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InstructionDue Date: 6 pm on October 28 (Wed) Part IProbability and Sampling Distributions1.Thinking about probability statements. Probability is measure of how likely an event is to occur. Match one of probabilities that follow with each statement of likelihood given (The probability is usually a more exact measure of likelihood than is the verbal statement.)Answer0 0.01 0.3 0.6 0.99 1(a) This event is impossible. It can never occur.(b) This event is certain. It will occur on every trial.(c) This event is very unlikely, but it will occur once in a while in a long sequence of trials.(d) This event will occur more often that not.2. Spill or Spell? Spell-checking software catches "nonword errors" that result in a string of letters that is not a word, as when "the" is typed as "the." When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following distribution:Value of X01234Probability0.10.20.30.30.1(a) Check that this distribution satisfies the two requirements for a legitimate assignment of probabilities to individual outcomes.(b) Write the event "at least one nonword error" in term of X (for example, P(X >3)). What is the probability of this event?(c) Describe the event X ≤ 2 in words. What is its probability? 3. Discrete or continuous? For each exercise listed below, decide whether the random variable described is discrete or continuous and explains the sample space.(a) Choose a student in your class at random. Ask how much time that student spent studying during the past 24 hours.(b) In a test of a new package design, you drop a carton of a dozen eggs from a height of 1 foot and count the number of broken eggs.(c) A nutrition researcher feeds a new diet to a young male white rat. The response variable is the weight (in grams) that the rat gains in 8 weeks.4. Tossing Coins(a) The distribution of the count X of heads in a single coin toss will be as follows. Find the mean number of heads and the variance for a single coin toss.Number of Heads (Xi)01mean:Probability (Pi)0.50.5variance:(b) The distribution of the count X of heads in four tosses of a balanced coin was as follows but some missing probabilities. Fill in the blanks and then find the mean number of heads and the variance for the distribution with assumption that the tosses are independent of each other.Number of Heads (Xi)01234mean:Probability (Pi)0.06250.0625variance:(c) Show that the two results of the means (i.e. single toss and four tosses) are related by the addition rule for means. (d) Show that the two results of the variances (i.e. single toss and four tosses) are related by the addition rule for variances (note: It was assumed that the tosses are independent of each other). 5. Generating a sampling distribution. Let's illustrate the idea of a sampling distribution in the case of a very small sample from a very small .
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Question 1 Independent random samples taken on two university campuses revealed the following information concerning the average amount of money spent on non-textbook purchases at the university’s bookstore during the fall semester. University A University B Sample Size 50 40 Average Purchase $260 $250 Population Standard Deviation(σ) $20 $23 We want to determine if, on the average, students at University A spent more on non-textbook purchases at the university’s bookstore than the students at University B. a. Compute the test statistic. b. Compute the p-value. c. What is your conclusion? Let α = .05. Question 2 In a residual plot against x that does not suggest we should challenge the assumptions of our regression model, we would expect to see a horizontal band of points centered near zero a widening band of points a band of points having a slope consistent with that of the regression equation a parabolic band of points Question 3 If we are testing for the equality of 3 population means, we should use the test statistic t test statistics z test statistic χ 2 test statistic F Question 4 The expected value of mean equals to the mean of the population from which the sample is drawn only if the sample size is 100 or greater for any sample size only if the sample size is 50 or greater only if the sample size is 30 or greater Question 5 A simple random sample of size n from a finite population of size N is to be selected. Each possible sample should have a probability of 1/n of being selected the same probability of being selected a probability of 1/N of being selected a probability of N/n of being selected Question 6 Consider the following results for two samples randomly taken from two normal populations with equal variances. Sample I Sample II Sample Size 28 35 Sample Mean 48 44 Population Standard Deviation 9 10 a. Develop a 95% confidence interval for the difference between the two population means. b. Is there conclusive evidence that one population has a larger mean? Explain. Question 7 As a general guideline, the research hypothesis should be stated as the null hypothesis hypothesis the researcher wants to disprove alternative hypothesis tentative assumption Question 8 As the degrees of freedom increase, the t distribution approaches the uniform distribution p distribution exponential distribution normal distribution Question 9 Two approaches to drawing a conclusion in a hypothesis test are p-value and critical value Type I and Type II one-tailed and two-tailed null and alternative Question 10 In hypothesis testing, the alternative hypothesis is the maximum probability of a Type I error All of the answers are correct the hypothesis tentatively assumed true in the hypothesis-testing procedure the hypothesis concluded to be true if the null hypothesis is rejected Question 11 For a two-tailed hypothesis test about μ, we can use any of the ...
Question 1 Independent random samples taken on two university .docx
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1 Review and Practice Exam Questions for Exam 2 Learning Objectives: Chapter 17: Thinking about chance • Explain how random events behave in the short run and in the long run and how random and haphazard are not the same thing. • Perform basic probability calculations using die rolls and coin tosses. • Define probability, and apply the rules for probability. • Explain whether the law of averages is true. • Explain how personal probability differs from a scientific or experimental probability. Chapter 18: Probability models • Define a probability model. Create a probability model for a particular story’s events. • Apply the basic rules of probability to a story problem. • Calculate probabilities using a probability model, including summing up probabilities or subtracting probabilities from the total. • Define a sampling distribution. Chapter 20: The house edge: expected values • Define expected value, and calculate the expected value when given a probability model. • Define the law of large numbers, and explain how it is different from the mythical “law of averages.” • Explain how casinos and insurance companies stay in business and make money. Chapter 13: The Normal distribution • Identify data that is Normally distributed. • Discuss how the shape/position of the Normal curve changes when the standard deviation increases/decreases or when the mean increases/decreases. • Define the standardized value or Z-score. Calculate the Z-score, and use the Z-score to do comparisons. • Calculate probabilities and cut-off values using the 68%-95%-99.7% (Empirical) Rule. • Identify the mean, standard deviation, cut-off value, probability, and Z-score on a Normal curve. • Use the Normal table to get percentiles (probabilities) for forward problems and to get Z-scores in order to determine cut-offs for backward problems using both > and < in the inequalities. • Recognize whether a story is a forward or backward Normal distribution problem, and perform the appropriate calculations showing correct notation, the initial probability expression, and all necessary steps. 2 Chapter 21: What is a confidence interval? • Define statistical inference and explain when statistical inference is used. • Explain what the confidence interval means and whether the results refer to the population or the sample. • Calculate the margin of error and identify the margin of error in a confidence statement. Explain what type of error is covered in the margin of error. • Determine whether a story is better described with a proportion or a mean. • Use appropriate notation for proportions and means, both in the population and the sample. • Calculate a confidence interval for a proportion and for a mean. • Describe how increasing/decreasing the sample size or confidence level changes the margin of error (width of the confidence interval). • Apply cautions for using confidence inte ...
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Chapter 5
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Chapter 5: Probability
Distributions
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Example: Boredom Tolerance
Test 400 6 1,600 5 4,400 4 6,000 3 3,600 2 2,600 1 1,400 0 Number of Subjects Score Boredom Tolerance Test Scores for 20,000 Subjects
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