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Compatible with any classroom text, Schaum's let you study at your own pace and remind you of all the important facts you need to remember--fast! Stage 1: Economic theory Mathematical model Econometric stochastic model Stage 2: Collection of appropriate data Estimation of the parameters of the model Stage 3: An economic theory expressed in exact or deterministic mathematical form 1.
Stating the theory in the form of Eq. Collection of time-series data on I and R and estimation of Eq. This breaks up the data into groups or classes and shows the number of observations in each class. The number of classes is usually between 5 and A relative frequency distribution is obtained by dividing the number of observations in each class by the total number of observations in the data as a whole. The sum of the relative frequencies equals 1.
A histogram is a bar graph of a frequency distribution, where classes are measured along the horizontal axis and frequencies along the vertical axis. A frequency polygon is a line graph of a frequency distribution resulting from joining the frequency of each class plotted at the class midpoint.
A cumulative frequency distribution shows, for each class, the total number of observations in all classes up to and including that class. When plotted, this gives a distribution curve, or ogive. A student received the following grades measured from 0 to 10 on the 10 quizzes he took during a semester: These grades can be arranged into frequency distributions as in Table 2.
Table 2. Relative frequency histogram Relative frequency Absolute frequency Panel A: The cans in a sample of 20 cans of fruit contain net weights of fruit ranging from If we want to group these data into 6 classes, we get class intervals of 0.
Schaum's Outline of Statistics and Econometrics
The weights given in Table 2. Relative frequency histogram 7 2 Histogram 2 Ogive 20 18 Panel C: Frequency polygon 16 Cumulative frequency 8 Absolute frequency 7 6 5 4 3 14 12 10 8 6 2 4 1 2 0 0 The most important measures of central tendency are 1 the mean, 2 the median, and 3 the mode. We will be measuring these for populations i. The median for ungrouped data is the value of the middle item when all the items are arranged in either ascending or descending order in terms of values: The mode is the value that occurs most frequently in the data set.
Other measures of central tendency are the weighted mean, the geometric mean, and the harmonic mean see Probs. The mode for the ungrouped data is 6 the value that occurs most frequently in the data set. We can estimate the mean for the grouped data given in Table 2. P fX The most important measures of dispersion are 1 the average deviation, 2 the variance, and 3 the standard deviation.
We will measure these for populations and samples, as well as for grouped and ungrouped data. Average deviation. Standard deviation. Other measures besides the variance and average deviation are the range, the interquartile range, and the quartile deviation see Probs.
A distribution has zero skewness if it is symmetrical about its mean. For a symmetrical unimodal distribution, the mean, median, and mode are equal.
A distribution is positively skewed if the right tail is longer. A distribution is negatively skewed if the left tail is longer. Symmetrical Mode Median Panel B: Positively skewed Panel C: Negatively skewed Fig. Skewness can also be measured by the third moment [the numerator of Eq. Kurtosis can be measured by the fourth moment [the numerator of Eq. The kurtosis for a mesokurtic curve is 3.
Leptokurtic Mesokurtic Platykurtic Fig. The comovement of two separate distributions can be measured by covariance: For kurtosis, see Prob. Note that since we are dealing here with discrete data i. Panel A: Histogram Panel B: Relative Frequency Distribution 8 0. Ogive 40 36 Panel C: These are shown in Table 2. Note that the class midpoints are obtained by adding together the lower and upper class boundaries and dividing by 2.
Relative frequency distribution 0. Frequency polygon Panel D: Ogive 0. Since they are both equal to 6, the median is 6. The mode is 7 the value that occurs most frequently in the data set. The median for the grouped data of Table 2. Thus the distribution is bimodal i. Sometimes the mode is simply given as the midpoint of the modal class. The disadvantages of the mean are 1 it does not use much of the information available, and 2 it requires that observations be arranged into an array, which is time-consuming for a large body of ungrouped data.
The disadvantages of the mode are 1 as for the median, the mode does not use much of the information available, and 2 sometimes no value of the data is repeated more than once, so that there is no mode, while at other times there may be many modes.
In general, the mean is the most frequently used measure of central tendency and the mode is the least used. See Table 2. Coding eliminates the problem of having to deal with possibly large and inconvenient class midpoints; thus it may simplify the calculations. Find the harmonic mean. Note that if the commuter had averaged Quartiles divide the data into 4 parts, deciles into 10 parts, and percentiles into parts.
The range for the ungrouped data in Table 2. For grouped data, the range extends from the lower limit of the smallest class to the upper limit of the largest class. For the grouped data in Table 2. Because of these disadvantages, the range is of limited usefulness except in quality control.
Find the interquartile range and the quartile deviation and b for the grouped data in Table 2. It is thus better than the range, but it is not as widely used as the other measures of dispersion.
Quartile deviation measures the average range of one-fourth of the data. It measures the average of the absolute deviation of each P observation from the mean. The standard deviation is by far the most widely used measure of absolute dispersion. Find the variance and the standard deviation for the grouped data in Table 2.
Furthermore, s2 and s for the grouped data are estimates for the true s2 and s that could be found for the ungrouped data because we use the estimate of X from the grouped data in our calculations. Coding also helps see Prob. This is to be contrasted with standard deviation and other measures of absolute dispersion, which are expressed in the units of the problem. For example, we can say that the dispersion of the data in Table 2.
When X and Y are both above or below their means, covariance is increased. When X and Y move in opposite directions relative to their means employee 5 , covariance is decreased.
Computations are Present the data in the form of a histogram, a relative-frequency histogram, a frequency polygon, and an ogive. Graph the data into a histogram, a relative-frequency histogram, a frequency polygon, and an ogive. What is the average price per gallon? What was its average speed? Probability and Probability Distributions 3.
In Fig. A head H and a tail T are the two equally possible outcomes in tossing a balanced coin. In rolling a fair die once, there are six possible and equally likely outcomes: A card deck has 52 cards divided into 4 suits diamonds, hearts, clubs, and spades with 13 cards in each suit 1, 2, 3,.
Suppose that in tosses of a balanced coin, we get 53 heads and 47 tails. For example, the relative frequency or empirical probability might be 0. Rule of addition for nonmutually exclusive events. Two events, A and B, are not mutually exclusive if the occurrence of A does not preclude the occurrence of B, or vice versa.
This can be seen with the Venn diagram in Fig. Rule of addition for mutually exclusive events.
Rule of multiplication for dependent events. Two events are dependent if the occurrence of one is connected in some way with the occurrence of the other. Rule of multiplication for independent events. Two events, A and B, are independent if the occurrence of A is not connected in any way to the occurrence of B.
On a single toss of a die, we can get only one of six possible outcomes: These are mutually exclusive events. The outcomes of two successive tosses of a balanced coin are independent events.
Problem 3. The set of all possible values of a random variable and its associated probabilities is called a probability distribution. The sum of all probabilities equals 1 see Example 9. One discrete probability distribution is the binomial distribution. Table 3. If we were not dealing with a coin and the trials were not dependent as in sampling without replacement , we would have had to use the hypergeometric distribution see Prob.
It is used to determine the probability of a designated number of successes per unit of time, when the events or successes are independent and the average number of successes per unit of time remains constant.
A police department receives an average of 5 calls per hour.
The probability that X falls within any interval is given by the area under the probability distribution or density function within that interval. The total area probability under the curve is 1 see Prob.
The normal distribution is a continuous probability distribution and the most commonly used distribution in statistical analysis see Prob. The normal curve is bell-shaped and symmetrical about its mean. Any normal distribution X scale in Fig. This gives the proportion of the area probability included under the curve between the mean and that z value.
We move down the z column in the table to 1. The value that we get is 0. This means that Another continuous probability distribution is the exponential distribution see Prob. Relative frequency or empirical probability is given by the ratio of the number of times an event occurs to the total number of actual outcomes or observations. As the number of experiments or trials such as the tossing of a coin increases, the relative frequency or empirical probability approaches the classical or a CHAP.
Subjective or personalistic probability refers to the degree of belief of an individual that the event will occur, based on whatever evidence is available to the individual. In realworld problems of economics and business, we often cannot assign probabilities a priori and the classical approach cannot be used. The relative-frequency or empirical approach overcomes the disadvantages of the classical approach by using the relative frequencies of past occurrences as probabilities.
These probabilities stabilize, or approach a limit, as the number of trials or experiments increases. These probabilities are easier to understand and illustrate for games of choice because objective probabilities can easily be assigned to various events. However, the primary reason for studying probability theory is to help us make intelligent decisions in economics, business, science, and everyday life when risk and uncertainty are involved.
What is the probability of a A head in one toss of a balanced coin? A tail? A head or a tail? Not a 2? What is the probability that, in picking up a single ball, the ball is a Red? Since there are 3 blue balls and 7 nonblue balls, the odds in favor of picking a blue ball are 3 to 7, or 3: If the die is fair, we expect the 3 to come up times in rolls of the die as compared with the actual, observed, or empirical times.
When one event takes place, the other s will not. Heads and tails are therefore mutually exclusive events. In a simple toss of a die, we get one and only one of six possible outcomes: The outcomes are therefore mutually exclusive. A card picked at random can be of only one suit: A child is born either a boy or a girl.
An item produced on an assembly line is either good or defective. The occurrence of one does not preclude the occurrence of the other s. For example, a card picked at random from a deck of cards can be both an ace and a club. Therefore, aces and clubs are not mutually exclusive events, because we could pick the ace of clubs. The same is true for two successive tosses of a pair of dice or picks of two cards from a deck with replacement.
For example, if we pick a card from a deck and do not replace it, the probability of picking the same card on the second pick is 0. Similarly, if the proportion of defective items is greater for the evening than for the morning shift, the probability that an item picked at random from the evening output is defective is greater than for the morning output. When one event occurs, the probability of the other occurring is 0.
Therefore, we subtract the probability of getting the ace of clubs in order to avoid this double counting. If the events are mutually exclusive, the probability that both events will occur simultaneously is 0, and no double counting is involved. This is why the rule of addition for mutually exclusive events does not contain a negative term.
The events are independent. Therefore CHAP. More than 4?
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In Table 3. The total of the 36 possible outcomes also can be shown by a tree or sequential diagram, as in Fig. These are 1, 4; 2, 3; 3, 2; and 4, 1. There are 6 possible and equally likely ways of rolling a total of 4 or less.
These are 1, 1; 1, 2; 1, 3; 2, 1; 2, 2; and 3, 1. The probability of getting a total of more than 4 equals 1 minus the probability of getting a total of 4 or less. What is the probability of a Picking a second red ball from the urn in Prob. During a h period, items are produced by the morning shift and by the evening shift. What is the probability that an item picked at random from the total of items produced during the h period a Was produced by the morning shift and is defective?
Thus we expect 5 defective items from the items produced during the h period. However, bayesian econometrics is becoming increasingly important. Combinations and permutations were not used in previous problems because those problems were simple enough without them. Thus the outcome from the roll of a die is a random variable. For example, the outcomes from rolling a die constitute discrete random variables because they are limited to the values 1, 2, 3, 4, 5, and 6.
The set of the 6 outcomes in rolling a die and their associated probabilities is an example of a discrete probability distribution. The sum of the probabilities associated with all the values that the discrete random variable can assume always equals 1. Because those probabilities are assigned a priori and without any experimentation, a probability distribution is often referred to as a theoretical relative frequency distribution.
Determine the expected number of applications processed and the variance and standard deviation. See Eqs. If there are 6 children in the family, what is the probability that half of them will have blond hair? Substituting these values into the binomial formula, we get 6! What is the probability that no more than 2 of the tubes picked are defective? What is the probability that 10 of the items picked are acceptable?
Since App. Then the hypergeometric distribution is used. This is the reason the binomial distribution was used in Prob. Why can this be useful? Find the probability that more than 1 bulb is defective in a random sample of 30 bulbs, using a the binomial distribution and b the Poisson distribution. Using App. Using Eq. A continuous variable can be measured with any degree of accuracy simply by using smaller and smaller units of measurement.
For example, if we say that a production process takes 10 h, this means anywhere between 9. If we used minutes as the unit of measurement, we could have said that the production process takes 10 h and 20 min. This means anywhere between 10 h and Time is thus a continuous variable, and so are weight, distance, and temperature. The probability distribution of a continuous random variable is often called a probability density function, or simply a probability function.
It is given by a smooth curve such that the total area probability under the curve is 1. However, we can measure the probability that a continuous random variable X assumes any value within a given interval say, between X1 and X2 by the area under the curve within that interval: Probability tables for some of the most used continuous probability distributions are given in the appendixes, thus eliminating the need to perform the integration ourselves.
What is its usefulness? As we move further away from the mean in both directions, the normal curve approaches the horizontal axis but never quite touches it. Many distributions actually found in nature and industry are normal. Some examples are the IQs intelligence quotients , weights, and heights of a large number of people and the variations in dimensions of a large number of parts produced by a machine.
The normal distribution often can be used to approximate other distributions, such as the binomial and the Poisson distributions see Probs. Distributions of sample means and proportions are often normal, regardless of the distribution of the parent population see Sec. This is accomplished by moving down the z column in the table to 1. Note that the table only gives detailed z values for up to 2. This is 0.
Because of symmetry, 0. Since 0.
What is the probability that a bulb picked at random will have a lifetime between and burning hours? Subtracting 0.
What is the probability that a family picked at random will have an income: What is the lowest grade point that can be designated an A on the midterm? Since the total area under the curve to the right of 78 is 0. We must look into the body of App. The X value the grade point that corresponds to the z value of 1. However, the number of people is a discrete variable. This means that 0. As n becomes even larger, the approximation becomes even closer.
Find the probability that 4 or less items are defective out of the output of a randomly chosen hour using a the Poisson distribution and b the normal approximation of the Poisson.
This means that 0: Because we are dealing with time, the exponential is a continuous probability distribution. The expo- The mean level of schooling for a population is 8 years and the standard deviation is 1 year. What is the probability that a randomly selected individual from the population will have had between 6 and 10 years of schooling?
Less than 6 years or more than 10 years? What is the probability that by picking a single ball we pick a A blue ball? Also d a 1 or not CHAP. As this process is repeated times, we obtain spades. How many claims can the company expect during a 1-year period?
Calculate expected number of lunch customers, b the variance, and c the standard deviation. What is the probability of: If persons are randomly selected from the national labor force: What is the probability that this random variable will assume a value a Between 67 and 70?
What is the lowest IQ score acceptable for the advanced training? What is the probability that after a defective item: Statistical inference refers to estimation and hypothesis testing Chap. Estimation is the process of inferring or estimating a population parameter such as its mean or standard deviation from the corresponding statistic of a sample drawn from the population.
To be valid, estimation and hypothesis testing must be based on a representative sample. This can be obtained by random sampling, whereby each member of the population has an equal chance of being included in the sample.
A random sample of 5 out of the 80 employees of a plant can be obtained by recording the name of each employee on a separate slip of paper, mixing the slips of paper thoroughly, and then picking 5 at random. A less cumbersome method is to use a table of random numbers App. Then starting at random say, from the third column and eleventh row in App.
For example, reading vertically we get 13, 54, 19, 59, and The probability distribution of these sample means is called the sampling distribution of the mean. Two important theorems relate the sampling distribution of the mean to the parent population. This is the central-limit theorem. If the parent population is normal, the sampling distributions of the mean are also normally distributed, even in small samples.
Assume that a population is composed of elements with a mean of 20 units and a standard deviation of A point estimate is a single number. Such a point estimate is unbiased if in repeated random samplings from the population, the expected or mean value of the corresponding statistic is equal to the population parameter. A random sample of with a mean of and a standard deviation of 60 is taken from a population of First, we note that this problem involves the binomial distribution see Sec.
This is obtained from App. The value we get is 2. When n Fig. What is its function and importance? Hypothesis testing? A population is the collection of all the elements people, parts produced by a machine, cars passing through a checkpoint, etc. A sample is a portion chosen from the population.
These problems can be overcome by taking a representative sample from a population and making inferences about the population from the sample.
A statistic is a descriptive characteristic of a sample. In statistical inference, we make inferences about parameters from their corresponding statistics. Estimation is the process of inferring or estimating a parameter from the corresponding statistic. For example, we may estimate the mean and the standard deviation of a population from the mean and standard deviation of a sample drawn from the population.
Hypothesis testing is the process of determining, on the basis of sample information, whether to accept or reject a hypothesis or assumption with regard to the value of a parameter.
We deal with estimation in this chapter and with hypothesis testing in Chap. What is meant by random sampling? What is its importance? Random sampling is a sampling procedure by which each member of a population has an equal chance of being included in the sample. Random sampling ensures a representative sample. There are several types of random sampling. In simple random sampling, not only each item in the population but each sample has an equal probability of being picked.
In systematic sampling, items are selected from the population at uniform intervals of time, order, or space as in picking every one-hundredth name from a telephone directory. Systematic sampling can be biased easily, such as, for example, when the amount of household garbage is measured on Mondays which includes the weekend garbage. Cluster sampling is used when the opposite is the case. In what follows, we assume simple random sampling. Appendix 4 lists digits in sets of 5 digits generated by a completely random process and such that each digit and sequence of digits has the same probability of occurring as every other digit and sequence of digits.
Starting at an arbitrary point in App. The probability distribution of these sample means is called the theoretical sampling distribution of the mean. For example, we CHAP. For simplicity, this section deals only with the sampling distribution of the mean. By convention, this is done whenever n 4. The number of combinations of 5 numbers taken 2 at a time without concern for the order is 5!
These 10 samples are 1; 3; 1; 5; 1; 7; 1; 9; 3; 5; 3; 7; 3; 9; 5; 7; 5; 9; and 7; 9. The theoretical sampling distribution of the mean is given in Table 4. In this case, the sampling distribution of the sample mean generated is referred to as the empirical sampling distribution of the mean.
Find the mean and standard error of the sampling distribution of the mean for sample sizes of a and b If the parent population is not normal? According to the central limit theorem, even if the parent population is not normal, the theoretical sampling distributions of the sample mean approach normality as sample size increases i. It allows us to use sample statistics to make inferences about population parameters without knowing anything about the shape of the parent population.
This will be done in this chapter and in Chap. This is analogous to what was done in Sec. The area under the curve within 1, 2, and 3 standard deviation units from the mean is When the estimate of an unknown population parameter is given by a single number, it is called a point estimate.
Another way of stating this is that an estimator is unbiased if its expected value see Probs. Other important criteria for a good estimator are discussed in Sec.
The mean test score for the sample is , and the standard deviation for the entire population of students is What is the minimum sample size required? Thus CHAP. Because in Prob. Since we were deciding on what sample size to take in Prob.
What is the minimum sample size required if other polls indicate that the proportion voting for this candidate is 0. In this and similar cases, trying to get an actual estimate of p does not greatly reduce the size of the required sample. When p is taken to be 0. For example, if we deal with a sample of 2 and we know that the sample mean for these two values is 10, we can freely assign the value to only one of these two numbers. If one number is 8, the other number must be 12 to get the mean of How do these t values compare with their corresponding z values?
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This gives the t value of 1. However, z values given in App. Because of symmetry, 5, 2. These coincide with the corresponding z values in App. Suppose that we know that the population from which the sample is taken is not normally distributed. However, it represents the only possibility short of increasing the sample size to at least 30 so that the normal distribution can be used. In reality, however, the t distribution is used even in these cases.
What is the mean and standard error of the theoretical sampling distribution of the mean for sample sizes of a 25 and b 81? The average weight for the sample of army recruits is lb, and the standard deviation of the entire population of army recruits is 40 lb. What is the minimum sample size required if previous experience indicates that the proportion of defective lightbulbs produced is 0.
The grades for the entire class are known to be normally distributed. In testing a hypothesis, we start by making an assumption with regard to an unknown population characteristic. We can make two types of errors in testing a hypothesis. First, on the basis of the sample information, we could reject a hypothesis that is in fact true. This is called a type I error. Second, we could accept a false hypothesis and make a type II error. This is represented by H0: The alternative hypotheses are then H1: Take a random sample from the population and compute X.
If X in standard deviation units falls in the acceptance region, accept H0 ; otherwise, reject H0 in favor of H1. Since the rejection region is in both tails, we have a two-tail test. For the entering class of 36, only 15 received their degrees by To test if the class CHAP. Since we would like to test if the class performed worse, we have H0: Problem 5. Problems 5. The hypotheses to be tested are H0: When the expected frequency of a category is less than 5, the category should be combined with an adjacent one see Prob.
For testing if the sampled distribution is binomial or normal, see Probs. Table 5. A car dealer has collected the data shown in Table 5. Thus Table 5. The populations are assumed to be independently normally distributed, and of equal variance. The steps are as follows: Estimate the population variance from the variance between the sample means MSA in Table 5. Estimate the population variance from the variance within the samples MSE in Table 5.
The preceding steps are formalized in Table 5. The sales for 5 months are given in Table 5. Sales data are normally distributed with equal variance. The preceding procedure is referred to as one-way, or one-factor, analysis of variance.
For two-way analysis of variance, see Probs. Usually the assumption in question is the normality of the distribution distribution of the data is unknown or the sample size is small. Nonparametric tests are often based on counting techniques that are easier to calculate and may be used for ordinal as well as quantitative data. To test a hypothesis about the median of a population analogous to test of population mean , the Wilcoxon signed rank test may be used: This is compared to the critical values in App.
The signed rank test can be adjusted slightly to test equality of medians of more than two samples analogous to ANOVA, but no assumption of normality in the Kruskal-Wallis test: Rank all data as if from a single sample. P Add ranks of each sample, Rj. The test statistic! Arrange data from smallest value to largest value.
The proportion of data below each value is compared with cumulative probability below that value from the hypothesized distribution. Since 4 CHAP. What is the general procedure? Type II error refers to the acceptance of a false hypothesis. In statistical analysis, we can control or determine the probability of type I or type II errors.
By specifying a smaller type I error, we increase the probability of a type II error. For example, we might toss the coin 20 times and get 9 heads instead of the expected This, however, does not necessarily mean that the coin is unbalanced. If, however, we get only 4 heads in 20 times, we are likely to be dealing with an unbalanced coin because the probability of getting 4 heads and 16 tails in 20 times with a balanced coin is very small indeed see Sec.
By accepting the hypothesis that the coin is balanced, we could thus be making a type I error. However, 4 heads in 20 tosses is very likely to mean an unbalanced coin. But by accepting the hypothesis that the coin is unbalanced, we must face the small probability that the coin is instead balanced, which would mean that we made a type II error. To do this, once again, the producer takes a random sample of the cable produced and tests the mean breaking strength X.
The more X falls short of lb, the more likely the producer is to accept the hypothesis that the breaking strength of the steel cables is less than the lb i. A breaking strength of less than lb would not be adequate, and to produce steel cables with breaking strengths of more than lb would unnecessarily increase production costs. Note CHAP. The relationship between this and the result obtained in Prob. How can this test be performed? With H1: That would be an unusual sample indeed.
The agency can set up H0 and H1 as follows: For the sample CHAP.
An antipollution advocate does not believe the government claim. The rejection region for H0 lies to the right 1. Note that increasing n and holding everything else the same increases the probability of accepting the government claim. What does this show? Note that even though brand 2 lasts longer than brand 1, brand 2 also has a greater standard deviation than brand 1.
In , the 81 students who apply have average GRE scores of with a standard deviation of A random sample is taken of 21 persons using each toothpaste. In the second group, the average number of cavities is 23 with a standard deviation of 4.
Note that as in the case of the t distribution, there is a 2 distribution for each degree of freedom. However, the 2 test is used here as a right-tail test only. Since for fe CHAP. Note also that because of the squaring, 2 can never be negative. Note that the 2 distribution is a continuous distribution as are the normal and t distributions.
From App. Since the calculated 2 is smaller than the tabular 2 , we accept the null hypothesis, H0 , that age is independent of sex in the occurrence of heart attacks. Note that the same adjustment indicated by Eq. Assume that the outputs with each fertilizer are normally distributed with equal variance.
Since the calculated value of F is smaller than the tabular value, we accept H0 , that the population means are the same. Since we were told in Prob. Note that the CHAP. Note that the F distribution is continuous and is used here for a right-tail test only. The results are shown in Table 5. From Table 5. The CHAP. The usual situation for using a nonparametric test in statistics is a small sample size. There are nonparametric tests appropriate for most scales of measurement, and for nonstandard functional forms and distributions.
The disadvantages of a nonparametric test focus around the loss of information. Nonparametric tests are based on counting rules, such as ranking, and therefore summarize magnitudes into a rank statistic. This only uses the relative position of values. A focus group of 10 individuals rate the taste on a scale of 1 to Results of the focus group are listed in Table 5. We proceed with the nonparametric test. The critical values for the signed rank test App.
Calculations are listed in Table 5. Since all African rankings fell above the rankings of South American and Asian countries, the rankings from Table 5. Since 41 5. To calculate the probability of being between values a and b, one can take the area under the density function: Since 0: Of rejecting a true hypothesis? What is another name for this? Thinner sheets would not be appropriate, and thicker sheets would be too heavy.
Of the 88 students applying to be admitted into the program in , 22 had GRE scores above By joining the values found in Prob. Yes 5. A ball is picked from the urn and its color is recorded. The ball is then replaced in the urn, the balls are thoroughly mixed, and another ball is picked. Does the urn contain an equal number of green, white, red, or blue balls?
Do rainy days in U. No Table 5. Accept H0 Table 5. Cities A. Assume that the miles per gallon for each octane is normally distributed with equal variance. Rejected Table 5. Table 1 gives the frequency distribution of the rate of unemployment in a sample of 20 large U.
The lifetime of an electronic component is known to be normally distributed with a mean of h and a standard deviation of 80 h. The average IQ of a random sample of 25 students at a college is For the sample in plant 2, the mean is g and the standard deviation is 60 g. Answers 1. Since the tabular value of t0: Simple Regression Analysis 6. Simple linear regression analysis usually begins by plotting the set of XY values on a scatter diagram and determining by inspection if there exists an approximate linear relationship: Table 6.
These are plotted in the scatter diagram of Fig. The relationship between X and Y in Fig. It involves minimizing the sum of the squared vertical deviations of points from the line: This gives the following two normal equations see Prob. Solving simultaneously Eqs. The values of y2i are obtained by squaring yi from Table 6. The total variation in Y is equal to the explained plus the residual variation: Figure shows the total, the explained, and the residual variation of Y. Thus OLS estimators are the best among all unbiased linear estimators [see Probs.
This is to be contrasted with multiple regression analysis, in which there are not one, but two or more independent or explanatory variables. Multiple regression analysis is discussed in Chap. This is to be contrasted with nonlinear regression analysis discussed in Sec. Its purpose is to determine by inspection if there exists an approximate linear relationship between the dependent variable Y and the independent or explanatory variable X.
Draw a scatter diagram for the data and determine by inspection if there exists an approximate linear relationship between Y and X. From Fig.
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In Eq. The second assumption is that the expected value of the error term or its mean equals zero: Since the average value of u is assumed to be 0, Eq. The third assumption is that the variance of the error term is constant in each period and for all values of X: However, the sum of the squared deviations is preferred so as to penalize larger deviations relatively more than smaller deviations. The reader without knowledge of calculus can skip this problem.
Start by P a Multiplying Eq. How does this regression line compare with the regression line plotted in Fig. See Fig. It measures the marginal propensity to consume MPC or the change in consumption per one-unit change in disposable income. Once again, the fact that 0 The income elasticity of consumption measures the percentage change in consumption resulting from a given percentage change in disposable income.
Since the elasticity usually changes at every point in the function, it is measured at the means: P 2 What is its range of values? However, correlation analysis implies no causality or dependence but refers simply to the type and degree of association between two variables.
Thus correlation analysis is a much less powerful tool than regression analysis and is seldom used by itself in the real world.
In fact, the main use of correlation analysis is to determine the degree of association found in regression analysis. Furthermore, with a great number of observations of large values, r 0 can be found as an estimate of r in order to avoid very time-consuming calculations however, easy accessibility to computers has practically eliminated this reason for using r 0. The mean of the sampling distribution is the expected value of the estimator. The hope is that the sample actually obtained is close to the mean of the sampling distribution of the estimator.
Why is this important? It is the unbiased estimator with the most compact or least spread-out distribution. This is very important because the researcher would be more certain that the estimator is closer to the true population parameter being estimated. It should be noted that minimum variance by itself is not very important, unless coupled with the lack of bias.
Is it superior to all other estimators? That is, among all unbiased linear estimators, it has the lowest variance. However, nonlinear estimators may be superior to the OLS estimator i. The OLS estimator, being linear, is also easier to use than nonlinear estimators. Why and when is the rule to minimize the meansquare error useful?
The researcher is then likely to choose the estimator with the lowest MSE. This rule penalizes equally for the larger variance or for the square of the bias of an estimator. World Bank World Development Indicators.
P 2 e Regress Yi on Xi and report your results in summary form. Ordinary least-squares OLS parameter estimates for Eq. Table 7. Using Eqs.The value we get is 2. Harry Potter.
What is the minimum sample size required if previous experience indicates that the proportion of defective lightbulbs produced is 0. This is common in time-series analysis. The CHAP. While causality is an elusive concept that can never be proved with certainty, time-series econometrics can help sort out these timing issues. With H1: Analyzing the ACF, we see a large positive correlation at 4 lags, and subsequently smaller correlations at intervals of every 4 lags 8, 12, 16 lags.
Then starting at random say, from the third column and eleventh row in App. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited.