Probability and Statistics: Week 12 Learning Objectives for MATH-1530 Fall 2004 - Prof. De | Assignments Statistics

MATH-1530-04/07/15/17 Probability & Statistics Fall 2004 / Week 12

LEARNING OBJECTIVES (CH 9, 12, & 10, in that order)

I. Probability (See Ch. 9 in BPS)

A. Recognize that some phenomena are random and understand the idea of probability as long-term relative frequency.

B. Understand the meaning of the following terms: outcome, sample space, event, probability model (or distribution), at least, and at

most.

C. Understand the following notation: For some event A, P(A) means “the probability of A” and P(x = 4) means “the probability that x

equals four.”

D. Recognize that a probability is a number that always takes a value between zero and one. Thus, for some event A, 0  P(A)  1.

E. Identify a sample space for simple random phenomena. (Examples: Know how to enumerate the sample space for given coin,

dice, and card exercises.)

F. Decide whether assignments of probabilities to individual outcomes do or do not satisfy the basic rules of probability, and

understand that assignments that do not satisfy these rules cannot be legitimate.

G. Calculate the probability of an event, from an assignment of probabilities to individual outcomes (i.e., a discrete probability

distribution) by adding the probabilities of the outcomes that make up the given event.

H. Use the addition rule to find the probability that one or the other of two disjoint events occur.

I. Use the complement rule to find the probability that an event does not occur.

II. Random Variables

A. Recognize when it is reasonable to assume that events are independent.

B. Recognize that a probability distribution of a random variable X tells us what the possible values of X are and how probabilities

are assigned to those values.

C. Understand how the distribution of a discrete random variable assigns probabilities to individual outcomes that make up the

sample space.

D. Understand how the distribution of a continuous random variable assigns probabilities as areas under a density curve.

III. The Binomial Setting (See Ch. 12 in BPS)

A. Understand that a binomial distribution is a distribution of COUNTS and that binomial distributions comprise a special class of

discrete probability models.

B. Recognize when a binomial distribution is appropriate (i.e., when a problem fits the four conditions of the Binomial Setting) and

be able to identify n (# of trials or observations) and p (probability of success on each trial).

C. Find binomial probabilities for specified numbers of successes using the binomial table.

D. Use n and p to calculate the mean (or expected) number of successes and the standard deviation in the number of successes

for a binomial distribution.

IV. Sampling Distributions

A. Understand the meanings of, and recognize the differences between the following terms: Sample, Statistic, Population, and

Parameter

B. Recognize that the sampling distribution of a statistic (such as the sample mean x-bar or the sample proportion p-hat) is the

distribution of values taken by some statistic calculated for all possible samples of the same size (n) from the same population.

C. The Sampling Distribution of the Sample Sean (See Ch. 10 in BPS): Recognize that if x-bar is the mean of an SRS of size n from

a large population with mean mu and standard deviation sigma, then the mean (center) of the sampling distribution of x-bar is also

mu, but its standard deviation (spread) is sigma divided by the square root of the sample size.

D. The sampling distribution of the sample proportion (see pp. 470 – 473 in BPS): Recognize that if p-hat is the proportion of

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MATH-1530-04/07/15/17 Probability & Statistics Fall 2004 / Week 12

LEARNING OBJECTIVES (CH 9, 12, & 10, in that order) I. Probability (See Ch. 9 in BPS) A. Recognize that some phenomena are random and understand the idea of probability as long-term relative frequency. B. Understand the meaning of the following terms: outcome, sample space, event, probability model (or distribution), at least, and at most. C. Understand the following notation: For some event A, P(A) means “the probability of A” and P(x = 4) means “the probability that x equals four.”

D. Recognize that a probability is a number that always takes a value between zero and one. Thus, for some event A, 0  P(A)  1.

E. Identify a sample space for simple random phenomena. (Examples: Know how to enumerate the sample space for given coin, dice, and card exercises.) F. Decide whether assignments of probabilities to individual outcomes do or do not satisfy the basic rules of probability , and understand that assignments that do not satisfy these rules cannot be legitimate. G. Calculate the probability of an event, from an assignment of probabilities to individual outcomes (i.e., a discrete probability distribution) by adding the probabilities of the outcomes that make up the given event. H. Use the addition rule to find the probability that one or the other of two disjoint events occur. I. Use the complement rule to find the probability that an event does not occur. II. Random Variables A. Recognize when it is reasonable to assume that events are independent. B. Recognize that a probability distribution of a random variable X tells us what the possible values of X are and how probabilities are assigned to those values. C. Understand how the distribution of a discrete random variable assigns probabilities to individual outcomes that make up the sample space. D. Understand how the distribution of a continuous random variable assigns probabilities as areas under a density curve. III. The Binomial Setting (See Ch. 12 in BPS) A. Understand that a binomial distribution is a distribution of COUNTS and that binomial distributions comprise a special class of discrete probability models. B. Recognize when a binomial distribution is appropriate (i.e., when a problem fits the four conditions of the Binomial Setting) and be able to identify n (# of trials or observations) and p (probability of success on each trial). C. Find binomial probabilities for specified numbers of successes using the binomial table. D. Use n and p to calculate the mean (or expected ) number of successes and the standard deviation in the number of successes for a binomial distribution. IV. Sampling Distributions A. Understand the meanings of, and recognize the differences between the following terms: Sample, Statistic, Population, and Parameter B. Recognize that the sampling distribution of a statistic (such as the sample mean x-bar or the sample proportion p-hat) is the distribution of values taken by some statistic calculated for all possible samples of the same size (n) from the same population. C. The S ampling Distribution of the Sample Sean (See Ch. 10 in BPS): Recognize that if x-bar is the mean of an SRS of size n from a large population with mean mu and standard deviation sigma, then the mean (center) of the sampling distribution of x-bar is also mu, but its standard deviation (spread) is sigma divided by the square root of the sample size. D. The sampling distribution of the sample proportion (see pp. 470 – 473 in BPS): Recognize that if p-hat is the proportion of

successes in an SRS of size n from a large population with population proportion of successes p, then the mean (center) of the sampling distribution of p-hat is p and the standard deviation (spread) is the square root of (p (1 – p) / n).

The density curve for a continuous random variable X has which of the following properties? a. The probability of any event is the area under the density curve and above the values of X that make up that event. b. The total area under the density curve for X must be exactly 1. c. The probability of any event with the form X = a constant is 0 (that is, P(x = c) = 0 for any constant c). d. All of the above.
A chemist repeats a solubility test 10 times on the same substance. Each test is conducted at a temperature 10 degrees higher than the previous test. She counts the number of times (X) that the substance dissolves completely. Is it reasonable to use a binomial distribution for the random variable X? a. Yes, since the substance either dissolves or it does not dissolve. There are only two possible outcomes for each trial. b. No, because a higher temperature will probably improve the chance that the substance will dissolve on successive trials. c. Yes, because there are a fixed number of trials (n = 10). d. No, since p, the probability of success, is not given for the problem. In a test for ESP (extrasensory perception), a subject is told that cards the researcher can see but the subject connot, contain either a star, a circle, a half-moon, or a square on the face of the card. There are 5 cards showing each shape in the well-shuffled deck of 20 cards. As the experimenter looks at each of the 20 cards in turn, the subject names the shape on the card.
A subject who is just guessing has what probability of correctly naming the shape on a single card? a. 0.20 b. 0.50 c. 0.25 d. 0.
The count of correct guesses in 20 cards has a binomial distribution. What is the mean and standard deviation of correct guesses in many repetitions?
What is the probability of a subject guessing the correct names for half of the cards?
What is the probability of a subject getting at most 7 correct guesses?
What is the probability that a subject misnames the card on at least 16 guesses?

A factory produces plate glass with a mean thickness of 4mm and a standard deviation of 1.1 mm. For quality control, a simple random sample of n = 100 sheets of glass is to be measured, and the sample mean thickness of the 100 sheets (x-bar) is to be computed.

We know the random variable x-bar has approximately a normal distribution because of a. the law of large numbers. b. the central limit theorem. c. the rules of probability. d. the fact that probability is the long run proportion of times an event occurs.
For the factory described above, the probability that the average thickness of the 100 sheets of glass is less than 4.1 mm [i.e., P(x- bar < 4.1)] is a. 0.8186. b. 0.3183. c. 0.1814. d. 0.

The distribution of values taken by a statistic in all possible samples of the same size from the same population is a. the probability that the statistic is obtained. b. the population parameter. c. the variance of the values. d. the sampling distribution of the statistic.
The variability of a statistic is described by a. the spread of its sampling distribution.

b. the amount of bias present. c. the vagueness in the wording of the questions used to collect the sample data. d. the stability of the population it describes.

An assignment of probability must obey which of the following? a. The probability of any event in the sample space must be a number between 0 and 1, inclusive. b. The sum of the probabilities of all outcomes in the sample space must be exactly 1. c. The addition rule. d. All of the above. The College Alcohol Study interviewed an SRS of 14,941 college students about their drinking habits. Suppose it is known that half of all college students “drink to get drunk” at least once in a while. That is, the population proportion is p = 0.5.
What are the mean and standard deviation of the distribution of the proportion (p-hat) of the sample that drink to get drunk?
Use the Normal approximation to find the probability that p-hat is between 0.49 and 0.51 in the sample of students participating in this study.

Probability and Statistics: Week 12 Learning Objectives for MATH-1530 Fall 2004 - Prof. De, Assignments of Statistics

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MATH-1530-04/07/15/17 Probability & Statistics Fall 2004 / Week 12

D. Recognize that a probability is a number that always takes a value between zero and one. Thus, for some event A, 0  P(A)  1.