STATS 101 WK7 NOTE.pptx

May. 28, 2023
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
STATS 101 WK7 NOTE.pptx
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STATS 101 WK7 NOTE.pptx

Editor's Notes

  1. The absolute frequencies or cumulative frequencies are placed on the y-axis while the categories, class midpoints( frequency polygons) or class boundaries (ogive) are placed on the x-axis.
  2. It is not wrong to truncate an axis of the graph; many times it is necessary to do so. However, the reader should be aware of this fact and interpret the graph accordingly. Do not be misled if an inappropriate impression is given.
  3. Another common method used to organize data is the stem and leaf plot. It is a combination of sorting and graphing. Unlike grouped frequency distribution, a stem and leaf plot retains the actual data while showing them in graphical form.
  4. When the data values are in the tens such as 39, the stem is 3 (leading digit) and the leaf is 9 (trailing digit) When the data values are in hundreds, such as 435, the stem is 43 (leading digits) and the leaf is 5 (trailing digit).
  5. When the data values are in the tens such as 39, the stem is 3 (leading digit) and the leaf is 9 (trailing digit) When the data values are in hundreds, such as 435, the stem is 43 (leading digits) and the leaf is 5 (trailing digit).
  6. Most of Bomi citizens selected are less than 10 years old. Few Bomi citizens are in their 55s. Like Bomi, most of Bassa citizens selected are less than 10 years old. Unlike Bomi, there are more Bassa citizens who are in their late 20s.
  7. Most of Bomi citizens selected are less than 10 years old. Few Bomi citizens are in their 55s. Like Bomi, most of Bassa citizens selected are less than 10 years old. Unlike Bomi, there are more Bassa citizens who are in their late 20s.
  8. When the data values are in the tens such as 39, the stem is 3 (leading digit) and the leaf is 9 (trailing digit) When the data values are in hundreds, such as 435, the stem is 43 (leading digits) and the leaf is 5 (trailing digit).
  9. In addition to knowing the average, you must know how the data values are dispersed. That is, do the data values cluster around the mean, or are they spread more evenly throughout the distribution? Measures of position tell where a specific data value falls within the dataset or its relative position in comparison with other data values.
  10. The mean, in most cases, is not an actual data value.
  11. The mean, in most cases, is not an actual data value.
  12. The mean, in most cases, is not an actual data value.
  13. The modal class is 20.5–25.5, since it has the largest frequency. Sometimes the midpoint of the class is used rather than the boundaries; hence, the mode could also be given as 23 miles. An extremely high or extremely low data value in a data set can have a striking effect on the mean of the data set. These extreme values are called outliers. This is one reason why when analyzing a frequency distribution, you should be aware of any of these values.
  14. In this example, the mean is much higher than the median or the mode. This is so because the extremely high salary of the owner tends to raise the value of the mean. In this and similar situations, the median should be used as the measure of central tendency.
  15. If the data set contains one extremely large value or one extremely small value, a higher or lower midrange value will result and may not be a typical description of the middle.
  16. If the data set contains one extremely large value or one extremely small value, a higher or lower midrange value will result and may not be a typical description of the middle.
  17. If the data set contains one extremely large value or one extremely small value, a higher or lower midrange value will result and may not be a typical description of the middle.
  18. When a distribution is extremely skewed, the value of the mean will be pulled toward the tail, but the majority of the data values will be greater than the mean or less than the mean (depending on which way the data are skewed); hence, the median rather than the mean is a more appropriate measure of central tendency. An extremely skewed distribution can also affect other statistics.
  19. The mean, in most cases, is not an actual data value.
  20. In the Previous example, the range for brand A shows that 50 months separate the largest data value from the smallest data value. For brand B, 20 months separate the largest data value from the smallest data value, which is less than one-half of brand A’s range.
  21. For example, Chebyshev’s theorem shows that, for any distribution, at least 75% of the data values will fall within 2 standard deviations of the mean. A note of caution should be mentioned here. The range rule of thumb is only an approximation and should be used when the distribution of data values is unimodal and roughly symmetric.
  22. When we have sample that are measure in the same units (usually the same variable), we can compare the variance and standard deviations directly. But what happens if the units are not the same?
  23. Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results. For example, a saleswoman can compute the probability that she will make 0, 1, 2, or 3 or more sales in a single day. An insurance company might be able to assign probabilities to the number of vehicles a family owns. A self-employed speaker might be able to compute the probabilities for giving 0, 1, 2, 3, or 4 or more speeches each week. Once these probabilities are assigned, statistics such as the mean, variance, and standard deviation can be computed for these events. With these statistics, various decisions can be made. The saleswoman will be able to compute the average number of sales she makes per week, and if she is working on commission, she will be able to approximate her weekly income over a period of time, say, monthly. The public speaker will be able to blu34978_ch05.qxd 8/5/08 1:30 PM Page 252 plan ahead and approximate his average income and expenses. The insurance company can use its information to design special computer forms and programs to accommodate its customers’ future needs. This chapter explains the concepts and applications of what is called a probability distribution. In addition, special probability distributions, such as the binomial, multinomial, Poisson, and hypergeometric distributions, are explained.
  24. Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results. For example, a saleswoman can compute the probability that she will make 0, 1, 2, or 3 or more sales in a single day. An insurance company might be able to assign probabilities to the number of vehicles a family owns. A self-employed speaker might be able to compute the probabilities for giving 0, 1, 2, 3, or 4 or more speeches each week. Once these probabilities are assigned, statistics such as the mean, variance, and standard deviation can be computed for these events. With these statistics, various decisions can be made. The saleswoman will be able to compute the average number of sales she makes per week, and if she is working on commission, she will be able to approximate her weekly income over a period of time, say, monthly. The public speaker will be able to blu34978_ch05.qxd 8/5/08 1:30 PM Page 252 plan ahead and approximate his average income and expenses. The insurance company can use its information to design special computer forms and programs to accommodate its customers’ future needs. This chapter explains the concepts and applications of what is called a probability distribution. In addition, special probability distributions, such as the binomial, multinomial, Poisson, and hypergeometric distributions, are explained.
  25. Before probability distribution is defined formally, the definition of a variable is reviewed. In Chapter 1, a variable was defined as a characteristic or attribute that can assume different values. Various letters of the alphabet, such as X, Y, or Z, are used to represent variables. Since the variables in this chapter are associated with probability, they are called random variables. For example, if a die is rolled, a letter such as X can be used to represent the outcomes. Then the value that X can assume is 1, 2, 3, 4, 5, or 6, corresponding to the outcomes of rolling a single die. If two coins are tossed, a letter, say Y, can be used to represent the number of heads, in this case 0, 1, or 2.
  26. The word counted means that they can be enumerated using the numbers 1, 2, 3, etc. For example, the number of joggers in Riverview Park each day and the number of phone calls received after a TV commercial airs are examples of discrete variables, since they can be counted.
  27. A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation. Discrete probability distributions can be shown by using a graph or a table. Probability distributions can also be represented by a formula.
  28. Refer to text book for solution.
  29. Objective: Find the mean, variance, standard deviation, and expected value for a discrete random variable
  30. Note: X∗P(X) means to sum the products.
  31. Refer to text book for solution.
  32. Find the variance of a probability distribution by multiplying the square of each outcome by its corresponding probability, summing those products, and subtracting the square of the mean. Remember that the variance and standard deviation cannot be negative.
  33. Refer to text book for solution.
  34. Refer to text book for solution.
  35. The formula for the expected value is the same as the formula for the theoretical mean. The expected value, then, is the theoretical mean of the probability distribution. That is, E(X) = 𝜇. When expected value problems involve money, it is customary to round the answer to the nearest cent.
  36. Refer to text book for solution.
  37. Note that the expectation is $0.65. This does not mean that you lose $0.65, since you can only win a television set valued at $350 or lose $1 on the ticket. What this expectation means is that the average of the losses is $0.65 for each of the 1000 ticket holders. Here is another way of looking at this situation: If you purchased one ticket each week over a long time, the average loss would be $0.65 per ticket, since theoretically, on average, you would win the set once for each 1000 tickets purchased.
  38. Refer to text book for solution.
  39. Note that the expectation is $0.65. This does not mean that you lose $0.65, since you can only win a television set valued at $350 or lose $1 on the ticket. What this expectation means is that the average of the losses is $0.65 for each of the 1000 ticket holders. Here is another way of looking at this situation: If you purchased one ticket each week over a long time, the average loss would be $0.65 per ticket, since theoretically, on average, you would win the set once for each 1000 tickets purchased.
  40. Refer to text book for solution.
  41. Refer to text book for solution.
  42. The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.
  43. The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution. In binomial experiments, the outcomes are usually classified as successes or failures. For example, the correct answer to a multiple-choice item can be classified as a success, but any of the other choices would be incorrect and hence classified as a failure.
  44. Note that 0 ≤ X ≤ n and X = 0, 1, 2, 3, . . . , n.
  45. In a binomial experiment, the probability of exactly X successes in n trials is
  46. This situation fits the binomial experiment because the four conditions are satisfied There are a fixed number of trials (three). There are only two outcomes for each trial, heads or tails. The outcomes are independent of one another (the outcome of one toss in no way affects the outcome of another toss). The probability of a success (heads) is in each case. Refer to text book for solution.
  47. If X consists of events E1, E2, E3, . . . , Ek, which have corresponding probabilities p1, p2, p3, . . . , pk of occurring, and X1 is the number of times E1 will occur, X2 is the number of times E2 will occur, X3 is the number of times E3 will occur, etc., then the probability that X will occur is given by the formula above.
  48. Refer to the textbook for solution.
  49. Refer to textbook for solution.
  50. Refer to textbook for solution.
  51. Given a population with only two types of objects (females and males, defective and nondefective, successes and failures, etc.), such that there are a items of one kind and b items of another kind and a + b equals the total population, the probability P(X) of selecting without replacement a sample of size n with X items of type a and n - X items of type b is given by the formula above. The basis of the formula is that there are aCX ways of selecting the first type of items, bCn-X ways of selecting the second type of items, and a+bCn ways of selecting n items from the entire population.