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  • Quiz 2
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  • @@ -394,7 +395,7 @@ 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 128.
    πŸ”—
    +
    Table 132.
    πŸ”—
    @@ -445,7 +446,7 @@ 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 129. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 133. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    @@ -480,7 +481,7 @@ 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 130. Probability Distribution for an Even-biased Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 134. Probability Distribution for an Even-biased Die
    πŸ”—
    @@ -674,7 +675,7 @@ 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 131. Example Distribution
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 135. Example Distribution
    πŸ”—
    @@ -799,7 +800,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    21.

    Consider \(X, Y\) with the joint distribution table below. Are \(X, Y\) independent?
    πŸ”—
    -
    Table 132. Joint distribution for \(X, Y\)
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 136. Joint distribution for \(X, Y\)
    πŸ”—
    @@ -834,7 +835,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—
    Answer.
    -
    Table 133. Joint Distribution
    πŸ”—
    \(X = 0\)
    +
    Table 137. Joint Distribution
    πŸ”—
    @@ -857,7 +858,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    23.

    Consider a random variable \(X\) with probability distribution below. Find \(\E(X)\text{.}\)
    πŸ”—
    -
    Table 134.
    πŸ”—
    \(X = 0\)
    +
    Table 138.
    πŸ”—
    @@ -966,7 +967,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    29.

    Consider a random variable \(X\) with probability distribution below. Find \(\Var(X)\text{.}\)
    πŸ”—
    -
    Table 135.
    πŸ”—
    \(x\) \(\Pr(X = x)\)
    +
    Table 139.
    πŸ”—
    diff --git a/course/Exam-2-Review.html b/course/Exam-2-Review.html index 4b6c346..bdfaabf 100755 --- a/course/Exam-2-Review.html +++ b/course/Exam-2-Review.html @@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
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  • @@ -390,7 +391,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—
    πŸ”—

    6.

    The weights of five mice are measured and recorded below. Give a 95% confidence interval for the sample mean weight of mice. (Pretend 5 measurements is large enough for the CLT to apply.)
    πŸ”—
    -
    Table 136.
    πŸ”—
    • \(x\) \(\Pr(X = x)\)
    +
    Table 140.
    πŸ”—
    @@ -423,7 +424,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—

    7.

    The heights of five plants are measured and recorded below. Give a 95% confidence interval around the sample mean for the heights of the plants. (Pretend 5 measurements is large enough for the CLT to apply.)
    πŸ”—
    -
    Table 137.
    πŸ”—
    mouse \(i\)
    +
    Table 141.
    πŸ”—
    @@ -513,7 +514,7 @@ eBookConfig.enable_chatcodes = false;
    πŸ”—

    19.

    A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. They survey their customers about their favorite drinks. Is the data below consistent with the null hypothesis that each type of drink will be equally preferred?
    πŸ”—
    -
    Table 138.
    πŸ”—
    plant \(i\)
    +
    Table 142.
    πŸ”—
    @@ -534,7 +535,7 @@ eBookConfig.enable_chatcodes = false;
    A particular drug is administered in 100 independent trials. In each trial, the drug is administered to four people, and we count how many respond to the drug. The table below shows how many trials have each different count of people who respond to the drug.
    πŸ”—
    -
    Table 139.
    πŸ”—
    drink type water
    +
    Table 143.
    πŸ”—
    diff --git a/course/Final-Exam-Review.html b/course/Final-Exam-Review.html index b679f40..40b0e68 100755 --- a/course/Final-Exam-Review.html +++ b/course/Final-Exam-Review.html @@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
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  • @@ -394,7 +395,7 @@ 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 140.
    πŸ”—
    • # who respond to drug 0
    +
    Table 144.
    πŸ”—
    @@ -445,7 +446,7 @@ 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 141. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 145. Probability Distribution for a Linearly Scaled Die
    πŸ”—
    @@ -480,7 +481,7 @@ 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 142. Probability Distribution for an Even-biased Die
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 146. Probability Distribution for an Even-biased Die
    πŸ”—
    @@ -674,7 +675,7 @@ 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 143. Example Distribution
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 147. Example Distribution
    πŸ”—
    @@ -799,7 +800,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    21.

    Consider \(X, Y\) with the joint distribution table below. Are \(X, Y\) independent?
    πŸ”—
    -
    Table 144. Joint distribution for \(X, Y\)
    πŸ”—
    \(x\) \(\Pr(x)\)
    +
    Table 148. Joint distribution for \(X, Y\)
    πŸ”—
    @@ -834,7 +835,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—
    Answer.
    -
    Table 145. Joint Distribution
    πŸ”—
    \(X = 0\)
    +
    Table 149. Joint Distribution
    πŸ”—
    @@ -857,7 +858,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    23.

    Consider a random variable \(X\) with probability distribution below. Find \(\E(X)\text{.}\)
    πŸ”—
    -
    Table 146.
    πŸ”—
    \(X = 0\)
    +
    Table 150.
    πŸ”—
    @@ -966,7 +967,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    29.

    Consider a random variable \(X\) with probability distribution below. Find \(\Var(X)\text{.}\)
    πŸ”—
    -
    Table 147.
    πŸ”—
    \(x\) \(\Pr(X = x)\)
    +
    Table 151.
    πŸ”—
    @@ -1114,7 +1115,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—
    πŸ”—

    44.

    The weights of five mice are measured and recorded below. Give a 95% confidence interval for the sample mean weight of mice. (Pretend 5 measurements is large enough for the CLT to apply.)
    πŸ”—
    -
    Table 148.
    πŸ”—
    \(x\) \(\Pr(X = x)\)
    +
    Table 152.
    πŸ”—
    @@ -1147,7 +1148,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    45.

    The heights of five plants are measured and recorded below. Give a 95% confidence interval around the sample mean for the heights of the plants. (Pretend 5 measurements is large enough for the CLT to apply.)
    πŸ”—
    -
    Table 149.
    πŸ”—
    mouse \(i\)
    +
    Table 153.
    πŸ”—
    @@ -1237,7 +1238,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    57.

    A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. They survey their customers about their favorite drinks. Is the data below consistent with the null hypothesis that each type of drink will be equally preferred?
    πŸ”—
    -
    Table 150.
    πŸ”—
    plant \(i\)
    +
    Table 154.
    πŸ”—
    @@ -1258,7 +1259,7 @@ A\cap B \amp = \{HHH, HHT\}
    A particular drug is administered in 100 independent trials. In each trial, the drug is administered to four people, and we count how many respond to the drug. The table below shows how many trials have each different count of people who respond to the drug.
    πŸ”—
    -
    Table 151.
    πŸ”—
    drink type water
    +
    Table 155.
    πŸ”—
    @@ -1288,7 +1289,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—
    πŸ”—

    59.

    Calculate the covariance of \(X\) and \(Y\text{:}\)
    πŸ”—
    -
    Table 152.
    πŸ”—
    # who respond to drug 0
    +
    Table 156.
    πŸ”—
    @@ -1306,7 +1307,7 @@ A\cap B \amp = \{HHH, HHT\}
    \(X = 1\)
    πŸ”—

    60.

    Calculate the covariance of \(X\) and \(Y\text{:}\)
    πŸ”—
    -
    Table 153.
    πŸ”—
    +
    Table 157.
    πŸ”—
    @@ -1331,7 +1332,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    62.

    Calculate the correlation of \(X\) and \(Y\text{:}\)
    πŸ”—
    -
    Table 154.
    πŸ”—
    \(X = 0\)
    +
    Table 158.
    πŸ”—
    @@ -1349,7 +1350,7 @@ A\cap B \amp = \{HHH, HHT\}
    \(X = 1\)
    πŸ”—

    63.

    Calculate the correlation of \(X\) and \(Y\text{:}\)
    πŸ”—
    -
    Table 155.
    πŸ”—
    +
    Table 159.
    πŸ”—
    @@ -1374,7 +1375,7 @@ A\cap B \amp = \{HHH, HHT\}
    πŸ”—

    65.

    A survey of local companies collects information about marketing budgets \(X\) and revenue \(Y\) (each measures in thousands of dollars), shown below. A linear regression gives the best linear fit as \(Y = 18.28 X + 29.69\text{.}\) What is the coefficient of determination \(r^2\text{?}\)
    πŸ”—
    -
    Table 156.
    πŸ”—
    \(X = 0\)
    +
    Table 160.
    πŸ”—
    @@ -1426,7 +1427,7 @@ A\cap B \amp = \{HHH, HHT\}
    \(x_i\) \(y_i\)
    πŸ”—

    66.

    A sample of 100 measurements are taken and a best fit line is calculated, resulting in the data below (the final line shows the sums for each column). Find the coefficient of determination.
    πŸ”—
    -
    Table 157.
    πŸ”—
    +
    Table 161.
    πŸ”—
    diff --git a/course/boxes.html b/course/boxes.html index c946705..8e2aef4 100755 --- a/course/boxes.html +++ b/course/boxes.html @@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
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  • Exam 2
    • πŸ”—
    -PrevTopNext +PrevTopNext
    diff --git a/course/frontmatter.html b/course/frontmatter.html index f817203..4cb4cff 100755 --- a/course/frontmatter.html +++ b/course/frontmatter.html @@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
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  • diff --git a/course/index.html b/course/index.html index 6a8b10b..a0f1910 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 bbf655b..feed6a1 100755 --- a/course/lunr-pretext-search-index.js +++ b/course/lunr-pretext-search-index.js @@ -1197,6 +1197,33 @@ var ptx_lunr_docs = [ "title": "", "body": " Suppose we have a coin that we're told is not fair and has a probability of of coming up heads. We suspect the probability is even higher than this. Throughout the following parts, do not use a normal approximation. We flip the coin 10 times and see 9 heads. Should we use a 1-tailed test or a 2-tailed test? Find the -value. Do we have strong enough evidence to reject the null hypothesis at a 0.05 significance level? Using a 1-tailed test, the -value is : The -value is the sum: , so we can reject the null hypothesis. What is the minimum number of heads we would need to see in 10 flips to reject the null hypothesis? We saw in part (a) that at least 9 heads is enough to reject. Then: Adding to the previous probabilities, . So 8 heads would not be enough to reject the null hypothesis, therefore the minimum number of heads to reject would be 9. What is the power of the test if the true value of the parameter is ? With : So the power of the test is . " }, +{ + "id": "quiz-05", + "level": "1", + "url": "quiz-05.html", + "type": "Worksheet", + "number": "", + "title": "Quiz 5", + "body": " Quiz 5 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. The joint and marginal distributions of and are given below. Find and . Joint Distribution Table Distribution Table for Distribution Table for . We have: Since is an indicator random variable, we have . For : So the correlation is: A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination. Sample Data best fit predicted res sum: The RSS is the sum of res , , and the SST is the sum of , . Then the coefficient of determination is: " +}, +{ + "id": "quiz-05-3-1", + "level": "2", + "url": "quiz-05.html#quiz-05-3-1", + "type": "Worksheet Exercise", + "number": "1", + "title": "", + "body": " The joint and marginal distributions of and are given below. Find and . Joint Distribution Table Distribution Table for Distribution Table for . We have: Since is an indicator random variable, we have . For : So the correlation is: " +}, +{ + "id": "quiz-05-4-1", + "level": "2", + "url": "quiz-05.html#quiz-05-4-1", + "type": "Worksheet Exercise", + "number": "2", + "title": "", + "body": " A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination. Sample Data best fit predicted res sum: The RSS is the sum of res , , and the SST is the sum of , . Then the coefficient of determination is: " +}, { "id": "exam-01", "level": "1", @@ -1400,7 +1427,7 @@ var ptx_lunr_docs = [ "level": "2", "url": "recitation-calculus-review.html#thm-FTC", "type": "Theorem", - "number": "127", + "number": "131", "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 8340ff4..796e916 100755 --- a/course/notes-01-13.html +++ b/course/notes-01-13.html @@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
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    +

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

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    Andy Eisenberg

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    +Worksheet Quiz 5
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    +
    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.
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    Show all work unless instructed otherwise.
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    1.

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    The joint and marginal distributions of \(X\) and \(Y\) are given below. Find \(\Cov(X, Y)\) and \(\rho_{X, Y}\text{.}\)
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    Table 127. Joint Distribution Table
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    \(x_i\) \(y_i\)
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    \(Y = 1\)\(Y = 2\)\(Y = 3\)
    \(X = 0\)\(0.1\)\(0.15\)\(0.1\)
    \(X = 1\)\(0.15\)\(0.2\)\(0.3\)
    Table 128. Distribution Table for \(X\)
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    \(x\)\(0\)\(1\)
    \(\Pr(X = x)\)\(0.35\)\(0.65\)
    Table 129. Distribution Table for \(Y\)
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    \(y\)\(1\)\(2\)\(2\)
    \(\Pr(Y = y)\)\(0.25\)\(0.35\)\(0.4\)
    Solution.
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    +
    +\(\Cov(X, Y) = \E(XY) - \E(X)\E(Y)\text{.}\) We have:
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    +\begin{align*} +\E(X) \amp = 0(0.35) + 1(0.65) = 0.65 \\ +\E(Y) \amp = 1(0.25) + 2(0.35) + 3(0.4) = 2.15 \\ +\E(XY) \amp = (0)(1)(0.1) + (0)(2)(0.15) + (0)(3)(0.1) + \\ +\amp \quad\, (1)(1)(0.15) + (1)(2)(0.2) + (1)(3)(0.3) = 1.45 \\ \\ +\Cov(X, Y) \amp = 1.45 - (0.65)(2.15) = \boxed{0.0525} +\end{align*} +
    +
    Since \(X\) is an indicator random variable, we have \(\Var(X) = (0.65)(0.35) = 0.2275\text{.}\) For \(\Var(Y)\text{:}\) +
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    +\begin{align*} +\E(Y^2) \amp = 1^2(0.25) + 2^2(0.35) + 3^2(0.4) = 5.25 \\ +\Var(Y) \amp = \E(Y^2) - (\E(Y))^2 = 5.25 - (2.15)^2 = 0.6275 +\end{align*} +
    +
    So the correlation is:
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    +\begin{align*} +\rho_{X, Y} \amp = \frac{\Cov(X, Y)}{\sqrt{\Var(X)\Var(Y)}} = \frac{0.0525}{\sqrt{(0.2275)(0.6275)}} \approx \boxed{0.139}. +\end{align*} +
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    πŸ”—

    2.

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    A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination.
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    Table 130. Sample Data
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    \(x_i\)\(y_i\)best fit predicted \(y_i\) +res\(^2\) +\((y - \textrm{avg }y)^2\)
    \(1.94\)\(3.31\)\(10.53\)\(52.13\)\(886.55\)
    \(2.63\)\(10.38\)\(17.37\)\(48.76\)\(515.34\)
    \(\vdots\)\(\vdots\)\(\vdots\)\(\vdots\)\(\vdots\)
    \(3.94\)\(19.47\)\(30.31\)\(117.35\)\(185.24 \)
    \(6.35\)\(53.42\)\(54.22\)\(0.64\)\(413.51\)
    sum:\(992.55\)\(992.55\)\(953.03\)\(38397.35\)
    Solution.
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    +
    The RSS is the sum of res\(^2\text{,}\) \(953.03\text{,}\) and the SST is the sum of \((y - \textrm{avg})^2\text{,}\) \(38397.35\text{.}\) Then the coefficient of determination is:
    +
    +\begin{gather*} +r^2 = 1 - \frac{\text{RSS}}{\text{SST}} = 1 - \frac{953.03}{38397.35} \approx \boxed{0.975} +\end{gather*} +
    +
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    • πŸ”—
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    -Theorem 127. Fundamental Theorem of Calculus. +Theorem 131. Fundamental Theorem of Calculus.

    If \(F(x)\) is an antiderivative of \(f(x)\)β€”i.e., if \(F'(x) = f(x)\)β€”then:
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    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 \(F(x)\text{.}\)
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    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 \(F(x)\text{.}\)
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    5.

    Practice with the following integrals:
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