Stats Final
There is a strong correlation between the temperature and the number of skinned knees on playgrounds. Does this tell us that warm weather causes children to trip?
No. In warm weather, more children will go outside and play.
Fifty percent of people in a certain country like to cook and 56% of people in the country like to shop, while 27% enjoy both activities. What is the probability that a randomly selected person in the country enjoys cooking or shopping or both?
50+56-27 = .79
A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that the trucks will average 1.2 tickets a month, with a standard deviation of 0.8 tickets. a) If they have 21 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month? b) What assumption did you make in answering?
A) 1.2 x 21 = 25.2 b)It was assumed that each truck gets tickets independently.
A nutritionist wants to study the effect of storage time (6, 12, and 18 months) on the amount of vitamin C present in freeze dried fruit when stored for these durations. Vitamin C is measured in milligrams per 100 milligrams of fruit. Six fruit packs were randomly assigned to each of the three storage times. The treatment, experimental unit, and response are respectively:
A specific storage time, a fruit pack, amount of vitamin C
What is the correlation?
It measures the strength and direction of linear relationships.
A study was conducted on shoe sizes of students, reported in European sizes. For the men, the mean size was 44.71 with a standard deviation of 2.02. To convert European shoe sizes to U.S. sizes for men, use the equation shown below. USsize=EuroSize×0.7911−23.2 a) What is the mean men's shoe size for these responses in U.S. units? b) What is the standard deviation in U.S. units?
Mean: 12.17 Sx: 1.60
A citrus farmer has observed the following distribution for the number of oranges per tree. How many oranges does he expect on average?
Multiply the oranges by the prob and add them up
In a certain population of students, the number of calculators a student owns is a random variable X described by the following probability distribution: X 0 1 2 P(X) 0.5 0.2 0.3 What is the expected number of calculators owned by students from that population?
0.8
A basketball player makes 75% of his free throws. We want to estimate the probability that he makes 4 or more frees throws out of 5 attempts (we assume the shots are independent). To do this, we use the digits 1, 2, and 3 to correspond to making the free throw and the digit 4 to correspond to missing the free throw. If the table of random digits begins with the digits below, how many free throw does he hit in our first simulation of five shots? 19223 95034 58301
5
IQs among undergraduates at Mountain Tech are approximately Normally distributed. 95% of undergraduates have IQs between 100 and 120. What is the standard deviation of these IQs among graduates?
5
A study gathers data on the outside temperature during the winter, in degrees Fahrenheit, and the amount of natural gas a household consumes, in cubic feet per day. Call the temperature x and gas consumption y. The house is heated with gas, so x helps explain y. The least-squares regression line for predicting y from x is: y=1344−19x. On a day when the temperature is 20°F, the regression line predicts that gas used will be about
964 cubic feet.
Which of the following are true statements? I. The area under a normal curve is always 1, regardless of the mean and standard deviation. II. The mean is always equal to the median for any normal distribution. III. The interquartile range for any normal curve extends from μ+σ to μ−σ
I and II
Satellites send back nearly continuous data on the Earth's land masses, oceans, and atmosphere from space. How might researchers use this information in both the short and long term to help study changes in the Earth's climate?
In the short term, researchers can more accurately report weather patterns, including hurricanes and tsunamis. In the long term, this rise and fall of temperature and water levels can help in planning for future problems and guide public policy to protect our safety.
An experiment investigates a dietary approach to treating insomnia. Researchers randomly assigned 30 subjects to two treatment groups, one taking a high dose of calcium supplements and the other a placebo. Suppose some of the 30 subjects were very active people who walked a lot or got vigorous exercise several times a week, while others tended to be more sedentary, working office jobs and watching a lot of TV. Why might researchers choose to block the subjects by activity level before randomly assigning them to the calcium supplements and placebo groups?
People who engage in regular exercise might differ from others with respect to insomnia, and that additional variability could obscure the effectiveness of this treatment.
A grocery supplier believes that in a dozen eggs, the mean number of broken eggs is 0.6 with a standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them. a) How many broken eggs do you expect to get? b) What's the standard deviation? c) What assumptions did you have to make about the eggs in order to answer this question?
a) 0.6+0.6+0.6 = 1.8 b) 0.5squared + 0.5squared + 0.5squared = 0.75. Square root = .87 c) The cartons are independent of each other.
In a study appearing in a science journal, a research team reports that plants in southern England are flowering earlier in the spring. Records of the first flowering dates for 382 species over a period of 46 years show that flowering has advanced an average of 12 days per decade, an indication of climate warming, according to the authors. a) Identify the Who b) Identify the What c) Identify the When d) Identify the Where e) Identify the Why f) Identify the How g) The first flowering date variable is h) The year variable is i) The flower species variable is
a) 382 plant species in southern England b) The first flowering dates for 382 plant species in southern England c) The When for this situation cannot be determined from the given information. d) Southern England e) The study was conducted to determine whether plants are flowering earlier in the spring. f) The How for this situation cannot be determined from the given information. g) quantitative w/ units of days h) quantitative w/ units of years i) categorical
Public museums sometimes ask patrons to donate a minimum admission fee, but some people cheat. Suppose researchers alternately taped two posters over the donation point at a certain public museum. During one week, it was a picture of a mountain scene and during the other, it was a pair of staring eyes. They found that the average donation was significantly higher when the eyes poster was up than when the mountain scene was there. a) Identify the Who described by the data. b) Identify the What described by the data c) Identify the population of interest.
a) Each instance of a patron who used the donation point at the particular public museum b) The total donations received and the poster present for each case c) All people in honor system payment situations
Determine if the following statements are True or False. If False, explain briefly. a) Choose the linear model that passes through the most data points on the scatterplot. b) The residuals are the observed y-values minus the y-values predicted by the linear model. C) Least squares means that the square of the largest residual is as small as it could possibly be.
a) False. The line usually touches none of the points. Minimize the sum of the squared errors. B) True. The residuals are the observed y-values minus the y-values predicted by the linear model. c) False. Least squares means that the sum of the squares of all the residuals is minimized.
A study begun in 2011 examines the use of stem cells in treating two forms of leukemia. Each of the 20 patients entered one of two separate trials in which embryonic stem cells were to be used to treat the condition. a) Identify the Who described by the data. b) Identify the What described by the data c) Identify the population of interest.
a) The 20 leukemia patients b) The effects the treatments have on leukemia c) All people with these two forms of leukemia
The pie chart summarizes the genres of 130 first-run movies released one year. a) Is this an appropriate display for the genres? Why / why not? b) Which genre was least common?
a) Yes, because each movie falls into only one category and no categories overlap.
The pie chart shows the ratings assigned to 996 first-run movies released in a recent year. a) Is this an appropriate display for these data? Explain. b) Which was the most common rating?
a) Yes, because each movie falls into only one category and no categories overlap.
Researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side effect of drowsiness. To investigate this question, the researchers give the new medication to 50 adult volunteers who suffer from allergies. Forty-four of these volunteers report a significant reduction in their allergy symptoms without any drowsiness. This study could have been improved by
using a control group
Which of the following quantities is minimized by the least-squares regression line?
The sum of the squared differences between observed values of the response variable and values of the response variable predicted by the model.
A pizza delivery driver, always trying to increase tips, runs an experiment on his next 40 deliveries. He flips a coin to decide whether or not to call a customer from his mobile phone when he is five minutes away, hoping this slight bump in customer service will lead to a slight bump in tips. After 40 deliveries, he will compare the average tip percentage between the customers he called and those he did not. Is this experiment blinded? Can it be double-blinded? Explain. a) Is this experiment blinded? b) Can this experiment be double-blinded?
a) Yes, because the customers are not aware of the differing treatments. b) No, because the driver must know who he calls.
A copy machine dealer has data on the number x of copy machines at each of 89 customer locations and the number y of service calls in a month at each location. Summary calculations give x = 8.4, sx= 2.1, y= 14.2, sy= 3.8, and r = 0.86. What is the slope of the least squares regression line of number of service calls on number of copiers?
1.56
A college basketball player makes 5/6 of his free throws. Assuming free throw attempts are independent, the probability that he makes exactly three of his next four free throws is
4(1/6)^1 (5/6)^3
The heights of American men aged 18 to 24 are approximately normally distributed with mean 68 inches and standard deviation 2.5 inches. Only about 5% of young men have heights outside the range:
63 inches to 73 inches
The heights of American men aged 18 to 24 are approximately normally distributed with mean 68 inches and standard deviation 2.5 inches. Half of all young men are shorter than:
68 inches
The amount of cereal that can be poured into a small bowl varies with a mean of 1.9 ounces and a standard deviation of 0.5 ounces. A large bowl holds a mean of 2.7 ounces with a standard deviation of 0.3 ounces. You open a new box of cereal and pour one large bowl and one small bowl. a) How much more cereal do you expect to be in the large bowl than in the smaller bowl? b) What is the standard deviation of this difference? (c) If the difference follows the Normal model, what's the probability that the small bowl contains more cereal than the large one?
a) 0.8 b).58 c)0.085
The length of time the battery in a cell phone lasts before needing charging is a ________________ random variable.
Continuous
Which of the following best describes a response variable? A. The variable that might change in response to a change in the explanatory variable. B. A response variable is the variable you hope to predict or explain. C. A response variable is the dependent variable D. All of the above.
D. All of the above.
An introductory statistics student asks all her friends what their hourly pay is. When she later looks at her dataset, she is surprised to find someone who is paid $800 per hour. Which of the following is NOT a likely explanation?
One of her friends has an excellent job for a college student.
A citrus farmer has observed the following distribution for the number of oranges per tree. The farmer expects 32 oranges per tree on average. What is the standard deviation for this distribution?
Subtract each x value from the mean. Square each of those values and add together. Square root to get the answer
A pharmaceutical company conducts an experiment in which a subject takes 200 mg of a substance orally. The researchers measure how many hours it takes for half of the substance to exit the bloodstream. What kind of variable is the company studying?
quantitative variable
Following are the SAT math scores for an AP Statistics class of 20 students: 664, 658, 610, 670, 640, 643, 675, 650, 676, 575, 660, 661, 520, 667, 668, 635, 671, 673, 645, and 650. The distribution of scores is
skewed to the left
In measuring the center of the data from a right skewed distribution, the median might be preferred over the mean because:
the mean may be too heavily influenced by the larger observations and this gives too high an indication of the center
For the following density curve, which answer is true?
the proportion of outcomes exceeding 1.5 is equal to 0.25
You want to use simulation to estimate the probability of getting exactly one head and one tail in two tosses of a fair coin. You assign the digits 0, 1, 2, 3, 4 to heads and 5, 6, 7, 8, 9 to tails. Using the following random digits to execute as many simulations as possible, what is your estimate of the probability? 19226 95034 05756 07118
6/10
Will Guess takes a true-false test of 6 questions and has absolutely no idea of any of the answers. So, true to his name, he guesses on all of them. If 4 questions correct is passing, what is the probability that he passes the exam
.344
You play tennis regularly with a friend, and from past experience, you believe that the outcome of each match is independent. For any given match you have a probability of 0.6 of winning. The probability that you win the next two matches is:
0.36
A least squares regression line was fitted to the following data: A(1,11), B(3,7), C(5,2), D(7,1), and E(9,-2). What is the residual associated with point D
0.4
The area under the standard normal curve corresponding to -0.3 < z < 1.6 is:
0.5631
If P(A) = 0.24 and P(B) = 0.52 and A and B are independent, what is P(A or B)?
0.6352
Suppose that sixteen-ounce bags of chocolate chip cookies are produced with an actual mean weight of 16.1 ounces and a standard deviation of 0.1 ounces and is approx normal. The percentage of bags that will contain between 16.0 and 16.1 ounces is:
34
If we want to study the relationship between two quantitative variables such as fat and calorie content of food, what type of graphical display should be made?
A scatterplot
Gasoline: number of miles you drove since filling up, gallons remaining in your tank.
Number of miles driven would best be used as the explanatory variable, and gallons of gas remaining the response variable, to predict gas remaining based on miles driven. The scatterplot should have a negative, straight (linear), and moderate correlation.
A sociologist is studying the relationship between early childhood nutrition and academic achievement in middle school among children in a certain city. Which of the following statements about the variable "early childhood nutrition" is correct?
Early childhood nutrition is an explanatory variable.
Apples: weight in grams, weight in ounces.
Either weight in grams or weight in ounces can be the explanatory value. Then the other will be the response variable, because weight in ounces can be predicted by weight in grams, and weight in grams can also be predicted by weight in ounces. The scatterplot should have a positive, straight (linear), and strong correlation.
In a scatterplot, we try to describe the direction of an association. A positive direction is one that can be described how?
Generally runs from the lower left to the upper right.
An employer pays a mean salary for a 5-day workweek of $1250 with a standard deviation of $115. On the weekends, his salary expenses have a mean of $450 with a standard deviation of $62. What are the mean and standard deviation of his total weekly salaries?
Mean: 1250+350 = 1700 Sx = 115squared + 62squared = 17069. Square root = 131
A golfer keeps track of his score for playing nine holes of golf (half a normal golf round). His mean score is 88 with a standard deviation of 9. Assuming that the second 9 has the same mean and standard deviation, what are the mean and standard deviation of his total score if he plays a full 18 holes?
Mean: 176 Sx = 9squared + 9squared = 162. Square root = 12.7
The probability of a randomly selected adult having a rare disease for which a diagnostic test has been developed is 0.001. The diagnostic test is not perfect. The probability the test will be positive (indicating that the person has the disease) is 0.99 for a person with the disease and 0.02 for a person without the disease. If a randomly selected person is tested and the result is positive, the probability the individual has the disease is
0.047
Many online retailers keep data on products that customers buy, and even products they look at. What do online retailers hope to gain from such information?
B. Guidance in pricing and offers to help maximize profit C. Revenue from selling the data to marketing companies D. Guidance in designs and placement of links to personalize the Web site experience
Which of the following best describes the explanatory variable?
The explanatory variable explains or predicts changes in the response variable.
Which of the following statements are true? I. Correlation and regression require identified explanatory and response variables. II. Scatterplots require that both variables be quantitative. III. Every least-square regression line passes through x,y.
II and III
The expected value of the sum of two random variables, E(X+Y), is the sum of the expected values, E(X)+E(Y).
true
Consider n pairs of numbers . The following apply: x=5 and y=10 sx=4 and sy=10
yhat = 8.5 + 0.3x
Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume X is Normal with a mean of $360 and standard deviation $50. The value of P(X > $400) is:
0.2119
A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let X denote the number in the sample that say they support the increase. Suppose that 40% of all adults in Ohio support the increase. Which of the following is the approximate standard deviation σX ?
19
Every morning, my neighbor goes out walking. I observe that 20% of the time she walks with her beagle, 70% of the time she walks with her golden retriever, and 30% of the time she walks alone. What is the probability she walks both dogs at the same time? Hint: use the general addition rule.
20%
A recent newspaper article reported on Americans' eating habits. The article reported that only about one third of American adults ate 2 or more servings of fruit per day, and only 26% ate 3 or more servings of vegetables per day. Californians ate the most fruit, while Tennesseans ate the most vegetables. Which of the following is NOT a variable reported in this study?
American adults
The relationship between any two variables can be studied with a scatterplot.
False
The first sentence in Henry James's novel The Turn of the Screw has 62 words. The five number summary for the lengths of those words is 1,2,3.5,6,12. According to the 1.5 x IQR rule for identifying outliers, does this distribution have any outliers?
No, there are no outliers
Which of the following is not necessarily an outlier?
The maximum value in a dataset
If most of the measurements in a large data set are of approximately the same value except for a few measurements that are quite a bit larger, how would the mean and median of the data set compare and what shape would a histogram of the data set have?
The mean would be larger than the median and the histogram would be skewed with a long right tail.
Under which of the following conditions is it preferable to use stratified random sampling rather than simple random sampling?
The population can be divided into strata so that the individuals in each stratum are as much alike as possible.
A study begun in 2011 examines the use of stem cells in treating two forms of paraplegia. Each of the 30 patients entered one of two separate trials in which embryonic stem cells were to be used to treat the condition. a) Identify the Who described by the data. b) Identify the What described by the data c) Identify the population of interest.
a) The 30 paraplegia patients b) The effects the treatments have on paraplegia c) All people with these two forms of paraplegia
The variance of the sum of two random variables, Var(X+Y), is the sum of the variances, Var(X)+Var(Y).
false
A researcher for a consumer products company is field testing a new formula for laundry detergent. He has contracted with 60 families, each with two children, who have agreed to test the product. He randomly assigns 30 families to the group that will use the new formula and 30 to the group that will use the company's current detergent formula. The most important reason for this random assignment is that
randomization is a good way tocreate two groups of 30 families that are as similar as possible, so that comparisons can be made between the two groups.
You ask a sample of 370 people, "Should clinical trials on issues such as heart attacks that affect both sexes use subjects of just one sex?" The responses are in the table below. Suppose you choose one of these people at random. yes no Male 34 105 Female 46 185 What is the probability that the person said "Yes," given that she is a woman?
0.20
The probability model below describes the number of repair calls that an appliance repair shop may receive during an hour. On average, the shop receives 1.4 calls an hour, with a standard deviation of 0.92 calls. Find the mean and standard deviation of the number of repair calls the appliance shop should expect during a 7-hour day. Assume that a variation of 1 standard deviation is typical.
A typical 7-hour day will have about 9.8 to nothing repair calls. 1.4 x 7 = 9.8 0.92squared = 0.8464. 0.8464 x 7 = 5.9248. Squared = 2.43 9.8-2.43=7.37 9.8+2.43 = 12.23
Noting a recent study predicting the increase in cell phone costs, a friend remarks that by the time he's a grandfather, no one will be able to afford a cell phone. Explain where his thinking went awry.
He is extrapolating into the future. It is impossible to know if a trend like this will continue so far into the future.
In the U.S., shoe sizes are defined differently for men and women, but in Europe, both sexes use the same shoe size scale. The accompanying histogram shows the European shoe sizes of 269 male and female college students, converted from their reported U.S. shoe sizes. What might be the problem with either the mean or the median as a measure of center?
The distribution is bimodal, so it is more useful to report the modes than the mean or median.
Pollsters are interested in predicting the outcome of elections. Give an example of how they might model whether someone is likely to vote.
The pollsters might consider whether a person voted previously or whether he or she could name the candidates, which indicates a greater interest in the election.
b) Using a side-by-side bar chart, compare the proportions of the men and women who worked at or below minimum wage at each Age group. Summarize what you see. Which statement below summarizes what the side-by-side bar chart shows?
The proportion of female workers who work at minimum wage or less is greater than that of men at every age group.
Standardizing the variables will make the correlation 0.
The statement is false. Changing the units of one or both of the variables (or in this case, removing the units by standardizing the variable) will not affect the value of the correlation between those variables.
Which of the following is not a random variable?
The suit of a card randomly-selected from a 52-card deck.
A friend says "I flipped five heads in a row! The next one has to be tails!" Explain why this thinking is incorrect.
There is no law of averages for the short run. The first five flips do not affect the sixth flip.
Which of the following is TRUE about the correlation coefficient?
We should only find correlation between two variables if they are quantitative, the association is straight enough, and there are no outliers.
A bicycle shop plans to offer 2 specially priced children's models at a sidewalk sale. The basic model will return an income of $100 and the deluxe model $120. Past experience indicates that sales of the basic model will have a mean of 5.3 bikes with a standard deviation of 1.3, and sales of the deluxe model will have a mean of 3.9 bikes with a standard deviation of 0.5 bikes. The cost of setting up for the sidewalk sale is $200.
a) B=number of basic bicycles sold, D=number of deluxe bicycles sold, Income = 100B +120D -200
We need to survey a random sample of the 300 passengers on a flight from San Francisco to Tokyo. Name each sampling method described below: a) Pick the first 10 people that board the plane. b) Randomly generate 30 seat numbers and survey the passengers who sit there.
a) convenience b)srs
Some companies offer 401(k) retirement plans to employees, permitting them to shift part of their before-tax salaries into investments such as mutual funds. Employers typically match 50% of the employees' contribution up to about 6% of salary. One company, concerned with what it believed was a low employee participation rate in its 401(k) plan, sampled 45 other companies with similar plans and asked for their 401(k) participation rates. a) Identify the Who described by the data. b) Identify the What described by the data c) Identify the population of interest.
a) 45 companies with similar 401(k) plans b) 401(k) employee participation rates c) All companies with similar 401(k) plans
In a certain population of students, the number of calculators a student owns is a random variable X described by the following probability distribution: X 0 1 2 P(X) 0.5 0.2 0.3 What is the standard deviation of X?
0.8718
Forty volunteers suffering from insomnia agreed to participate in a month-long test. Half of the people in each of these groups were randomly assigned to an exercise program, while the others did not exercise. Those who engaged in exercise showed more improvement in their insomnia symptoms. Suppose some of the 40 subjects had maintained a healthy weight, but others were quite overweight. Why might researchers choose to block the subjects by weight level before randomly assigning some of each group to the exercise program?
B. People who are overweight could potentially lose more weight to exercise than those with a healthy weight, and thus are more likely to be thrown off their usual sleep patterns. This additional variability could be mistaken as effectiveness of the treatment. C. People who are overweight could be less responsive to exercise. This additional variability could obscure the effectiveness of the treatment. Your answer is correct. D. People who are overweight might have different sleep patterns than those with a healthy weight. This additional variability could obscure the effectiveness of the treatment.
A sociologist studying the relationship between early childhood nutrition and academic achievement in middle school among children in a certain city finds that the correlation between these two variables is 0.86. Which of the following conclusions can he draw from this study?
Children in this city who have a healthy diet in early childhood tend to do better in middle school.
An experiment investigates a dietary approach to treating anxiety disorder. Researchers randomly assigned 23 subjects to two treatment groups, one taking a high dose of vitamin C and the other a placebo. Suppose some of the 23 subjects were very active people who walked a lot or got vigorous exercise several times a week, while others tended to be more sedentary, working office jobs and watching a lot of TV. Why might researchers choose to block the subjects by activity level before randomly assigning them to the vitamin C and placebo groups?
People who engage in regular exercise might differ from others with respect to anxiety disorder, and that additional variability could obscure the effectiveness of this treatment.
Meteorologists utilize sophisticated models to predict the weather up to ten days in advance. Give an example of how they might assess their models.
They can use the models to predict the average temperature ten days in advance and compare their predictions to the actual temperatures.
d) Do the variables sex and politics appear to be independent? Explain.
They do not appear to be independent because the conditional distribution of politics is significantly different for at least one group for the two sexes.
Hotels sometimes ask guests to serve and record their own drinks, but some people cheat. Suppose researchers alternately taped two posters over the bar at a certain hotel. During one week, it was a picture of flowers and during the other, it was a pair of staring eyes. They found that the average payment was significantly higher when the eyes poster was up than when the flowers were there. a) Identify the Who described by the data. b) Identify the What described by the data c) Identify the population of interest.
a) Each instance of a guest who used the bar at the particular hotel b) The total payments received and the poster present for each case c) All people in honor system payment situations
A casino knows that people play slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose that a certain slot machine pays out an average of $0.89 with a standard deviation of $120. a) Why is the standard deviation so large? b) If the machine is played 5 times, what are the mean and standard deviation of the casino's profit? c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino's profit? d) Is the casino likely to be profitable?
a) Players lose a relatively small amount most of the time, but there are several possible payouts that are much larger. B) 0.
A rock concert producer has scheduled an outdoor concert. If it is warm that day, she believes she'll make a $20,000 profit. If it is cool that day, she thinks she'll make a $5,000 profit. If it is very cold that day, she thinks she'll suffer a $12,000 loss (consider these to be the only 3 weather options). Based upon historical records, the weather office has estimated the chances of a warm day to be .60; the chances of a cool day to be .25. What is the producer's expected profit?
$11,450
Items produced by a manufacturing process are supposed to weigh 90 grams. The manufacturing process is such, however, that there is variability in the items produced and they do not all weigh exactly 90 grams. The distribution of weights can be approximated by a Normal distribution with mean 90 grams and a standard deviation of 1 gram. About what percentage of the items will either weigh less than 87 grams or more than 93 grams? ibution with mean 90 grams and a standard deviation of 1 gram. About what percentage of the items will either weigh less than 87 grams or more than 93 grams?
0.3%
Sulé's job is just a few bus stops away from his house. While it can be faster to take the bus to work, it's more variable, because of variations in traffic. He estimates that the commute time to work by bus is approximately Normally distributed with a mean of 12 minutes and a standard deviation of 4 minutes. The commute time if he walks to work is also approximately Normally distributed with a mean of 16 minutes with a standard deviation of 1 minute. What is the probability that the bus will be faster than walking?
0.8340
We wish to draw a random sample of 5 unique numbers from a population of 50 households. Suppose the households are numbered 01-50, and that the relevant line of the random number table is: 11362 35692 96237 90842 46843 62719 64049 17823 The households selected are:
11, 36, 23, 08, 42
If a peanut M&M is chosen at random, the chances of it being of a particular color are shown in the table below. Color brown red yellow green orange blue Probability .3 .2 .2 .2 .1 ? The probability of randomly drawing a blue peanut M&M is
According to this distribution, it's impossible to draw a blue peanut M&M.
Scuba diving: depth, water pressure
Depth would be the explanatory water pressured would be response depth would most likely be used to predict water pressure A strong positive linear association
Scuba diving: depth, visibility
Depth would be the response visibility would be explanatory depth would most likely be predicted by using visibility A weak to moderate negative, possibly linear association
Every morning, my neighbor goes out walking. I observe that 20% of the time she walks with her beagle, 70% of the time she walks with her golden retriever, and 30% of the time she walks alone. Determine whether the following statement is true or false. We could find the probability of walking with a dog by adding P(beagle) and P(golden retriever).
False
When I feed my parrot breakfast in the morning, he either grunts happily and eats it, screams petulantly and eats it, or screams and flings it out of the cage. These events are disjoint. Are they also independent?
No
An experiment to measure the effect of giving growth hormones to girls affected by Turner's Syndrome was carried out recently in Vancouver. All 34 girls in the study were given the growth hormone and their heights were measured at the time the hormone was given and again one year later. No measurements were made on their final adult heights. Which of the following is NOT a problem with this experiment:
Nonresponse bias
A television news editor would like to know how local registered voters would respond to the question, "Are you in favor of the school bond measure that will be voted on in an upcoming special election?" A television survey is conducted during a break in the evening news by listing two telephone numbers side by side on the screen, one for viewers to call if they approve of the bond measure, and the other to call if they disapprove. This survey method could produce biased results for a number of reasons. Which of the following is the most plausible reason?
People who feel strongly about the issue are more likely to respond.
T-shirts at a store: price each, number sold
Price would be the explanatory variable Number sold would be the response variable. price would most likely be used to predict the number sold. A moderate to strong negative linear association
A study of body fat on 250 men collected measurements of 12 body parts as well as the percentage of body fat that the men carried. The accompanying display is a dotplot of their bicep circumferences (in centimeters). For the bicep measurements, would you report the mean, the median, or both? Explain briefly.
Report both. The distribution of bicep circumferences is unimodal and symmetric, so the mean and median should be very similar.
The toy hoop, a popular children's toy in the 1950s, has gained popularity as an exercise in recent years. But does it work? To answer the question, an exercise council conducted a study to evaluate the cardio and calorie-burning benefits of using this children's toy. Researchers recorded heart rate and oxygen consumption of adult participants, as well as their individual ratings of perceived exertion, at regular intervals during a 30-minute workout. Identify Who and What were investigated and the Population of interest.
The Who is the participants in the study and the What is the recorded heart rate, oxygen consumption, and perceived exertion. The population of interest is all adults.
Comment briefly on what you see from the display in part a.
The proportions of females and males for conservatives are significantly different from the proportions for the other categories.
A farmer has 80 lbs of apples and 100 lbs of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.8 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.6 dollars and the standard deviation is 0.2 dollars. It also costs him 5 dollars to bring all the apples and potatoes to the market. The market is busy with shoppers, so assume that he'll be able to sell all of each type of produce at that day's price. a) Define your random variables, and use them to express the farmer's net income. b) mean c) sx d) Do you need to make any assumptions in calculating the mean? e) Do you need to make any assumptions in calculating the standard deviation?
a) A=price per pound ofapples, P=price per pound of potatoes, Profit = 80A + 100P - 5 b) 80(.8)+100(.6) - 5 = 119 c) 0.2(80) + 0.2(100) = 656. Square root = 25.61 d) No; no assumptions are made in calculating the mean. e) Yes; the prices must be independent.
An insurance company estimates that it should make an annual profit of $150 on each homeowner's policy written, with a standard deviation of $4000. a) Why is the standard deviation so large? b) If the company writes only five of these policies, what are the mean and standard deviation of the annual profit? c)If the company writes 10,000 of these policies, what are the mean and standard deviation of the annual profit? d) Do you think the company is likely to be profitable? e)What assumptions underlie your analysis? f) Can you think of circumstances under which those assumptions might be violated?
a) There will be many gains of $150 with a few large losses. b) Mean: 150x5 = 750 Sx = 4000squared = 16000000. 16000000x5 = 80000000. Square root = 8944.27 c) Mean : 1500000 sx = 400000 d) Yes. $0 is 3.8 standard deviations below the mean for 10,000 policies. e) Losses are independent of each other. f) A major catastrophe with many policies in an area
The baseball hat size of a randomly chosen student in your statistics class is a _________ random variable
discrete
The zip code of a randomly selected person in the US is a ______________ random variable
discrete
Which of the following is NOT a characteristic of stratified random sampling?
every possible subset of the population, of the desired sample size, has an equal chance of being selected.
Another student surveys members of her student club and finds that the balance on their credit cards is strongly skewed. She would be better off reporting which statistic as representative of a "typical" balance (the center)?
median
The life span of a calculator battery is normally distributed with a mean of 35 hours and a standard deviation of 5 hours. What is the probability that a battery lasts more than 41 hours?
normalcdf(41,999,35,5) = 0.115
A researcher wishes to test a new drug developed to treat hypertension (high blood pressure). A group of 40 hypertensive men and 50 hypertensive women is to be used. The experimenter randomly assigns 20 of the men and 30 of the women to the placebo and assigns the rest to the treatment. The major reason for separate assignment for men and women is that
the new drug may affect men and women differently
Suppose that sixteen-ounce bags of chocolate chip cookies are produced with an actual mean weight of 16.1 ounces and a standard deviation of 0.1 ounces. Approximately what percentage of the bags will likely be underweight (i.e. less than 16 ounces)? Assume normality.
16
A factory makes silicon chips for use in computers. It is known that about 90% of the chips meets specifications. Every hour a sample of 18 chips is selected at random for testing. Assume a binomial distribution is valid. Suppose we collect a large number of these samples of 18 chips and determine the number meeting specifications in each sample. What is the approximate mean of the number of chips meeting specifications?
16.20
Suppose that 33% of people have a dog, 25% of people have a cat, and 13% of people own both. What is the probability that someone owns a dog or a cat?
33+25-13 = .45
A medical researcher collects health data on many women in each of several countries. One of the variables measured for each woman in the study is her weight in pounds. The following list gives the five-number summary for the weights of women in one of the countries. Country A: 100, 110, 120, 160, 200 About what percentage of Country A women weight between 110 and 200 pounds?
75%
Many grocery store chains offer customers a card they can scan when they check out and offer discounts to people who do so. To get the card, customers must give information, including a mailing address and e-mail address. The actual purpose is not to reward loyal customers but to gather data. What data do these cards allow stores to gather, and why would they want that data?
A. Items a customer purchases at the grocery store D. Frequency with which a customer makes purchases at the grocery store E. Amount of money a customer spends at the grocery store F. The geographical location of a customer's home B. To help determine the most profitable locations for new grocery stores C. To determine how best to market and advertise to different customers D. To analyze the effects of price changes on individual shoppers' purchases
All elementary school students: age, score on a reading test
Age would be the explanatory Score on reading test would be response age would most likely be used to predict score on reading test A moderate positive, possibly linear association
Which of the following are random variables? A. The closing price of a particular stock B. The sum of two rolled dice in a gambling game such as craps C. Whether the stoplight at a street corner is red or green, with 1=red and 2=green D. The shoe size of a randomly selected person in your statistics class E. All of the above
All of the above
A recent newspaper article reported on Americans' eating habits. The article reported that only about one third of American adults ate 2 or more servings of fruit per day, and only 26% ate 3 or more servings of vegetables per day. Californians ate the most fruit, while Tennesseans ate the most vegetables. Which of the following is NOT a variable reported in this study?
American adults
Apples: circumference (inches), weight (ounces).
Circumference in inches would best be used as the explanatory variable, and weight in ounces the response variable, to predict the weight based on circumference. The scatterplot should have a positive, straight (linear), and moderately strong correlation.
Exercise psychologists are investigating the relationship between lean body mass (in kilograms) and the resting metabolic rate (in calories per day) in sedentary males. Based on the following computer output, which of the following is the best interpretation of the value of the slope of the regression line? Predictor Coef StDev T P Constant 264.0 276.9 0.95 0.363 Mass 22.563 6.360 3.55 0.005 S=144.9 R-Sq = 55.7% R-sq (adj) = 51.3%
For each additional kilogram of lean body mass, the resting metabolic rate increases on average by 22.563 calories per day.
Which of the following are true statements? I. Two students working with the same set of data may come up with correct histograms that look different. II. Displaying outliers is less problematic when using histograms than when using stemplots. III. Histograms are more widely used than stemplots or dotplots because histograms display the values of individual observations.
I only
Which of the following statements are true? I. A sample survey is an example of an experimental study. II. An observational study requires fewer resources than an experiment. III. The best method for investigating causal relationships is an observational study.
II only
A national consumer magazine reported the following correlations: The correlation between car weight and car reliability (measured on a numeric scale) is -0.30. The correlation between car weight and annual maintenance cost is 0.20. Which of the following conclusions apply? I. Heavier cars are less reliable. II. Heavier cars cost more to maintain. III. Car weight is related more strongly to reliability than to maintenance cost.
III only
College freshmen: shoe size, grade point average.
Shoe size would best be used as the explanatory variable, and grade point average the response variable, to predict grade point average based on shoe size. The scatterplot should have no direction, no form, and a weak correlation.
Sensors in parking lots are able to detect and communicate when spaces are filled in a large covered parking garage next to an urban shopping mall. How might the owners of the parking garage use this information both to attract customers and to help the store owners in the mall make business plans?
The owners of the parking garage can advertise about the availability of parking. They can also communicate with businesses about hours when more spots are available and when they should encourage more business.
Cans of soft drinks cost $0.30 in a certain vending machine. What is the expected value and variance of daily revenue (Y) from the machine, if X, the number of cans sold per day has E(X)=125, and Var(X) = 50 ?
E(Y) = 37.5, Var(Y) = 4.5
Ian Walker, a psychologist at the University of Bath, wondered whether drivers treat bicycle riders differently when they wear helmets. He rigged his bicycle with an ultrasonic sensor that could measure how close each car was that passed him. He then rode on alternating days with and without a helmet. Out of 2500 cars passing him, he found that when he wore his helmet, motorists passed 3.35 inches closer to him, on average, than when his head was bare. [NY Times, Dec. 10, 2006]
Helmet; Categorical Distance; Quantitative
On a certain ship that sank, the probability of survival was 0.397. Among first class passengers, it was 0.397. Were survival and ticket class independent? Explain.
Yes, because the probability of survival and the probability of survival given a first class passenger are the same.