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Merged
Ishayu1 merged 1 commit into from
Dec 3, 2025
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fixed bugs #6

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<title>Discussion 8: The Central Limit Theorem and Hypothesis Testing</title>
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code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
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code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
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Expand Down Expand Up @@ -3310,8 +3310,7 @@ <h5>Difficulty:
<p><br></p>
<hr />
<h2 id="problem-22">Problem 22</h2>
<p>Suppose you have a DataFrame called <code>plum</code> with 31
rows.</p>
<p>Recall, <code>plum</code> has 31 rows.</p>
<p>Consider the function <code>df_choice</code>, defined below.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode py"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> df_choice(df):</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> df[np.random.choice([<span class="va">True</span>, <span class="va">False</span>], df.shape[<span class="dv">0</span>], replace<span class="op">=</span><span class="va">True</span>)]</span></code></pre></div>
Expand Down Expand Up @@ -3388,6 +3387,11 @@ <h1 class="title"> </h1>
missing}) \cdot P(\text{row 1 missing}) \cdot ... \cdot P(\text{row 30
missing}) \\ &amp;= \frac{1}{2} \cdot \frac{1}{2} \cdot ... \cdot
\frac{1}{2} \\ &amp;= \boxed{\frac{1}{2^{31}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 83%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3439,6 +3443,11 @@ <h1 class="title"> </h1>
is missing}) \\ &amp;= \frac{1}{2^{31}} + \frac{1}{2^{31}} + ... +
\frac{1}{2^{31}} \\ &amp;=
\boxed{\frac{31}{2^{31}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 48%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3485,6 +3494,11 @@ <h1 class="title"> </h1>
just row 0 is <span class="math inline">\frac{1}{2^{31}}</span>, and
this is the same as the answer to the first subpart (<span class="math inline">\frac{1}{2^{31}}</span>); in both situations, we are
calculating the probability of one specific subset.</p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 63%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3555,14 +3569,20 @@ <h1 class="title"> </h1>
\frac{1 + 31 + 31 + 1}{2^{31}} \\ &amp;= \frac{64}{2^{31}} \\ &amp;=
\frac{2^6}{2^{31}} \\ &amp;= \frac{1}{2^{31 - 6}} \\ &amp;=
\boxed{\frac{1}{2^{25}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 35%.</p>
</div>
</div>
</div>
</div>
<p><br></p>
<hr />
<h2 id="problem-23">Problem 23</h2>
<p>Recall, <code>plum</code> has 31 rows.</p>
<p>Suppose you have a DataFrame called <code>plum</code> with 31
rows.</p>
<p>Consider the function <code>df_choice</code>, defined below.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode py"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> df_choice(df):</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> df[np.random.choice([<span class="va">True</span>, <span class="va">False</span>], df.shape[<span class="dv">0</span>], replace<span class="op">=</span><span class="va">True</span>)]</span></code></pre></div>
Expand Down Expand Up @@ -3639,11 +3659,6 @@ <h1 class="title"> </h1>
missing}) \cdot P(\text{row 1 missing}) \cdot ... \cdot P(\text{row 30
missing}) \\ &amp;= \frac{1}{2} \cdot \frac{1}{2} \cdot ... \cdot
\frac{1}{2} \\ &amp;= \boxed{\frac{1}{2^{31}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 83%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3695,11 +3710,6 @@ <h1 class="title"> </h1>
is missing}) \\ &amp;= \frac{1}{2^{31}} + \frac{1}{2^{31}} + ... +
\frac{1}{2^{31}} \\ &amp;=
\boxed{\frac{31}{2^{31}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 48%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3746,11 +3756,6 @@ <h1 class="title"> </h1>
just row 0 is <span class="math inline">\frac{1}{2^{31}}</span>, and
this is the same as the answer to the first subpart (<span class="math inline">\frac{1}{2^{31}}</span>); in both situations, we are
calculating the probability of one specific subset.</p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 63%.</p>
</div>
</div>
</div>
Expand Down Expand Up @@ -3821,11 +3826,6 @@ <h1 class="title"> </h1>
\frac{1 + 31 + 31 + 1}{2^{31}} \\ &amp;= \frac{64}{2^{31}} \\ &amp;=
\frac{2^6}{2^{31}} \\ &amp;= \frac{1}{2^{31 - 6}} \\ &amp;=
\boxed{\frac{1}{2^{25}}} \end{aligned}</span></p>
<hr/>
<h5>Difficulty: ⭐️⭐️⭐️⭐️</h5>
<p>
</p>
<p>The average score on this problem was 35%.</p>
</div>
</div>
</div>
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code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style>
<link rel="stylesheet" href="..\..\assets\theme.css" />
<link rel="stylesheet" href="../../assets/theme.css" />
<script defer=""
src="https://cdn.jsdelivr.net/npm/katex@latest/dist/katex.min.js"></script>
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