Probability
A fair coin comes up tails 4 times in a row. What is the probability that the coin will come up heads on the next flip?
50%
A marketing survey compiled data on the number of personal computers in households. If X = the number of computers in a randomly-selected household, and we omit the rare cases of more than 5 computers, then X has the following distribution:
X 0 1 2 3 4 5
P(X) 0.24 0.37 0.20 0.11 0.05 0.03
What is the probability that a randomly chosen household has at least two personal computers?
0.20 + 0.11 + 0.05 + 0.03 = 0.59
How do we check the Normal condition for the sampling distribution of sample means?
Either the Population is Normal (regardless of sample size) OR the Central Limit Theorem n>=30 is met (regardless of the shape of the population)
Security workers at an airport randomly choose one of the first 50 people to pass through a checkpoint for extra security screening. After that person, they choose every 50th person who passes through for extra screening as well. What kind of sample is this?
Systematic random sample
A factory overseer selects 404 threaded rods at random from those produced that week at the factory, then she tests their tensile strength.
Identify the population and sample in this setting.
The population is threaded rods produced at the factory that week; the sample is the 404 threaded rods selected.
The population is the entire group of people or things we want to study.
The sample is the part of the population that we actually collect data from.
What are two ways to check mathematically that two events are independent?
1) P(A|B)=P(A)
2) P(A and B)=P(A)*P(B)
A large bakery uses a machine to dispense frosting onto its cookies. A certain type of cookie gets 4 dots of frosting on top of each cookie. The dots of frosting have a mean mass of μ = 3 grams and a standard deviation of σ = 0.25 grams. We can assume that the masses of the dots of frosting are independent from each other. Let T be the total mass of the dots of frosting on a randomly chosen cookie. What is μT?
When do we check the 10% condition?
When sampling without replacement.
How do the shape, center, and spread change when you add/subtract the same value to every score?
shape: unchanged
center: add/subtract the same value
spread: unchanged
How do you test for an outlier?
Must fall more than 1.5 X IQR above the third quartile or below the first quartile
What is the difference between disjoint events and independent events?
Disjoint means the events cannot occur at the same time (there is no intersection). Independent means the outcome of one event won't affect the outcome of the other.
Casey ran out of time while taking a multiple-choice test and plans to guess on the last 10 questions. Each question has 5 possible choices, one of which is correct. Let X = the number of answers Casey correctly guesses in the last 10 questions.
What is the formula to calculate P(X=2)?
(10 2) (1/5)^2 * (4/5)^8
Explain the difference between accuracy and precision in statistics.
Accuracy = unbiased (the center of the sampling distribution is equal to the parameter)
Precision = low variability (larger sample size reduces variability of sampling distributions)
What are both of the important features of a density curve?
1. total area under curve has to equal 1
2. all of it is above or on the horizontal axis
The distribution for the number of pairs of shoes that a random sample of 20 male students own is Right Skewed with a mean of 11.6 and a standard deviation of 9.3. Jackson owns 22 pairs of shoes.
Calculate and interpret the z-score for Jackson.
z=1.12
Jackson's number of pairs of shoes is 1.12 standard deviations above the mean of 11.6 shoes.
If P(A)=0.18, P(B)=0.41, and P(A∩B)=0.24, then what is P(A∪B)?
P(A∪B) = P(A) + P(B) - P(A∩B) = 0.18 + 0.65 - 0.24 = 0.59
Jeremiah makes 25% of the three-point shots he attempts. For a warm up, Jeremiah likes to shoot three-point shots until he successfully makes one. Let M be the number of shots it takes Jeremiah to successfully make his first three-point shot. Assume that the results of each shot are independent.
Find the probability that Jeremiah's first successful shot occurs on his 3rd attempt.
The answer is : 0.140625
Big Town Fisheries recently stocked a new lake in a city park with 2,000 fish of various sizes. The distribution of the lengths of these fish is approximately normal. They claim that the mean length of the fish is 8 inches. If this is true, which is more likely? Justify your answer.
a) A random sample of 15 fish having a mean length that is greater than 10 inches
b) A random sample of 50 fish having a mean length that is greater than 10 inches
(a) is more likely because the sampling distribution of sample mean lengths of the smaller sample size (15) is more spread out than that of the larger sample size (50) therefore sample mean lengths further from the mean (8 inches) are more likely to occur
What is the purpose of random assignment?
The purpose of random assignment in an experiment is to be able to establish cause and effect by creating roughly equivalent groups so that any difference in responses can be attributed to the treatments.
When do we check the Large Counts Condition?
When we are using the Normal distribution to approximate the Binomial distribution, or when finding the shape of the sampling distribution of sample proportions.
According to the National Telecommunication and Information Administration, 56.5% of U.S. households owned a computer in 2001. What is the probability that of three randomly selected U.S. households at least one owned a computer in 2001?
P(at least 1 household owns computer) = 1 - P(none own computer) = 1 - P(.435)3 = .918 or 91.8%
Ravi sells real estate. Based on previous data, he knows that 5%, percent of home tours result in a sale. Assume that the results of these tours are independent from each other.
Which of the following choices are binomial random variables? (could be more than 1 right answer)
A) Take a random sample of 3 tours and let K = the number of tours that result in a sale.
B) Take a random sample of 3 tours and let M = equal the amount of sales (in dollars) generated by the tours.
C) Take a random sample of 30 tours and let L = equal the number of tours that result in a sale.
A and C are correct
A is correct because : Each trial has two outcomes (sale or not), results of each trial are independent, there is a fixed number of trials (3), and the probability of success is the same for each trial (5%).
B is wrong because : M isn't binomial since each tour is being recorded as a number that isn't being categorized as a success or failure
C is also correct because : Each trial has two outcomes (sale or not), results of each trial are independent, there is a fixed number of trials (30), and the probability of success is the same for each trial (5%).
Suppose that 75% of young adult Internet users (ages 18 to 29) watch online videos. A polling organization contacts an SRS of 1000 young adult Internet users and calculates the proportion in this sample who watch online videos.
Calculate and interpret the standard deviation of sample proportions.
sigma_p^ = 0.0137
The proportion of young adults in a sample of 1000 who watch online videos typically varies by 0.0137 from the true proportion of 0.75
There are three children in a room, ages three, four, and five. If a four-year-old child enters the room, what would happen to the mean and variance?
Mean age will stay the same but the variance will decrease
A shipping company claims that 90% of its shipments arrive on time. Suppose this claim is true. If we take a random sample of 20 shipments made by the company, what's the probability that at least one of them arrives late?
0.8784