-up on this part. g doesn't change with n, and t does.

Why: they're in grade 9, two or three grades above most of the 619. The test can only say that the gap is too big for sampling error.

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# True or false? Hands up

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:center;">   </th>
   <th style="text-align:left;"> Statement </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> A </td>
   <td style="text-align:left;"> p = 0.03 means there's a 3% chance that H₀ is true </td>
  </tr>
  <tr>
   <td style="text-align:center;"> B </td>
   <td style="text-align:left;"> α is the chance of a Type 1 error when H₀ is true </td>
  </tr>
  <tr>
   <td style="text-align:center;"> C </td>
   <td style="text-align:left;"> Maintaining H₀ shows the program had no effect </td>
  </tr>
  <tr>
   <td style="text-align:center;"> D </td>
   <td style="text-align:left;"> A bigger sample raises power </td>
  </tr>
  <tr>
   <td style="text-align:center;"> E </td>
   <td style="text-align:left;"> A tiny p means a big effect </td>
  </tr>
</tbody>
</table>

.box-8.small[Hands up for true, then for false. Then fix each false one with a neighbor]

???

About 5 minutes. A false: p is how surprising our sample would be if H₀ were true. B true. C false: it means the sample can't tell, maybe because power was low. D true: so does a bigger effect, less spread, or a larger α. E false: p also depends on n, which is why we report g.

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# The slips we see most

.box-inv-8.small[Forgetting the last square root]

.box-inv-8.small[Using σ where the test needs σ<sub>M</sub> or σ̂<sub>M</sub>]

.box-inv-8.small[Using n for df, when it's n − 1]
]

.box-inv-8.small[Writing "accept H₀". It's maintain]

.box-inv-8.small[Saying the test shows why]

.box-inv-8.small[A decision with no numbers. Report t(df) = …, p = …]
]

???

These come from the try-its across Topics 3 to 8. Each is quick to check on the exam, so suggest students save two minutes at the end for this list.

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# How to get ready

.box-inv-8.small.sp-after-half[Redo the try-its in the decks, and the by-hand problems from Labs 2, 4 and 5]

.box-inv-8.small.sp-after-half[MindTap for Chapters 1 to 4 is open as practice, all quarter]

.box-inv-8.small.sp-after-half[Make your own notes: you can bring them, so build one page per topic]

.box-8.small[Practise with the tables we attach: z first, then t]

???

Open book can make students study less. Remind them there's no time to learn from the book during the exam: notes help only if they wrote them.

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# Next steps

Quiz, as midterm review. Bring a calculator
]

The midterm, in class. Bring a calculator and your notes
]

.box-3.small[Sunday, February 14 at 6 PM: Mastery and Problem Set for Chapter 9]

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