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  • @@ -369,7 +370,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—

    5.

    Consider the sample space \(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\) with probability distribution below. Calculate the probabilities of \(A = \{1, 3, 7, 8\}\text{,}\) \(B = \{2, 3, 6, 7\}\text{,}\) \(A\cup B\text{,}\) and \(A \cap B\text{.}\)
    πŸ”—
    -
    Table 127.
    πŸ”—
    +
    Table 128.
    πŸ”—
    @@ -420,7 +421,7 @@ eBookConfig.enable_chatcodes = false;
    Suppose a die has the values \(1, 2, 3, 4, 5, 6\) on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.
    πŸ”—
    Answer.
    -
    Table 128. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 129. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    @@ -455,7 +456,7 @@ eBookConfig.enable_chatcodes = false;
    Suppose a die has the values \(1, 2, 3, 4, 5, 6\) on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.
    πŸ”—
    Answer.
    -
    Table 129. Probability Distribution for an Even-biased Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 130. Probability Distribution for an Even-biased Die
    πŸ”—
    @@ -649,7 +650,7 @@ A\cap B \amp = \{HHH, HHT\}
    Let \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\) be events in the sample space \(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\) Create a probability distribution for \(\Omega\) so that \(A, B\) are independent.
    πŸ”—
    Answer.
    -
    Table 130. Example Distribution
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 131. Example Distribution
    πŸ”—
    @@ -774,7 +775,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    21.

    Consider \(X, Y\) with the joint distribution table below. Are \(X, Y\) independent?
    πŸ”—
    -
    Table 131. Joint distribution for \(X, Y\)
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 132. Joint distribution for \(X, Y\)
    πŸ”—
    @@ -809,7 +810,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—
    Answer.
    -
    Table 132. Joint Distribution
    πŸ”—
    \(X = 0\)
    +
    Table 133. Joint Distribution
    πŸ”—
    @@ -832,7 +833,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    23.

    Consider a random variable \(X\) with probability distribution below. Find \(\E(X)\text{.}\)
    πŸ”—
    -
    Table 133.
    πŸ”—
    \(X = 0\)
    +
    Table 134.
    πŸ”—
    @@ -941,7 +942,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    29.

    Consider a random variable \(X\) with probability distribution below. Find \(\Var(X)\text{.}\)
    πŸ”—
    -
    Table 134.
    πŸ”—
    \(x\) \(\Pr(X = x)\)
    +
    Table 135.
    πŸ”—
    diff --git a/course/ch-notes.html b/course/ch-notes.html index a9b524d..9d36a67 100755 --- a/course/ch-notes.html +++ b/course/ch-notes.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/course-documents.html b/course/course-documents.html index e14fb93..5514820 100755 --- a/course/course-documents.html +++ b/course/course-documents.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/exam-01.html b/course/exam-01.html index 42a021a..b274204 100755 --- a/course/exam-01.html +++ b/course/exam-01.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/exams.html b/course/exams.html index 0a23e9a..ef0fef1 100755 --- a/course/exams.html +++ b/course/exams.html @@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false; -PrevUpNext +PrevUpNext
  • @@ -276,7 +277,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—
    -PrevTopNext +PrevTopNext
    diff --git a/course/frontmatter.html b/course/frontmatter.html index 1f63942..e72ff61 100755 --- a/course/frontmatter.html +++ b/course/frontmatter.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/index.html b/course/index.html index 7c87e57..407130b 100755 --- a/course/index.html +++ b/course/index.html @@ -2,7 +2,7 @@ - + diff --git a/course/lunr-pretext-search-index.js b/course/lunr-pretext-search-index.js index 41342ff..7578616 100755 --- a/course/lunr-pretext-search-index.js +++ b/course/lunr-pretext-search-index.js @@ -1134,6 +1134,51 @@ var ptx_lunr_docs = [ "title": "", "body": " Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . " }, +{ + "id": "quiz-03", + "level": "1", + "url": "quiz-03.html", + "type": "Worksheet", + "number": "", + "title": "Quiz 3", + "body": " Quiz 3 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. Suppose a parameter has likelihood function with . Then the maximum likelihood estimation is . True. Suppose are random variables, and . Then, for sufficiently large , . False. Suppose we collect data to estimate the value of a parameter . Based on the collected data, we find a 95% confidence interval and a 90% confidence interval . Then . True. The heights of five plants are measured and recorded below. Find the sample mean, sample variance, and a 95% confidence interval around the sample mean for the heights of the plants. (Pretend that 5 measurements is enough for the CLT to apply.) Plant (in) Let be the average height of the sample. Then: So the 95% confidence limits are: Suppose a coin comes up heads with probability 0.6. We flip the coin 80 times, and let be the number of heads. Estimate . (Use a continuity correction if appropriate.) , so and . By the CLT, , so: Values 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 βˆ’1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 βˆ’1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 βˆ’1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 βˆ’1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 βˆ’1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 " +}, +{ + "id": "quiz-03-3-1", + "level": "2", + "url": "quiz-03.html#quiz-03-3-1", + "type": "Worksheet Exercise", + "number": "1", + "title": "", + "body": " Write either True or False for each of the following statements. No justification is required. Suppose a parameter has likelihood function with . Then the maximum likelihood estimation is . True. Suppose are random variables, and . Then, for sufficiently large , . False. Suppose we collect data to estimate the value of a parameter . Based on the collected data, we find a 95% confidence interval and a 90% confidence interval . Then . True. " +}, +{ + "id": "quiz-03-3-2", + "level": "2", + "url": "quiz-03.html#quiz-03-3-2", + "type": "Worksheet Exercise", + "number": "2", + "title": "", + "body": " The heights of five plants are measured and recorded below. Find the sample mean, sample variance, and a 95% confidence interval around the sample mean for the heights of the plants. (Pretend that 5 measurements is enough for the CLT to apply.) Plant (in) Let be the average height of the sample. Then: So the 95% confidence limits are: " +}, +{ + "id": "quiz-03-4-1", + "level": "2", + "url": "quiz-03.html#quiz-03-4-1", + "type": "Worksheet Exercise", + "number": "3", + "title": "", + "body": " Suppose a coin comes up heads with probability 0.6. We flip the coin 80 times, and let be the number of heads. Estimate . (Use a continuity correction if appropriate.) , so and . By the CLT, , so: " +}, +{ + "id": "quiz-03-4-2", + "level": "2", + "url": "quiz-03.html#quiz-03-4-2", + "type": "Table", + "number": "126", + "title": "\\(\\Phi(z)\\)<\/span> Values", + "body": " Values 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 βˆ’1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 βˆ’1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 βˆ’1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 βˆ’1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 βˆ’1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 " +}, { "id": "exam-01", "level": "1", @@ -1265,7 +1310,7 @@ var ptx_lunr_docs = [ "level": "2", "url": "recitation-calculus-review.html#thm-FTC", "type": "Theorem", - "number": "126", + "number": "127", "title": "Fundamental Theorem of Calculus.", "body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: " }, diff --git a/course/notes-01-13.html b/course/notes-01-13.html index 36646fa..abba301 100755 --- a/course/notes-01-13.html +++ b/course/notes-01-13.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/notes-01-15.html b/course/notes-01-15.html index 2966157..5e4e891 100755 --- a/course/notes-01-15.html +++ b/course/notes-01-15.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/notes-01-20.html b/course/notes-01-20.html index 60cc674..fdda7a1 100755 --- a/course/notes-01-20.html +++ b/course/notes-01-20.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
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  • @@ -433,7 +434,7 @@ f(x) = F'(x) = 4x - 4x^3.
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    πŸ”—
    πŸ”—
    -PrevTopNext +PrevTopNext
    diff --git a/course/quiz-03.html b/course/quiz-03.html new file mode 100755 index 0000000..aa878f3 --- /dev/null +++ b/course/quiz-03.html @@ -0,0 +1,561 @@ + + + + + + + + + + + + + + + + +Quiz 3 + + + + + + + + + + + + + + + + + + + + +Skip to main content
    +

    Introduction to Probability and Statistics for the Life Sciences Spring 2026 MATH 1044

    +

    Andy Eisenberg

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    +Worksheet Quiz 3
    +

    +
    The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.
    πŸ”—
    +
    +
    Show all work unless instructed otherwise.
    πŸ”—
    +

    1.

    +
    Write either True or False for each of the following statements. No justification is required.
    πŸ”—
    +
    +

    (a)

    +
    Suppose a parameter \(\theta \in (-\infty, \infty)\) has likelihood function \(\mathcal{L}(\theta)\) with \(\L'(\theta) = (1 - \theta)e^{\theta}\text{.}\) Then the maximum likelihood estimation is \(\widehat{\theta} = 1\text{.}\)
    πŸ”—
    +
    +
    Solution.
    +
    True.
    πŸ”—
    +
    +
    πŸ”—
    +
    +
    πŸ”—

    (b)

    +
    Suppose \(X_1, \dotsc, X_n\) are random variables, and \(S = X_1 + \dotsb + X_n\text{.}\) Then, for sufficiently large \(n\text{,}\) \(S \approx \operatorname{N}(0, 1)\text{.}\)
    πŸ”—
    +
    +
    Solution.
    +
    False.
    πŸ”—
    +
    +
    πŸ”—
    +
    +
    πŸ”—

    (c)

    +
    Suppose we collect data to estimate the value of a parameter \(\theta\text{.}\) Based on the collected data, we find a 95% confidence interval \([a, b]\) and a 90% confidence interval \([c, d]\text{.}\) Then \(a \leq c \leq d \leq b\text{.}\)
    πŸ”—
    +
    +
    Solution.
    +
    True.
    πŸ”—
    +
    +
    πŸ”—
    +
    +
    πŸ”—
    πŸ”—

    2.

    +
    The heights of five plants are measured and recorded below. Find the sample mean, sample variance, and a 95% confidence interval around the sample mean for the heights of the plants. (Pretend that 5 measurements is enough for the CLT to apply.)
    πŸ”—
    +
    • \(x\) \(\Pr(X = x)\)
    + + + + + + + + + + + + + + + + + + + + + + + + +
    Plant \(i\) + +\(H_i\) (in)
    \(1\)\(13\)
    \(2\)\(14\)
    \(3\)\(10\)
    \(4\)\(15\)
    \(5\)\(17\)
    +
    Solution.
    +
    +
    Let \(A = \frac{H_1 + \dotsb + H_5}{5}\) be the average height of the sample. Then:
    +
    +\begin{align*} +A \amp = 13.8 \amp s^2 \amp = \frac{\sum H_i^2 - n(A^2)}{n - 1} = \frac{979 - 5(13.8^2)}{4} = 6.7 \\ +\sum H_i^2 \amp = 979 \amp \sqrt{\frac{s^2}{n}} \amp = \sqrt{\frac{6.7}{5}} \approx 1.16 +\end{align*} +
    +
    So the 95% confidence limits are:
    +
    +\begin{align*} +\mu_{\ell} \amp = 13.8 - 1.96(1.16) \approx \boxed{11.53} \\ +\mu_{h} \amp = 13.8 + 1.96(1.16) \approx \boxed{16.07} +\end{align*} +
    +
    πŸ”—
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    πŸ”—
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    +
    πŸ”—

    3.

    +
    Suppose a coin comes up heads with probability 0.6. We flip the coin 80 times, and let \(S\) be the number of heads. Estimate \(\Pr(42 \leq S \leq 50)\text{.}\) (Use a continuity correction if appropriate.)
    πŸ”—
    +
    +
    Solution.
    +
    +
    +\(S \sim \Bin(80, 0.6)\text{,}\) so \(\E(S) = (80)(0.6) = 48\) and \(\Var(S) = (80)(0.6)(0.4) = 19.2\text{.}\) By the CLT, \(S \approx \Norm(48, 19.2)\text{,}\) so:
    +
    +\begin{align*} +\Pr(42 \leq S \leq 50) \amp \approx \Pr\left( \frac{41.5 - 48}{\sqrt{19.2}} \leq Z \leq \frac{50.5 - 48}{\sqrt{19.2}}\right) \\ +\amp \approx \Pr(-1.48 \leq Z \leq 0.57) \\ +\amp = \Phi(0.57) - \Phi(-1.48) \\ +\amp \approx 0.7157 - 0.0694 \\ +\amp = \boxed{0.6463} +\end{align*} +
    +
    πŸ”—
    +
    +
    πŸ”—
    +
    +
    +
    πŸ”—
    Table 126. \(\Phi(z)\) Values
    πŸ”—
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    \(z\)0.000.010.020.030.040.050.060.070.080.09
    βˆ’1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455
    βˆ’1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559
    βˆ’1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681
    βˆ’1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823
    βˆ’1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985
    0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
    0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
    0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
    0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
    0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
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    +PrevTopNext +
    + +
    + +
    + + diff --git a/course/quizzes.html b/course/quizzes.html index 50ebd47..d878dd3 100755 --- a/course/quizzes.html +++ b/course/quizzes.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • @@ -277,6 +278,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—
    PrevTopNext diff --git a/course/recitation-calculus-review.html b/course/recitation-calculus-review.html index f614448..307353f 100755 --- a/course/recitation-calculus-review.html +++ b/course/recitation-calculus-review.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • @@ -347,7 +348,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—
    πŸ”—
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    πŸ”—
    +
    πŸ”—
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    πŸ”—

    5.

    Practice with the following integrals:
    πŸ”—
    diff --git a/course/recitations.html b/course/recitations.html index 8b71d5d..bdf4b70 100755 --- a/course/recitations.html +++ b/course/recitations.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
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  • diff --git a/course/root-1-2.html b/course/root-1-2.html index b368fec..f8a2006 100755 --- a/course/root-1-2.html +++ b/course/root-1-2.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;
  • diff --git a/course/syllabus.html b/course/syllabus.html index 73abb1b..5467e13 100755 --- a/course/syllabus.html +++ b/course/syllabus.html @@ -250,6 +250,7 @@ eBookConfig.enable_chatcodes = false;