From 6997eb083c26886bbf0d182bd50b8d516dbbf816 Mon Sep 17 00:00:00 2001 From: andyeisenberg Date: Tue, 31 Mar 2026 07:18:41 -0400 Subject: [PATCH] Update to PreTeXt project source. --- source/main.ptx | 1 + source/review/Exam-2-Review.ptx | 485 ++++++++++++++++++++++++++++++++ 2 files changed, 486 insertions(+) create mode 100644 source/review/Exam-2-Review.ptx diff --git a/source/main.ptx b/source/main.ptx index 7422001..10f5a8d 100644 --- a/source/main.ptx +++ b/source/main.ptx @@ -63,6 +63,7 @@ Exam Review + diff --git a/source/review/Exam-2-Review.ptx b/source/review/Exam-2-Review.ptx new file mode 100644 index 0000000..11777a6 --- /dev/null +++ b/source/review/Exam-2-Review.ptx @@ -0,0 +1,485 @@ + + +
+ Exam 2 Review + + +

+ Use the following problems to prepare for the exam. + There will be in-class review on Thursday, February 12. + Your recitation this week will also be exam review. +

+ + + + + Allowed Materials + +

+ You will be allowed to use a scientific calculator (not a graphing calculator, not a calculator app on your phone). + You may not share a calculator with another student; you must use your own calculator. +

+ +

+ You may bring a standard 3 in x 5 in index card with prepared notes. + You may use both sides of the notecard. + You must put your full name in the top right corner of the card, and turn it in along with your exam. +

+ + + + + + +

+ Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. + Let S be the number of heads. + Estimate the probability that 34 \leq S \leq 44. +

+ + + +

+ \Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294. +

+ + + + + +

+ Suppose a fair die is rolled 100 times, and let m be the average value of the rolls. + Estimate the probability that 3.45 \leq m \leq 3.55. +

+ + + +

+ \Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510. +

+ + + + + +

+ The heights of men in the US have a mean of 69 in and a variance of about 9 in^2, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in^2. +

+ + + + + +

+ Suppose the heights of 30 men are sampled, and a sample mean m is taken. + Find the expected value and variance of m. +

+ + + +

+ \E(m) = 69 and \Var(m) = 0.3. +

+ + + + + + +

+ Estimate the probability that m \geq 69.5. +

+ + + +

+ \Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814. +

+ + + + + + +

+ If the sampled group was women, estimate the probability that m \geq 64. +

+ + + +

+ \E(m) = 63.5 and \Var(m) \approx 0.208. \Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357. +

+ + + + + + + +

+ Suppose we flip a coin 100 times and count 60 heads. + Let p be the (unknown) probability that the coin comes up heads on a flip. + Give an approximate 95% confidence interval for the value of p. +

+ + + +

+ [0.504, 0.696]. +

+ + + + + +

+ Suppose in a sample of 100 people, 12 are left-handed. +

+ + + + + +

+ Give a 95% confidence interval for the proportion p of left-handed people. +

+ + + +

+ [0.056, 0.184]. +

+ + + + + + +

+ Give a 90% confidence interval. +

+ + + +

+ [0.066, 0.174]. +

+ + + + + + +

+ 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.) +

+ + + + + + + mouse i + weight W_i (g) + + + + 1 + 26 + + + + 2 + 32 + + + + 3 + 33 + + + + 4 + 20 + + + + 5 + 29 + + +
+ + + +

+ [23.39, 32.61]. +

+ + + + + +

+ 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.) +

+ + + + + + + plant i + height H_i (in) + + + + 1 + 15 + + + + 2 + 14 + + + + 3 + 18 + + + + 4 + 21 + + + + 5 + 17 + + +
+ + + +

+ [14.61, 19.39]. +

+ + + + + +

+ In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test. +

+ + + + + +

+ We have a coin which we've flipped many times, seeing an above-average number of heads. + We suspect the coin comes up heads more often than a fair coin would. +

+ + + + + + +

+ We find a coin on the street and wonder whether or not it's a fair coin. +

+ + + + + + +

+ We suspect there will be a difference in average weight of mice caught during the summer versus during the winter. +

+ + + + + + +

+ In a medical study, a group of patients are gathered and the proportion experiencing particular symptoms is measured. + A new drug intended to eliminate these symptoms is administered, after which the proportion experiencing symptoms is measured again. +

+ + + + + + +

+ Suppose we find a coin and wonder whether it's fair. + As a first test, we decide to flip the coin 200 times and count the number of heads. + If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05? +

+ + + + + +

+ Suppose we have a coin which we suspect comes up heads more often than a fair coin would. + As a first test, we decide to flip the coin 200 times and count the number of heads. + If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05? +

+ + + + + +

+ Suppose we find a six-sided die and wonder whether it's fair. + As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. + If we roll 22 1's, should we accept or reject the null hypothesis of a fair die at a significance level of 0.05? +

+ + + + + +

+ Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in^2. + We suspect that growing this plant in a greenhouse will increase its height. + A sample of 50 plants grown in a greenhouse has an average height of 40 in. + Is this significant enough data to reject the null hypothesis of equal means at the p = 0.05 significance level? +

+ + + + + + +

+ A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety. + A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams^2 (for an individual apple). + An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams^2. +

+ + + + + +

+ We have an established factory which produces coins that are close to fair. + We're opening up a second factory, and we'd like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. + We pick one sample coin from each factory, and flip each sample coin 100 times. + The coin from the established factory flips 52 heads in 100 flips. + The coin from the new factory flips 62 heads in 100 flips. + Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level? +

+ + + + + + +

+ 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? +

+ + + + + + + drink type + water + soda + tea + coffee + energy drinks + + + + favorite + 28 + 17 + 15 + 26 + 14 + + +
+ + + + + +

+ 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. +

+ + + + + + + # who respond to drug + 0 + 1 + 2 + 3 + 4 + + + + # of trials + 3 + 11 + 31 + 34 + 21 + + +
+ + + + + +

+ Is the data consistent with a binomial distribution with parameter \theta = 0.7? +

+ + + + + + +

+ What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation \widehat{\theta} for the probability that a person will respond to the drug? +

+ + + + + + +

+ Is the data consistent with a binomial distribution with the MLE value of \widehat{\theta}? +

+ + + + + + +
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