Modules 4-6 Quizzes

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. After keeping a record for several months, she developed following probability mass function (PMF) for the number of red lights she hits. X = # of red light] 0, 1, 2, 3, 4, 5 P(X = x)] 0.05, 0.25, 0.35, 0.15, 0.15, 0.05 Find P(2 < X ≤ 4). Report your answer to 1 decimal place.

0.3

A water hydrant dispenses water at a rate described by a uniform continuous distribution over the interval 50 to 70 gallons per minute. Find the probability that at most 65 gallons are dispensed during a randomly selected minute. Report your answer to 2 decimal places. Do NOT report your answer as a fraction. Such as if you get ¼, report it 0.25.

0.75

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. After keeping a record for several months, she developed following probability mass function (PMF) for the number of red lights she hits. X = # of red light] 0, 1, 2, 3, 4, 5 P(X = x)] 0.05, 0.25, 0.35, 0.15, 0.15, 0.05 Find P(0 < X < 5). Report your answer to 1 decimal place.

0.9

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. After keeping a record for several months, she developed following probability mass function (PMF) for the number of red lights she hits. X = # of red light] 0, 1, 2, 3, 4, 5 P(X = x)] 0.05, 0.25, 0.35, 0.15, 0.15, 0.05 Find the expected value of X. Report your answer to 2 decimal places.

2.25

Adding or subtracting a constant (c) from the random variable shifts the mean but doesn't change the variance or standard deviation. Assume c ≠ 0. a. True b. False

a

Choose the best statement that is true about the standardized z-score of a value of a normal random variable X, which has mean µ and standard deviation σ. a. The z-score has a mean equal to 0, the z-score has a standard deviation equal to 1, the distribution of z-scores is a normal distribution, and the z-score tells us by how many multiples of σ the original X observation fall away from the mean. b. The z-score has a mean = 1 and variance = 0. c, The z-score has a mean = 1 and variance = 1 d. The z-score has a mean equal to 0, the z-score has a standard deviation equal to 1, the distribution of z-scores is a normal distribution, and the z-score tells us by how many multiples of σ the original X observation fall away from the zero.

a

In most situations, the true mean and true standard deviation of a population are unknown (unobserved) quantities that have to be estimated from sample data. a. True b. False

a

Label the random variable G as either discrete or continuous based on its description. Let G = The amount of milk produces yearly by a particular cow (gallons). a. Variable G is a continuous random variable. b. If the cow is brown in color the variable G is a discrete random variable. c. All of the possible answers presented in this question are correct answers on the context. d. Variable G is a discrete random variable.

a

Probability mass function (PMF) is used to describe the behavior of which type of a random variable? a. Discrete random variable. b. Continuous random variable. c. Unknown random variable if probability = 1.99. d. Continuous random variable if probability = - 1.99.

a

Suppose studentsʹ ages follow a right skewed distribution with a mean of 25 years old and a standard deviation of 5 years. If we randomly sample 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect? Formula: Standard error of sample mean =σ/sqrt(n) a. The standard error of the sampling distribution is equal to 5 years. b. The shape of the sampling distribution of sample mean is approximately normal. c. The mean of the sampling distribution of sample mean is 25 years. d. The sample size is 100.

a

The Central Limit Theorem is considered powerful in statistics because a. it works for any population distribution provided the sample size from a random sample is sufficiently large. b. it works for any sample size provided the population is normal. c. it works for any population distribution provided the population mean is known. d. it works for any sample provided the population distribution is known.

a

The natural tendency of randomly drawn samples to differ, from another is known as a. sampling variability or sampling error. b. biased sample. c. standard error. d. census.

a

The theoretical long-run average value of a random variable which measures the center of the probability distribution is known as a. Expected value. b. Cumulative distribution function (CDF). c. Probability mass function (PMF). d. Probability density function (PDF).

a

In general, which of the following statement is true about the sampling distribution of the sample mean? Formula: Standard error of sample mean =σ /sqrt(n) a. Increasing the sample size increases the standard error. b. Increasing the sample size decreases the standard error. c. Standard error will be negative by increasing the sample size. d. Increasing the sample size does not change the standard error.

b

Indiana University administration reported that 56% of all faculty and staff members donated to the United Way campaign. A survey of a random sample of 100 faculty and staff members found that 60% have donated to United Way campaign. In this setting, a. 60% is a parameter value and 56% is a statistic value. b. 56% is a parameter value and 60% is a statistic value. c. 60% and 56% are both parameter values. d. 56% and 60% are both statistic values.

b

The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following condition is the necessary for the Central Limit Theorem to be used? a. The population from which we are sampling must have a known variance. b. The sampling must be done randomly and the sample size must be large (e.g., at least 30). c. The sample size must be very small such as less than 10. d. The population size must be large such as at least 30.

b

To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find and interpret the lowest possible score to qualify for a police academy. Assume the test scores are normally distributed. Two R codes and outputs are given below but only one of them is correct on the context of the question. Your job is first identify the right R code and interpret the result. You do NOT need to run R code on your machine. R Code 1: > qnorm(0.90,mean=200,sd=20) [1] 225.63 R code 2: > qnorm(0.10,mean=200,sd=20) [1] 174.369 a. The average possible score to qualify for a police academy for candidates is 225.63 on a general abilities test. b. The lowest possible score to qualify for a police academy for candidates is 225.63 on a general abilities test. c. The highest possible score to qualify for a police academy for candidates is 225.63 on a general abilities test. d. The lowest possible score to qualify for a police academy for candidates is 174.369 on a general abilities test.

b

A weapons manufacturer uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A statistician found that 23% of the cartridges in a particular lot were contaminated. Suppose you randomly sample (without replacement) gun cartridges from this lot until you find the first contaminated one. Let X be the number of cartridges sampled until the first contaminated one is found. Which distribution best describes the context? a. Normal distribution b. Binomial distribution c. Geometric distribution d. Poisson distribution

c

Assume each newborn baby had a probability of approximately 0.54 of being female and 0.46 of being male. For a family of four children, let X = number of children who are female. Which of the following statement correctly describes the required conditions of a binomial distribution? a. The n trials are independent, each trial has at least two possible outcomes, and each trial has the same probability of a success. b. The n trials are dependent, there are two trials, and each trial has two possible outcomes. c. The n trials are independent, each trial has the same probability of a success, and each trial has two possible outcomes. d. The n trials are dependent, each trial has the same probability of a success, and each trial has two possible outcomes.

c

Continuous random variables that appear to have equally likely outcomes (evenly distributed) over their range of possible values should be modeled using which distribution? a. Binomial distribution b. Bernoulli distribution c. Uniform continuous distribution d. Gamma distribution

c

Label the random variable F as either discrete or continuous based on its description. Let F = Number of defective parts of a machine. a. Variable F is a continuous random variable. b. All of the possible answers presented in this question are correct answers on the context. c. Variable F is a discrete random variable. d. If the part is made in USA it is continuous random variable.

c

The central limit theorem says that when a random sample of size n is drawn from any population with mean μ and standard deviation σ (read as sigma), then when n is sufficiently large (n ≥ 30) a. the distribution of the sample mean is exactly normal. b. the standard error of the sample mean is: (sigma square)/n c. the distribution of the sample mean is approximately normal. d. the distribution of the population is exactly normal.

c

The sampling distribution of a statistic is a probability distribution a. of all values in the population. b. the maximum value the statistic can take in all possible samples. c. calculated from all possible random samples of a specific size (n) taken from a population. all values the statistic can take in all possible samples of size n. d. of all values in the sample.

c

Which distribution should be used to model for predicting the number of events that occur over a given interval of time? a. Gamma distribution b. Normal distribution c. Poisson distribution d. Z score

c

A special type of discrete random variable that results in dichotomy (two) outcomes is known as a. Probability density function (PDF). b. Continuous Random Variable. c. Cumulative distribution function (CDF). d. Bernoulli Random Variable.

d

As the number of observations of a random variable increases, the average of the observations converges to the expected value. In other words, if we repeat an experiment a large number of times our observed average should be fairly close to the expected value. This gives the definition of a. Chebyshev's law. b. Law of permutation. c. Law of very small numbers. d. Law of large numbers.

d

Choose an incorrect relationship among mean, median, and mode for a normal distribution. a. Mean = Mode b. Mean = Median c. Mode = Median d. Mean < Median

d

What is the total area or probability under any probability density function curve for a continuous random variable such as normal distribution curve? a. Total area or probability = 0.5 b. Total area or probability = 0 c. Total area or probability = 0.25 d. Total area or probability = 1

d

Which distribution should be used to model the waiting time until the first event occurs? a. Standard normal distribution b. Normal distribution c. Gamma distribution d. Exponential distribution

d

In a large population of high school students who participated in the Indiana University High School Math Contest, the mean IQ is 120 with a standard deviation of 20. The distribution of IQ scores is normal. Suppose that 25 participants are chosen at random to be invited to a reception. The distribution of the sample mean IQ of the invitees is Formula: Standard error of sample mean =σ/sqrt(n) a. normal with mean 120, standard error 0.8. b. normal with mean 120, standard error 20. c. normal with mean 120 , standard error 5.0. d. approximately normal with mean 120, standard error 20. e. normal with mean 120, standard error 4.0.

e