MATH 1680 Unit 2
The event there is an in-class math test on Tuesday is A and the event that math class is cancelled on Tuesday is B. If these events are mutually exclusive events, using P(A) = 0.32, and P(B) = 0.11, what is P(B|A)?
0 - Remember that because A and B are mutually exclusive, it is impossible for the math class to be cancelled on Tuesday if there is an in-class math test on Tuesday. Therefore, P(B|A) = 0
If A and B are independent events with P(A) = 0.1 and P(B)) = 0.8, find P(A and B)
0.08 - Remember for that independent events, P(A and B) = P(A) * P(B) So plugging in these values we are given, we find that P(A and B) = (0.10) * (0.80) = 0.08
Given the probability of a student spending time watching TV is 0.89, and the probability of a student spending time reading and watching tv is 0.11, what is the probability of a student spending time reading given that the student spends time watching TV? Be sure to round your answer to two significant digits.
0.12 - Remember the multiplication rule for conditional probability: P(A and B) = P(A|B)P(B) Rearranging this, we find P(A|B) = P(A and B)/P(B) So if we think of A as being the event of a student spending time reading and B as the event of a student spending time watching TV, then we can plug in the known information to find: P(A|B) = 0.11/0.89 = 0.12
If the probability of a family owning a cat is 0.10, the probability of owning a dog is 0.20, and the probability of owning a cat is 0.08, what is the probability of a family owning a cat or a dog.
0.22 - Let A be the event owning a cat and B be the event owning a dog. It is helpful to write down the addition rule for probabilities: P(A or B) = P(A) + P(B) - P(A and B) Now, plugging in the known values, we find P(A or B) = 0.1 + 0.2 - 0.08 = 0.22 So the probability of owning a cat or a dog is 0.22.
Identify the parameter, p, in the following binomial distribution scenario. A basketball player has a 0.463 probability of making a free throw and a 0.537 probability of missing. If the player shoots 17 free throws, we want to know the probability that he makes no more than 6 of them. (Consider made free throws as successes in the binomial distribution.) Do not include 'p =' in your answer.
0.463 - The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a made free throw, so p = 0.463.
At a certain school, statistics class and computer science class meet at the same time, so it is impossible for a student to take both. If the probability that a student is taking statistics class is 0.10, and the probability that a student is taking computer science class is 0.67, what is the probability that a student takes statistics class or computer science class?
0.77 - Because it is impossible for a student to take both statistics class and computer science class, we see that they are mutually exclusive events. Therefore, we know that P(A AND B) = 0 (this is the defenition of mutually exclusive events). So for mutually exclusive events, the probability addiition rule becomes P(A OR B) = P(A) + P(B) - P(A AND B) = P(A) + P(B) So we find that P(A OR B) = P(A) + P(B) = 0.10 + 0.67 = 0.77 The probability that a student takes statistics class or computer science class is the sum of the individual probabilities, which is 0.77.
Given that the probability of a dad mowing the lawn on sunday is 0.89, and the probability of a dad washing his car on sunday and mowing the lawn on sunday is 0.73, what is the probability of a dad washing his car on a sunday given that the dad mowed the lawn on sunday?
0.82 Remember the multiplication rule for conditional probability: P(B AND A)=P(B|A)P(A) Rearranging, we see that P(B|A)=P(B AND A)P(A) So if we think of A as being the event a dad is mowing the lawn on Sunday and B as being the event a dad is washing his car on Sunday, then we can plug in the known information to find P(B|A)=0.730.89≈0.82
In a standard 52-card deck of playing cards, each card has one of four suits: spade, heart, club, or diamond. There are 13 cards of each suit. Alison thoroughly shuffles a standard deck, draws a card, then returns it to the deck, and shuffles again. She repeats this process until she has drawn nine cards. Find the probability that she draws at most three spade cards. Use Excel to find the probability. Round your answer to three decimal places.
0.834 - Note that this is a binomial probability. In this case, we want to find the probability of 0 to 3 successes, inclusive, where a success is drawing a spade. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 3, 9, and 1352=0.25, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.834274, which is 0.834 rounded to three decimal places.
The table below represents the probability density function for the random variable X. Find the standard deviation of X. x P(X = x) 0 1/4 1 1/4 2 1/4 5 1/4
1.87 Note that the mean (also called expected value) μ is the sum of the entries in the third column: μ=14+12+54=2 This value is used to compute the values in the fourth column. For example, the first entry is (0−2)2⋅14=1 Summing the values in the fourth column gives the variance: 1.0+0.25+0.0+2.25=3.5 Taking the square root of the variance gives the standard deviation: σ=3.5−−−√≈1.87
A spinner has eight equally sized regions labeled from 1 to 8. We spin the spinner five times. What is the probability that all five spins are odd? Write your answer as a simplified fraction.
1/32 - These are independent events because the first spin does not influence the second spin. By the multiplication rule for independent events, we can take the probability of each event and multiply them. There are 4 numbers on the spinner that are odd (1, 3, 5, and 7). The probability that the first spin is odd is 4/8, and the same is true for the probabilities of the second, third, fourth, and fifth spins. So the probability that all five spins are odd is (4/8) * (4/8) * (4/8) * (4/8) * (4/8) = 1024/32768 = 1/32
Sarah owns a farm and has lots of dogs on her property. In the table below, she has listed each of her dogs by name and has classified them according to size and color. At random, she picks one of the dogs to take with her to the pet store. What is the probability the selected dog is medium and brown? Enter the answer as a reduced fraction.
1/6 Sarah owns a total of 12 dogs. Of these, 5 are medium and 5 are brown. However, only 2 of the dogs are medium AND brown. Thus, the probability that Sarah selected a medium-sized brown dog is 2/12, or 1/6.
A casino features a game in which a weighted coin is tossed several times. The table shows the probability of each payout amount. To the nearest dollar, what is expected payout of the game Payout amount: 200 | 3800 | 190000 Probability: 0.126 | 0.03 | 0.0002
177 - The table shows the probability density function where the random variable takes on the values 200, 3800, and 190,000. To find the expected value, multiply each payout amount by its probability and round to the nearest dollar: 200(0.126) + 3800(0.03) + 190000(0.0002) = $177
Choose all of the true statements below: 1. The more chocolate cinnamon flavored coffee a company makes, the more they sell. 2. The more sandwiches someone orders, the more the cost will be. 3. The fewer absences a student has, the higher their grade. 4. The more hours a person works, the less free time they have.
2. The more sandwiches someone orders, the more the cost will be. 3. The fewer absences a student has, the higher their grade. 4. The more hours a person works, the less free time they have. In a supply and demand model, there first needs to be a demand for a product, before the supply will increase. Just because a manufacturer makes more of a product does not mean that they will sell more of that product.
Thirty students took a calculus exam. Of the students who passed the exam, 8 students earned an A, 9 students earned a B, 6 students earned a C, and 3 earned a D. If a single student is randomly chosen, what is the probability that the student did NOT pass the exam? Provide the final answer as a reduced fraction.
2/15 - The number of students who passed the exam is 8 + 9 + 6 + 3 = 26. Since a total of 30 students took the exam, there are 30 - 26 = 4 students who failed. Thus, the probability that the selected student failed the exam is 4/30, or 2/15.
Given SSR and SST, find SSE SSR = 16 SST = 19
3 SST = SSR + SSE 19 = 16 + SSE 19 - 16 = SSE 3 = SSE
Kyle is shopping at a mall where there are 11 clothing stores, 3 home goods stores, and 5 electronics stores, all with entrances from the parking lot. If Kyle randomly selects a store to enter, what is the probability that the store is a home goods store? Give your answer as a fraction.
3/19 There are 3 homes good stores and a total of 11 + 3 + 5 = 19 stores. So, the probability of choosing a home goods store is 3/19
A group of 145 students at an elementary school were asked if they prefer the color orange to the color green. The results are shown in the table below. Given that a randomly selected survey participant is male, what is the probability that this student prefers the color green? Enter the answer as a simplied fraction Male Female Total Orange 36 66 102 Green 16 27 43 Total 52 93 145
4/13 The total number of those that are males are 52. Of these, 16 prefer green to orange. Therefore, the probability that a randomly selected male student that was surveyed prefers green is 16/52 = 4/13.
Data is collected on the relationship between time spent playing video games and time spent with family. The data is shown in the table and the line of best fit for the data is y = - 0.45x + 81.83. Assume the line of best fit is significant and there is a strong linear relationship between the variables. Video Games Family 40 64 55 56 70 51 85 43 According to the line of best fit, what would be the predicted number of minutes spent with family for someone who spent 79 minutes playing video games? Round the final answer to 2 decimal places.
46.28 - Substitute 79 for x into the line of best fit to estimate the number of minutes spent with family for someone who spent 79 minutes playing video games: y = -0.45(79) + 81.83 = 46.28
Will wanted to track the growth of various fruits in his garden, so he decided to label them. His garden had APPLES labeled 1, 2, 3, 4, 5, 6, LEMONS labeled 1, 2, and MELONS labeled 1, 2, 3. If a single fruit is picked at random, what is the probability that the fruit is an APPLE or has an EVEN number? Provide your answer as a fraction.
8/11 - There are a total of 6 + 2 + 3 = 11 fruits. Of these, there are 6 apples, 1 even labeled lemon, and 1 even labeled melon. Thus the total number of fruits we are interested in is 8, so the answer is 8/11.
Given the following information about events A, B, and C, determine which pair of events, if any, are independent and which pairs are mutually exclusive. P(A) = 0.26 P(A|B) = 0.26 P(B) = 0.5 P(B|C) = 0 P(C) = 0.45 P(C|A) = 0.26 1: B and C are independent 2: A and C are mutually exclusive 3: A and B are independent 4: A and C are independent 5: B and C are mutually exclusive 6: A and B are mutually exclusive
A and B are independent B and C are mutually exclusive Note that P(A|B) = P(A), so A and B are independent. Also, P(B|C) = 0, so B and C are mutually exclusive.
An event is _______.
A subset of the set of all outcomes of an experiment.
Which of the pairs of events below is dependent? 1: drawing a 2 and drawing a 4 with replacement from a standard deck of cards 2: rolling two 4's from two rolls of a standard die 3: drawing a heart and drawing another heart with replacement from a standard deck of cards 4: drawing a face card and then drawing a 3 without replacement from a standard deck of cards
Drawing a face card and then drawing a 3 without replacement from a standard deck of cards. Dependent events are events in which the knowledge of one event occuring has an effect on the chance of the other event occurring. When two cards are drawn without replacement, the result of the first draw can affect the probability of the second one.
A maze has 25 possible paths. Let x represent the number of paths a subject tries in order to traverse the maze. Find the mean and the standard deviation of the probability distribution using excel Enter the mean and round the standard deviation to 2 decimal places. x P(x) 1 0.0625 2 0.0625 3 0.05 4 0.0375 5 0.0375 6 0.0625 7 0.0625 8 0.05 9 0.0375 10 0.0375 11 0.05 12 0.05 13 0.04 14 0.03 15 0.03 16 0.0375 17 0.0375 18 0.03 19 0.0225 20 0.0225 21 0.0375 22 0.0375 23 0.03 24 0.0225 25 0.0225
Mean = 220.9625, Standard Deviation = 7.05 - The mean and the standard deviation of the probability distribution can be calculated using Excel. Step 1: Place the values of the random variable, x, in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Add the elements in the third column to find the mean, μ, which is 11.2. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 175.1. Step 4: Calculate the variance by subtracting the square of the mean from ∑x2P(x). The variance is 49.66. Step 1: Find the square root of the variance to find the standard deviation. The standard deviation, σ, rounded to two decimal places, is 7.05.
A deck of cards contains RED cards numbered 1, 2, and BLUE cards numbered 1, 2, 3, 4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card. Drawing the Blue 4 is an example of which of the following events? Select all correct answers: B' O' R or E R' R or O
O' R or E R' Because the card is blue and the number is even, the card is an example of B and E. Therefore, it is also an example of R', O' and R or E.
Let M be the event that a randomly chosen student is a girl. Identify the answer which expresses the following with correct notation: Of all the students who play a musical instrument, the probability that a randomly chosen student is a girl. P(M and G) P(G) and P(M) P(M|G) P(G|M)
P(G|M) Remember that in general P(A|B) is read as "The probability of A given B," or equivelantly, as "Of all the times B occurs, the probability that A occurs also." So in this case, the phrase "Of all the students who play a musical instrument" can be rephrased to mean "Given that a student plays a musical instrument," so the correct answer is P(G|M).
Let C be the event that a randomly chosen person went to college. Let J be the event that a randomly chosen person has a job. Identify the answer which expresses the following with correct notation: Given that the person went to college, the probability that a randomly chosen person has a job.
P(J|C) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the person went to college, so the correct answer is P(J|C)
A scientific study on black lung from coal mining gives the following data table: Months of experience: 56, 57, 60, 61, 62 % increase in chance: 143, 137, 117, 164, 123 Using Technology, it was determined that the total sum of squares (SST) was 1360.82 and the sum of squares due to error (SSE) was 1335.96. Calculate R-Squared and determine its meaning. Round your answer to four decimal places.
R-squared = 0.0183 Therefore 1.83% of the variation is observed by y-values can be explained by the estimated regression equation. R^2 = 1 - SSE/SST R^2 = 1 - 1335.96/1360.82 R^2 = 1 - 0.9817 R^2 = 0.0183 R^2 = 1.83%
A scientific study on bird migration distance gives the following data table. Migration Distance: 54, 60, 63, 71, 72 Average # Birds/Flock: 103, 133, 136, 169, 174 The least squares regression line was found. Using technology, it was determined that the total sum of squares (SST) was 3386 and the sum of squares of regression (SSR) was 3344. Calculate R-squared. Round your answer to four decimal places.
R-squared: 0.9876 R^2 = SSR/SST R^2 = 3344/3386 R^2 = 0.9876
Of the following pairs of events, which pair has mutually exclusive events?
Rolling a sum of 9 from 2 rolls of a standard die and rolling a 2 for the first roll- Mutually exclusive events are events that cannot occur together. In this case, rolling a sum of 9 from 2 rolls of a standard die and rolling a 2 for the first roll are two events that cannot possibly occur together.
GIven SSE and SST, find SSR. SSE 28 SST 63
SSR 35 SST = SSR + SSE 63 = SSR + 28 63 - 28 = SSR 35 = SSR
Data is collected on the relationship between time spent playing video games and time spent with family. The data is shown in the table and the line of best for the data is y = -0.93x + 89. Video Games Family 30 60 45 50 60 30 75 20 According to the line of best fit, the predicted number of minutes spent with family for someone who spent 33 minutes playing video games is 58.31, is it reasonable to use this line of best fit to make the above prediction?
The estimate, a predicted time of 58.31, is both reliable and reasonable - The data in the table only includes video game times between 30 and 75 minutes, so the line of best fit gives reliable and reasonable predictions for values of x between 30 and 75. Since 33 is between these values, the estimate is both reliable and reasonable.
Charlie owns a computer repair service. For each computer, they charge $50 plus $45 per hour of work. A linear equation that expresses the total amount of money Charlie earns per computer is y = 50 + 45x. What are the independent and dependent variables? What is the y-intercept and the slope?
The independent variable (x) is the amount of time Charlie fixes a computer. The dependent variable (y) is the amount, in dollars, Charlie earns for a computer. Charlie charges a one-time fee of $50 (this is when x=0), so the y-intercept is 50. Charlie earns $45 for each hour they works, so the slope is 45.
The statistician for an ice cream shop has produced a beast fit line for the relationship between average daily sales in 1000s of dollars, y, and the high temperature for the day in celcius, x. The equation for the line is y = 0.21x - 3.2. If the sales for a particular day were $3023 and the high temperature was 32C, were the sales above or below average
The sales were below average because the residual is negative.
True or False? A trial is one specific execution of an experiment.
True
Is the statement below true or false? Independent is the property of two events in which the knowledge that one of the events occurred does not affect the chance the other occurs.
True Independent is defined as the property of two events in which the knowledge that one of the events occurred does not affect the chance the other occurs.
In a large population, about 10% of people do not like the taste of cilantro, a herb used in cooking. A researcher takes a random sample of 15 people and surveys whether they like cilantro. Use the binomial distribution to compute the probability that exactly 6 of the people in the sample do not like cilantro. Identify the following information required to find the probability of people who do not like the taste of cilantro. Trials n= successes x= probability of not liking cilantro p=
n = 15 x = 6 p = 0.1 We determine n, the number of trials, by reviewing the context. The researcher randomly sampled 15 people. We determine x, the number of successes, by determining what results we are looking for. In this case, we'd like to know the probability for the number of times exactly 6 people in the sample that do not like the taste of cilantro. Finally, we determine the probability from the context, as we're told about 10% of people do not like the taste of cilantro.
Suppose you computed r = 0.261 using n = 23 data points. Using the critical values table below, determine if the value of r is significant or not.
r is not significant because it is between the positive and negative critical values
GIven that n= 19, data points are collecting when studying the relationship between average and daily temperature and time spent watching television, use the critical values table below to determine if a calculated value of r = 0.978 is significant or not.
r is significant because it is not between the positive and negative critical values
An art dealer is trying to determine whether the age of a painting plays a significant role in its price at auction. They collect the sale data for 20 paintings, including the painter, the title of the painting, the age of the painting at auction, and the sale at auction (in millions $). The data is provided below. Use Excel to calculate the correlation coefficient, r, between the year and price data sets, rounding to two decimal places.
r= -0.23 The correlation coefficient can be calculated easily with Excel using the built-in CORREL function. 1. Open the accompanying data set in Excel. 2. In an open cell, type "=CORREL(C2:C21,D2:D21)", and then hit ENTER You could label the result of this cell by writing "Correlation coefficient" or "r" in an adjacent open cell. The correlation coefficient, rounded to 2 decimal places, is r = -0.23
A superintendent of a school district wants to understand the link between attendance and grade point average, or GPA. The GPA is out of 4.0 points, with 4.0 being the highest possible. The percent attendance and GPA of 30 randomly sampled students from the superintendent's district were collected. This sample data is provided below. Use Excel to calculate the correlation coefficient r between the two data sets. Round your answer to two decimal places. GPA Attendance 3.29 81% 3.17 68% 3.35 88% 3.17 83% 3.6 93% 2.03 60% 1.54 62% 3.63 97% 3.17 81% 3.18 64% 1.58 75% 3.21 95% 3.87 97% 3.29 67% 1.74 75% 2.86 68% 2.05 52% 3.04 82% 3.97 88% 3.1 70% 3.48 98% 1.63 77% 2.16 59% 3.23 92% 1.19 72% 3.14 93% 3.55 81% 3.29 81% 2.89 86% 3.14 94%
r= 0.61 - The correlation coefficient can be calculated easily with Excel using the built-in CORREL function. 1. Open the accompanying data set in Excel. 2. In an open cell, type "=CORREL(A2:A31,B2:B31)", and then hit ENTER. You could label the result of this cell by writing "Correlation coefficient" or "r" in an adjacent open cell. The correlation coefficient, rounded to two decimal places, is r≈0.61.
Which of the following are feasible equations of a least squares regression line for the number of cars produced each month as a function of the number of months the plant has been open? The plant has been open for 2 years. Select all that apply y = -215 + 5x y = 5 + 215x y = -45 + 3005x y = 215 - 5x
y = 5 + 215x y = 215 - 5x A plant cannot produce a negative number of cars, so any equation that results in a negative number produced must not be feasible. Although the equation y = 215 - 5x has a negative slope, any month in the first two years still produces a positive number of cars.
The data below shows the duration of eruption (in seconds) of a geyser in a national park and find the height (in feet) of the eruptions for a typical day. Use Excel to find the best fit linear regression equation, where duration of eruption is the explanatory variable. Round the slope and intercept to one decimal place. Duration Height 240 140 237 154 122 140 267 140 113 160 258 140 232 150 105 150 186 160 248 155 243 125 241 136 214 140 114 155 272 130 227 125 237 125 238 139 203 125 270 140 218 140 226 135 250 141 245 140 120 139 267 110 103 140 270 135 241 140 239 135
y=-0.01x + 157.8 To find the best fit for the given data, use Excel. 1. Open Excel. Enter the values for duration in column A and height in column B. Highlight all the cells containing the data. 2. From the Insert tab, select Scatter. Use Scatter with only Markers, the first type of scatter chart. A simple plot is shown. 3. To add the linear fit to the graph, click anywhere inside the graph area. Select the Layout tab from the Chart Tools. Click on the Trendline icon and select the Linear Trendline option. The line of best fit is added to the graph. 4. For the equation of the line of best fit, click on Trendline and select More Trendline Options. Check the Display Equation on chart box. The equation of line of best fit is shown on the graph. Optional Instructions: Right click on the Trendline and select Format Trendline. Select Linear on the right side of the screen and select Display Equation on chart. 5. To change the number of decimal places in the trendline equation, right-click on the equation for the trendline and select Format Trendline Label.... 6. Select Number under Category, change the number of decimal places to 1, and click Close. Thus, the equation of line of best fit with slope and intercept rounded to one decimal place is yˆ=−0.1x+157.8.
