Statistics Homework Analysis - Assignment Example

STATISTICS Exercises: Topic 5 11. The events A and B are mutually exclusive. Suppose P(A) = .30 and P(B) = .20. What is the probability of either A or B occurring? What is the probability that neither A nor B will happen? Two events are mutually exclusive if both cannot happen simultaneously in a single trial, in other words, it is impossible for them to happen together. In this case then, the probability of either A or B happening, = P (A) +P (B) =0.30+0.20 =0.50 The probability of neither A nor B happening =P (A) +P (B)-[P (A) and P (B)] =0.5-0.06 =0.44 12. The events X and Y are mutually exclusive. Suppose P(X) = .05 and P(Y) = .02. What is the probability of either X or Y occurring? What is the probability that neither X nor Y will happen? P(X) + P(Y) =0.05+0.02 =0.07 is the probability of either X or Y occurring 0.07-(0.02*0.05) 0.07-0.001 0.069 is the probability that neither X nor Y will happening 13. A study of 200 advertising firms revealed their income after taxes: Income after taxes Number of firms Under $1 million 102 $1million to 20million 61 $20 million or more 37 a. What is the probability an advertising firm selected at random has under $1 million in income after taxes? 102/200=0.51 is the probability an advertising firm selected at random has under $1 million in income after taxes b. What is the probability an advertising firm selected at random has either an income between $1 million and $20 million, or an income of $20 million or more? What rule of probability was applied? =P (61/200) or P (37/200) =0.305+0.185 =0.49 is the probability an advertising firm selected at random has either an income between $1 million and $20 million, or an income of $20 million or more. The rule of mutually exclusive events was applied. 14. The chair of the board of directors says there are a 50% chance this company will earn a profit, a 30% chance it will break even, and a 20% chance it will lose money next quarter. a. Use an addition rule to find the probability the company will not lose money next quarter. Using the additional rule, the probability of the company not losing money =P (earning a profit) + P (break even) P (0.5+0.3) =0.80 b. Use the complement rule to find the probability it will not lose money next quarter. P (earning a profit) + P (break even) P (0.5+0.3) =0.80 Chapter 6 38. For each of the following indicate whether the random variable is discrete or continuous. a. The length of time to get a haircut. Discrete b. The number of cars a jogger passes each morning while running. Discrete c. The number of hits for a team in a high school girls’ softball game. Discrete d. The number of patients treated at the South Strand Medical Center between 6 and 10 p.m. each night. Discrete e. The distance your car traveled on the last fill-up. Discrete f. The number of customers at the Oak Street Wendy’s who used the drive-through facility. Discrete g. The distance between Gainesville, Florida, and all Florida cities with a population of at least 50,000. Discrete 39. An investment will be worth $1,000, $2,000, or $5,000 at the end of the year. The probabilities of these values are .25, .60, and .15, respectively. Determine the mean and variance of the worth of the investment. Mean=∑xP(x) = (1000*0.25) + (2000*0.60) + (5000*0.15) =$2,200 Variance=∑ [(x-µ)2P(x)] µ=2200 [(1000-2200)2*0.25+ (2000-2200)2*0.60+ (5000-2200)2*0.15]=360000+24000+11,760,000 =$12,144,000 40. Croissant Bakery Inc. offers special decorated cakes for birthdays, weddings, and other occasions. It also has regular cakes available in its bakery. The following table gives the total number of cakes sold per day and the corresponding probability. Compute the mean, variance, and standard deviation of the number of cakes sold per day. No. of cakes: Probability np 12 0.25 3 13 0.40 5.2 14 0.25 3.5 15 0.10 1.5 Mean=∑np=13.2 Sd= √xnp 3.62 Variance= (12-13.2) + (13-13.2) + (14-13.2) + (15-13.2) =1.2 43. In a recent survey, 35% indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of 10 people and ask them to name their favorite flavor of ice cream. a. How many of those in the sample would you expect to name chocolate? 35% of 10 people= (35/100)*10 3 people and above would name chocolate b. What is the probability exactly four of those in the sample name chocolate? 4/10*0.35 0.14 is the probability exactly four of those in the sample name chocolate Chapter 7 17. A normal distribution has a mean of 50 and a standard deviation of 4. a. Compute the probability of a value between 44.0 and 55.0. z= (55-50)/4=1.25 The z value of 1.25=0.3944 And z for 44=-6/4=1.5 The z-value for 1.5=0.4332 The probability of a value between 44.0 and 55.0 is 0.4332-0.3944=0.0388 b. Compute the probability of a value between 52.0 and 55.0. z= (55-50)/4=1.25 The z value of 1.25=0.3944 And z for 52=2/4=0.5 The z-value for 0.5=0.1915 The probability of a value between 52.0 and 55.0 is 0.3944-0.1915=0.2029 19. According to the Internal Revenue Service, the mean tax refund for the year 2013 was $3,000. Assume the standard deviation is $450 and that the amounts refunded follow a normal probability distribution. a. What percent of the refunds is more than $3,100? Z= (3100-3000)/450 =100/450=0.2222 Z value=0.0871 The percentage that would be more than $3100 is 22.22% b. What percent of the refunds are more than $3,100 but less than $3,500? Z for 3500= (3500-3000)/450=500/450 =1.1111 =0.3643 0.3643-0.0871=0.2772 The percentage would be 27.72% c. What percent of the refunds are more than $2,250 but less than $3,500? (3000-2250)/450=1.6667 whose value is 0.4525 The percentage would be (0.4525-0.3643) =0.0882=8.82% 23. A normal distribution has a mean of 50 and a standard deviation of 4. Determine the value below which 95% of the observations will occur. Z=(x-µ)/δ The value of Z at 5% confidence level=1.96 1.96=x-50/4 7.84=x-50 X=57.84 29. In economic theory, a hurdle rate is the minimum return that a person requires before he or she will make an investment. A research report says that annual returns from a specific class of common equities are distributed according to a normal distribution with a mean of 12% and a standard deviation of 18%. A stock screener would like to identify a hurdle rate such that only 1 in 20 equities is above that value. Where should the hurdle rate be set? Mean=12% and the standard deviation=18% r=20-12/18 0.44% The differences between probability and conditional probability and the explanation of Bayes Theorem The rationality of probability reduces the ideology of uncertainty and ignorance that are normally present during minor and major decision making. Probability is the ratio of the number of favorable cases to the total number of equally likely cases. In this case, therefore, if there are equally likely events that may happen, then the probability that any of these events will happen is the ratio of the number of these events that are favorable to its happening to the total number of all possible events. For instance, if an event can happen in k ways and fail in y ways, and each of these ways (k+y) are likely to happen in equal measure, then the probability (the chance) of its happening is k/(k+y) and of its failure is y/(k+y). Conditional probability of an event given another event, on the other hand, is the probability of the event given that the other event has occurred. I.e. if P (B)>0, then P (A/B) = P (A and B)/P (B), where the vertical bas, /, represent conditioning (given). This illustration therefore, means that a major difference that exist between the normal probability and conditional probability of an event occurring is that an event in a conditional probability case occurs if and only if another event has occurred whereas in a normal probability situation, the chance of an event occurring is based on the ratio of the number of favorable events to the total number of equally likely events. Based on the analysis of the conditional probability described above, the Bayes’ theorem shows how to invert the conditional probabilities. The theory, modeled by Bayes (1702-1761) is concerned with estimation of probabilities of the cases by which an observed event may have been produced. The theory states that if events A1, A2….AX become mutually exclusive events whose union is universe, and another arbitrary event, B, happens to be in the universe such that P(A)≠0, and given that P (B/A1) and P (B/A2) are known, the P (Aj/B) = [P (B/Aj) P (Aj)]/∑P (Aj) P (B/Aj) The main purposes of variance and standard deviation and how the probability principle would interact with the variance and standard deviation concepts: an empirical example from the business environment of usefulness of this concept in business decision making information The deviations scores from the mean of a distribution have become the main statistical tests of variations when making decision. Making decisions on the dispersions of the various sets of scores about the mean is very critical when determining the variability of the options available for making a choice. Deviations about the mean are expressed in terms of the variance or standard deviation. In this case therefore, it is important for the management of firms to be able to grasp the concept of deviation and standard deviation since they are influential in determining variability of decision variables. The statistical inferences such as variance and standard deviations are directly linked to the principle of probability in the sense that the processes of modeling a probability concept use variance and standard deviations. As argued above, it is clearly true that deviations scores from the mean of a distribution have become the main statistical tests of variations when making decision and are thus important in determining variability as well. In the same way, a probabilistic event or decision may be based on specific theory whose determination may be based on the past data and experience that are worth statistical determination. Even so, the data when summarized and measures of dispersions are determined from the mean, and then the probability trends may be well determined with utmost exactness. Uncertainty, which may be both scientifically and socially based, is an inherent feature of environmental management and arises at many points through the environmental decision-making process for an organization. Every management must be ready to make decisions on behalf of their organizations. Such decisions have effects that may take any direction based on the criticality and the approach of the decision. This situation means that decisions are made in a field of competitions and risky situations. Uncertainties and maybe ignorance may exist in the event that practicable variances and standard deviations are not properly propagated. In this case, statistical analysis of the past and present resources and variables becomes very important. More than that, based on the past performance, or the performance of the market or industry, the probability of making the decision made become possible. In addition to that, managers are constantly to evaluate alternatives and make decisions regarding wider range of matters that are surrounded by risks and uncertainties. Therefore, decision making requires both qualitative and quantitative analyses like variances, standard deviations, and probability analysis. The area under the normal distribution and how the concept of probability related to the normal distribution concept A normal distribution of a random variable is given by a mean and variance that have a density function of y=f(x) =1/δ√2π e-1/2({x-µ/δ})2 Based on the above model, a graph of a normal distribution gives a curve that is divided by the mean in equal halves and depends on the frequency in total that is equated to one. The area under the curve of a normal distribution will always occur at the mean of the distribution. The area under the normal curve can be 100% or 1.0, based on the percentage of an area under the curve or a proportion of the area under the curve. Areas under the curve can also be interpreted as probabilities. The area within plus one and minus one standard deviation of the mean constitutes about 68 percent of the area under the curve. The Z-score and its calculation The z-score is the standardized raw data in order to use a standardized normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. However, the normal system is that different scales of measures are normally used for statististical analyses that can go beyond 500 as a mean and 30 as the standard deviation. In this case, it would be difficult to determine some areas under the curve at certain intervals. The z-score is determined by, z=(Y-Ý)/δ Here, z is the standard value (Z score) that has been obtained from the original data by subtracting the mean of the original data and dividing by the standard deviation. Example 1. Suppose we have a batch of data with a mean of 90 and a standard deviation of 10. What is the z score corresponding to a raw score of 100? Z= (100-90)/10 =1 2. Suppose now the data are such that the mean is 1053.72 and is 105.69. What is the z score or standardized value corresponding to Y = 1950.18? Z= (1950.18-1053.72)/105.69 =8.48 References Johnson, R., & Kuby, P. (2008). Elementary statistics. Belmont, CA: Thomson Brooks/Cole. Pal, N., & Sarkar, S. (2005). Statistics: Concepts and applications. New Delhi: Prentice-Hall of India. Read More