Statistics: Random Variables & Distributions
30 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Is the braking time of a car a discrete or continuous random variable?

  • Categorical
  • Continuous (correct)
  • Nominal
  • Discrete

    • What is the standard deviation, σ, for a binomial distribution with n = 503 and p = 0.7, rounded to the nearest hundredth?

  • 10.28
  • 7.87
  • 13.55 (correct)
  • 14.40

    • What is the mean, μ, for a binomial distribution with n = 676 and p = 0.7, rounded to the nearest tenth?

  • 471.7
  • 473.2 (correct)
  • 474.9
  • 474.5

    • Given Mars, Inc. claims that 20% of its M&M candies are orange, find the mean (μ) and standard deviation (σ) for a sample of 100 candies.

    <p>μ = 20, σ = 4.0</p> Signup and view all the answers

    What is the mean value (μ) for rolling a loaded die, given the probabilities in the table?

    <p>3.8</p> Signup and view all the answers

    On a multiple-choice test of 17 questions with four possible answers, what is the mean number of correct answers for students who guess?

    <p>8.5 questions</p> Signup and view all the answers

    When a die is rolled nine times, what is the expected mean for the number of twos observed?

    <p>3.0</p> Signup and view all the answers

    What is the mean number of adults who do not worry about identity theft in a group of 1013 adults, given that 66% worry about it?

    <p>344 adults</p> Signup and view all the answers

    In a binomial distribution where the trial is repeated n = 30 times and the probability of success is p = 1/5, what is the probability of having exactly 5 successes?

    <p>0.172</p> Signup and view all the answers

    If a die is rolled nine times, what is the mean number of times the outcome is a two?

    <p>1.5 twos</p> Signup and view all the answers

    What is the mean number of students working full-time in a sample of size 16, given that 22% of all students work full time?

    <p>2.8 students</p> Signup and view all the answers

    Is rolling a single die 53 times keeping track of the 'fives' rolled a binomial distribution, and why?

    <p>Not binomial: there are more than two outcomes for each trial.</p> Signup and view all the answers

    What is the probability that a tennis player, who makes a successful serve 51% of the time, gets exactly 3 successful serves when serving 9 times?

    <p>0.133</p> Signup and view all the answers

    If a student guesses on a test consisting of 10 true/false questions, what is the probability that the student will pass the test by answering at least 6 questions correctly?

    <p>0.246</p> Signup and view all the answers

    What is the probability of getting three or more cars that fail among six cars tested?

    <p>0.046</p> Signup and view all the answers

    For a normally distributed population of loan ratings with a mean of 200 and standard deviation of 50, what is the probability of a rating between 200 and 275?

    <p>0.9332</p> Signup and view all the answers

    If z is a standard normal variable, what is P(z > 0.97)?

    <p>0.1660</p> Signup and view all the answers

    In a population of 210 women with heights normally distributed, what are the mean and standard deviation of the population of sample means for a random selection of 36 women?

    <p>64.4 inches, 0.44 inches</p> Signup and view all the answers

    What is the test score threshold (P81) that separates the bottom 81% from the top 19% of a normally distributed test with a mean of 63.2 and standard deviation of 11.7?

    <p>73.5</p> Signup and view all the answers

    For the standard normal variable z, what is the probability P(-0.73 < z < 2.27)?

    <p>0.7557</p> Signup and view all the answers

    What is the probability of getting exactly one car that fails among six that were tested?

    <p>0.399</p> Signup and view all the answers

    In a situation where a coin is tossed 20 times, which of the following is most relevant to assess the outcomes predicted by someone claiming extrasensory perception?

    <p>The binomial distribution rules</p> Signup and view all the answers

    What is the probability of correctly guessing 14 or more times?

    <p>0.0582, no</p> Signup and view all the answers

    What is the probability that a randomly selected college football player weighs between 170 and 220 pounds?

    <p>0.3811</p> Signup and view all the answers

    What z score corresponds to a shaded area of 0.4483 in the standard normal distribution?

    <p>0.3264</p> Signup and view all the answers

    What is the area of the shaded region when the graph depicts a standard normal distribution?

    <p>0.9656</p> Signup and view all the answers

    What is the probability that a randomly chosen woman has a red blood cell count above 4.2 million cells per microliter?

    <p>0.0158</p> Signup and view all the answers

    An unbiased estimator targets the population parameter such that the sampling distribution's mean equals which value?

    <p>mean</p> Signup and view all the answers

    What is the mean red blood cell count for women in millions of cells per microliter?

    <p>4.577</p> Signup and view all the answers

    Which of the following standard deviations closely reflects the data for college football players' weights?

    <p>50 pounds</p> Signup and view all the answers

    Study Notes

    Discrete vs. Continuous Random Variables

    • The braking time of a car is a continuous random variable because it can take on any value within a range.

    Binomial Distribution

    • The standard deviation (σ) of a binomial distribution is calculated using the formula: σ = √(np(1-p)), where n is the number of trials and p is the probability of success on a single trial.
    • The mean (μ) of a binomial distribution is calculated using the formula: μ = np, where n is the number of trials and p is the probability of success on a single trial.

    Calculating Mean and Standard Deviation

    • For a sample of 100 M&M plain candies, with 20% being orange, the mean (μ) number of orange candies is 20 and the standard deviation (σ) is 4. This is calculated using μ = np and σ = √(np(1-p)), where n = 100 and p = 0.2.

    Probability Distribution and Mean

    • The mean (μ) of a probability distribution is calculated by summing the products of each value (x) and its corresponding probability (P(x)).

    Mean of Correct Answers

    • A multiple-choice test with 17 questions, each with four possible answers, has a 1/4 (0.25) probability of a student guessing correctly on each question. The mean number of correct answers for students who guess is 4.3, calculated by μ = np, where n = 17 and p = 0.25.

    Mean of a Repeated Experiment

    • If a die is rolled nine times and the number of times a "two" appears is recorded, the mean number of "twos" is 1.5. This is calculated by μ = np, where n = 9 and p = 1/6 (probability of rolling a two on a single roll).

    Binomial Probability

    • For a binomial distribution with 30 trials (n = 30) and a probability of success on a single trial of 1/5 (p = 1/5), the probability of getting exactly 5 successes (x = 5) is approximately 0.172. This probability can be found using the binomial probability formula.

    Determining Binomial Distribution

    • Rolling a single die 53 times and keeping track of the number of "fives" rolled results in a binomial distribution because it meets the criteria of having a fixed number of trials (n = 53), independent trials, two possible outcomes (roll a five or not), and a constant probability of success (p = 1/6).

    Probability in Multiple Trials

    • A tennis player has a 51% chance of making a successful first serve. If she serves 9 times, the probability of getting exactly 3 successful serves is approximately 0.154. This can be calculated using the binomial probability formula.

    Probability of Success in a Test

    • A true/false test with 10 questions requires at least 6 correct answers to pass. If a student guesses on each question, the probability of passing is approximately 0.377. This can be determined using the binomial probability formula and summing the probabilities for 6, 7, 8, 9, and 10 correct answers.

    Probability of Failure in a Sample

    • The probability of getting three or more cars that fail roadworthiness tests among six tested is approximately 0.046. This can be calculated by summing the probabilities for 3, 4, 5, and 6 failures as provided in the probability distribution table.

    Normal Distribution and Probability

    • If credit scores are normally distributed with a mean of 200 and a standard deviation of 50, the probability of a randomly selected applicant having a credit score between 200 and 275 is approximately 0.4332. This probability can be determined by calculating the z-score for each value (200 and 275) and then using the standard normal distribution table to find the area between these z-scores.

    Z-score and Probability

    • For a standard normal distribution, with a mean of 0 and a standard deviation of 1, the probability that z is greater than 0.97 is approximately 0.1660. This can be found using the standard normal distribution table.

    Sampling Distribution of Sample Means

    • For a population of women with a mean height of 64.4 inches and a standard deviation of 2.9 inches, the mean (μ) of the sampling distribution of sample means for a sample of 36 women is 64.4 inches. The standard deviation (σ) of the sampling distribution of sample means is 0.44 inches, considering a finite population correction factor because sampling is without replacement.

    Percentiles in Normal Distribution

    • P81, the 81st percentile for a normally distributed test with a mean of 63.2 and a standard deviation of 11.7, is 73.5. This value separates the bottom 81% of scores from the top 19%.

    Probability Between Z-Scores

    • The probability that a standard normal variable (z) is between -0.73 and 2.27 is approximately 0.7557. This can be determined by using the standard normal distribution table to find the area between these z-scores.

    Normal Approximation to Binomial Distribution

    • A person claims to predict coin toss outcomes with extrasensory perception and gets 14 correct out of 20. The probability of getting 14 or more correct by guessing is approximately 0.0582. Using the normal distribution to approximate this probability, it suggests the claim of extrasensory perception is not supported by the data, as this probability is relatively low.

    Probability in Normal Distribution

    • The weights of college football players are normally distributed with a mean of 200 pounds and a standard deviation of 50 pounds. The probability of a randomly selected player weighing between 170 and 220 pounds is approximately 0.3811. This probability can be calculated by finding the z-scores for 170 and 220 and using the standard normal distribution table to find the area between them.

    Area under Normal Distribution Curve

    • For a standard normal distribution, the shaded area under the curve representing 0.4483 corresponds to a z-score of approximately 0.13. This can be determined using the standard normal distribution table or a statistical calculator.

    Area under Normal Distribution Curve

    • The area of the shaded region under the standard normal distribution curve representing a z-score of 1.75 is approximately 0.9656. This can be determined using the standard normal distribution table or a statistical calculator.

    Probability in a Normal Distribution

    • For women's red blood cell counts, which are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter, the probability of a randomly selected woman having a red blood cell count above the normal range of 4.2 to 5.4 is approximately 0.0158. This can be calculated by finding the z-scores for 4.2 and 5.4 and using the standard normal distribution table to find the area outside of the normal range.

    Unbiased Estimators

    • An unbiased estimator is a statistic that targets the value of the population parameter. The key characteristic of an unbiased estimator is that the mean of its sampling distribution is equal to the mean of the corresponding parameter. This means that, on average, the estimator accurately reflects the true value of the parameter.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Test 3a PDF

    Description

    This quiz covers the concepts of discrete and continuous random variables, specifically focusing on the binomial distribution. You'll learn how to calculate the mean and standard deviation using specific examples. Test your understanding of probability distributions and their key properties.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser