Exam 1 STAT - Practice Exam
A professional basketball player is a poor free-throw shooter. Consider situations in which he shoots a pair of free throws. The probability that he makes the first free throw is 0.51. Given that he makes the first, suppose the probability that he makes the second is 0.59. Given that he misses the first, suppose the probability that he makes the second one is 0.36. What is the probability that he makes one of the two free throws?
0.3855 Event A = he makes the first free throw, Ac=he misses the first one Event B=he makes the second free throw, Bc=he misses the second P(one of the two free throws) = P(A and Bc, or Ac and B) = P(A and Bc)+P(Ac and B) = P(A) x P(Bc | A) + P(Ac) x P(B|Ac) = 0.51x(1-0.59) + (1-0.51) x 0.36 = 0.3855
In a new split level class, there are 20 undergraduate students and 10 graduate students. The professor is planning on asking two students to complete a long survey about the course in the middle of the semester. If the two students are randomly selected without replacement, what is the probability that one undergraduate and one graduate student are chosen?
0.46 G: draw graduate, U: draw undergraduate P(one U and one G) = P(G and U , or U and G) = P(GandU)+P(U and G) = 10/30 x 20/29 + 20/30 x 10/29 =0.46
Given the probabilities shown below, what is P(A⋂B)? P(A) = 0.65, P(B) = 0.32, and P(B|A) = 0.75
0.4875 P(A and B) = P(A|B) xP(B) or P(A and B) = P(B|A) x P(A) = 0.75 x 0.65
A researcher is interested in learning how often college students watch TV. They randomly select 500 college students and ask, "In the past seven days, how many days did you watch television?" The random variable X represents the number of days an individual watched television. The probability distribution below shows the results of the survey. What is the expected value of X?
0x0.04 + 1x0.03 + 2x0.06 + 3x0.08 + 4x0.09 + 5x0.08 + 6x0.05 + 7x0.57 = 5.44
1380 randomly sampled registered voters from Tampa, Florida were asked if they thought workers who have illegally entered the US should be (i) allowed to keep their jobs and apply for US citizenship, (ii) allowed to keep their jobs as temporary guest workers but not allowed to apply for US citizenship, or (iii) lose their jobs and have to leave the country. The results of the survey by political ideology are shown below. What percent of these Tampa, Florida voters who identify themselves as conservatives are in favor of the citizenship option?
20.0% given condition: conservative P (citizenship | conservative) = 87/435 =0.2
Suppose you have a data set with the following five-number summary: Minimum = 26, Q1 = 32, Median = 34, Q3 = 38, Maximum = 58. What is the lower cutoff point for potential outliers? (By lower cutoff point, we mean the number such that anything smaller than it is an outlier).
23 IQR = Q3-Q1= 6, lower cutoff = Q1-1.5xIQR = 32-9 =23 upper cutoff = Q3+1.5xIQR = 38+9 =47
Jim is conducting a health survey of residents of Brazos County. He asks them their age, gender, height (in cm), weight (in kg), type of insurance, marital status, income, and the number of days they were sick in the last month. The possible values for marital status are married, widowed, divorced/separated, single/never married, and living with a partner but not married. How many of the variables in the health survey were categorical and how many were numeric?
3 categorical and 5 numerical
The boxplot below shows the distribution of heights of 16 undergraduate statistics students. Using this boxplot, approximately how many students are 69 inches or taller?
4 there are 25% of 16 observations fall within Q3 (69)and maximum --> 0.25 x 16=4
Suppose that you have a bag of 15 spheres and 15 cubes. Each of the shapes contains 4 red, 6 white, and 5 yellow. (So, for example, there are 4 red spheres, 6 white spheres, and 5 yellow spheres.) You randomly pick one item from a bag and see that it is a sphere. Then what is the chance that it is a red sphere? (Assume you're equally likely to grab a sphere or a cube - they're the same volume.)
4/15 R: draw a red one, S: draw a sphere P(R |S) = P(R and S)/P(S) = (4/30)/(15/30) or simply restrict the population as the spheres then P(R|S)=4/15
Below is a histogram of the IQ ("intelligence quotient") scores of 60 fifth-grade students chosen at random from one school. The range is defined as the maximum minus the minimum. What is the range of this distribution?
80
Eli wants to determine whether or not there is an association between gender and GPA among high school seniors in Ohio. He randomly selects 100 high school seniors at Amelia High School, a public school in southwest Ohio. He asks them to report their gender, as well as their current GPA. Out. of these 100 students, 57 were female and 43 were male. The average GPA of the 57 female high school seniors was 3.47. What is the population in this study?
All high school seniors in Ohio
A state governor wants to survey people in his state to determine their opinions on a particular law up for adoption. He decides to randomly select a few districts in his state and then survey all who live in those districts. Which of the following is a reason why the governor chooses to use a cluster sample?
Because it is impossible or not feasible to take a simple random sample or because the governor cannot get a list of every individual in the population.
A researcher wants to determine if a new exercise program helps people lose weight. She recruits over 1200 subjects (her statistician determined this number from sample size calculations) and randomly assigns them to either participate in the new exercise program or no exercise program. She tells those in the new program group to exercise 3 times a week and tells those in the no exercise group that they should not exercise for the duration of the study. Based on this information, which of the following principles of experimental design is not presented in this study?
Blinding
Which of the following study designs would give you the true value of the parameter?
Census
A researcher wants to determine if a new flu vaccine protects people from the flu better than the current vaccine. He has 100 subjects for his study. He believes that sex will affect how well the vaccine works, so within these 100 subjects, splits them up into males (48 subjects) and females (52 subjects). Within each sex, he randomly assigns half of the subjects to get the new flu vaccine and the other half to get the current flu vaccine. What type of study design is this?
Experimental Study: Block Design.
Consider the population of all students at your school. A certain proportion support removing vending machines. Your friend randomly samples 20 students from the school and uses the sample proportion who support removing vending machines to predict the population proportion at the school. You take your own separate random sample of 20 students and find the sample proportion that supports removing vending machines. How likely is it that the sample proportions are the same?
It is unlikely that the sample proportions will be exactly the same. However, they should be close to each other because the samples represent the same population.
Multiple myeloma is a cancer of the bone marrow currently without an effective cure. It affects primarily older individuals: It is almost never diagnosed in individuals under 40 years old, and its incidence rate (the number of diagnosed malignant cases per 100,000 individuals in the population) is highest among individuals 70 years of age and older. If you were to create a histogram to show the distribution of the incidence rate of multiple myeloma by age at diagnosis, how would you describe its skewness?
Left Skewed since 70 or higher are most common, and the maximum is close to 70 -->long left tail can be expanded to 0
Consider the following measures that can be used to describe a data set: mean, median, IQR, and standard deviation. Which of these following describes all the values that are not robust to outliers?
Mean and Standard Deviation. robust = not sensitive
For the distribution shown below, which point represents the mean, and which point represents the median?
Mean: Point B, Median: Point B. Median = 50% percentile, dividing data into half =B Mean = Median for a symmetric distribution
Below is shown a table of IRS income tax returns from 2013 by three income categories, along with whether or not that tax return was audited. Were having an income over $1,000,000 and being audited independent in 2013? Why or why not?
No, because 22/393 ≠ 1413 / 138242. Event A and B are independent, if and only if P(A and B)=P(A) x P(B) or P(A|B)=P(A) (or P(B|A)=P (B) ) From the contingency table, P(audited | income over 1000K) = approximately 22/393 P(audited) = approximately 1413/138242
Below is shown a table of IRS income tax returns from 2013 by three income categories, along with whether or not that tax return was audited. Were having an income over $1,000,000 and being audited independent in 2013? Why or why not?
No, because people with an income over $1 million were much more likely to be audited than everyone else. compare the probability of a certain event within the given condition, against the probability of the event within the entire population
Which of the following is an example of a voluntary response sample?
Radio asks people listening to call in and say whether or not they want a new stoplight at a busy intersection.
Taylor Swift is interested in learning what proportion of her fans watched her premiere Betty live at the ACM Awards on September 16, 2020. She takes a simple random sample of 400 fans and asks them. Of the 400 fans, 382 of them (95.5%) respond that they watched her perform. What is the statistic and what is its value?
The statistic is the proportion of the sampled Taylor Swift fans who watched the performance, which is 0.955.
A researcher wants to know more about the resting pulse rate of high school-aged children. She takes a random sample of 17 students at a local high school and measures their resting pulse rate. The boxplot below depicts the five-number summary for her data set. Based on this boxplot, what is the skewness of the data set?
Right Skewed the range of first 25% (min to Q1) < range of second 25% (Q1 to Q2) < range of the third 25% (Q2 to Q3) < range of the fourth 25% (Q3 to max) --> the individuals of lower pulse rate (located on the left) are more common
Because of the increasing nuisance of spam email messages, many start-up companies have emerged to develop e-mail filters. One such filter was recently advertised as being 81% accurate. This could mean one of four things. One thing it could mean is that 81% of valid email is allowed through. Let S denote the event that the message is spam and let B denote the event that the filter blocks the message. Which of the following expresses the statement, "81% of valid email is allowed through," as a conditional probability?
S: mail is a spam, Sc: valid mail B: mail is blocked, Bc: mail passes through % of valid mail is allowed through = P(Bc | Sc) (because 81% is the percentage of VALID MAILs that were not blocked P(Bc|Sc) = 0.81
Given the following situations, when would you use a histogram?
Showing the distribution of the GPA's of all of the students at Texas A&M. GPA is a numerical variable: histogram
A scientist is curious about whether tooth whitening gel actually worked randomly assigned 58 adults to use a whitening gel after brushing their teeth, while another 59 adults only brushed their teeth with fluoride toothpaste twice a day. Before the study started and at the end of two weeks of either brushing and using the gel or brushing only, the tooth whiteness was measured, and it was found that the average change in the shade was larger for the gel group. Which of the following can be concluded from the study?
There is evidence the gel causes teeth to get whiter, on average. Since it is experimental study, cause-effect association is established
A certain region has a population of 6.5 million. On any given day, the probability that a randomly selected resident decides to visit the amusement park is 1/5000. Assume that each resident's decision to visit is independent of the other residents. What is the probability that all the residents of this region will decide to go to the amusement park tomorrow (assuming the amusement park is open tomorrow)?
Very close to zero. (1/5000)^6.5million =approximately zero
Every year, the Higher Education Research Institute at UCLA conducts a large-scale survey of college freshmen on a variety of issues. From the 2013 survey, it was found that 44% of men and 20% of women reported feeling frequently stressed. Which of the following is the most plausible confounding variable that prevents us from concluding that female hormones might cause stressful feelings?
number of stressful events experienced Confounding variable: factor that may affect to dependent variable, but of which effect on outcome is not one's interest
