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# Introduction to Statistics I STAT 1000

UCONN

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This 39 page Class Notes was uploaded by Blair Williamson on Thursday September 17, 2015. The Class Notes belongs to STAT 1000 at University of Connecticut taught by Vladimir Pozdnyakov in Fall. Since its upload, it has received 9 views. For similar materials see /class/205898/stat-1000-university-of-connecticut in Statistics at University of Connecticut.

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Date Created: 09/17/15

14 Comparing Two Population Means lst data set X1 an withEXl u1 2quotd data set Y1 with EX ui How to compare two population means quot2 Independent Sampling Large Samples 1 Setup Null Hypothesis H0 1 2 D0 Alternative Hypothesis 0 HaLtl LLZgtD0 or o Hau1 uzltDoor Ha 1 2 7 D0 2 Test Statistic The test statistic is given by Y f D 0 2 2 Li 7 i4 71 1 2 3 Rejection Region 0 zaoo or o 00 Za or o oo Z1 le 00 1 A U 4 1 Assumptions The two samples are randomly selected in an independent manner from two populations Both 111 and 112 are greater than 30 Independent Sampling Small Samples 1 Setup Null Hypothesis H0 1 2 D 0 Alternative Hypothesis 0 HaLtl LLZgtD0 or o Haul uzltDoor H a i 1 2 i D 0 2 Test Statistic The test statistic is given by t Y D0 2 2 15 n n 3 Rejection Region The rejection region is given by 0 twice or o 00 tayv or rt I 39 IAV l where the number of degrees of freedom is given by V minn1 1 n2 1 Assumptions The two samples are randomly selected in an independent manner from two populations Both sampled populations are approximately normal Example 1 A processor of recycled aluminum cans is concerned about the level of impurities principally other metals contained in lots from two sources Laboratory analysis of sample lots yields the following data kilograms of impurities per hundred kilogram of product n1 12 J 3267 3X 676 n2 12 173617 sy 1365 Can the processor conclude using the con dence level of 5 that there is a nonzero difference in means Solution 1 Setup Let y z39 12 be the true means of all the lots from the 139 th source H0ul uz0 HaLtl LLZ 0 2 Test statistic Note that the sample sizes are small I X Y D0 3267 36l7 7956 s s 6762 13652 77 n1 r12 12 12 RR oo tyv U tyvoo 3 Rejection Region 00 1 025711Ut025711oo oo2201U2201oo 4 Conclusion The test statistic does not fall into the rejection region Therefore we cannot reject the null at signi cance level of 5 That is our data does not support the research hypothesis that there is a nonzero difference between two sources Paired Difference Experiment Large Samples ngt30 The same dogs in two measurements or pairs of matched before measurements dogs 1 Setup Null Hypothesis H0 IuD D0 Alternative Hypothesis 0 Ha yD gtD0 or 0 Ha yD ltD0 or 0 Ha yD 7E D0 2 Test Statistic The test statistic is given by where X D is the sample mean difference 3 D is the sample standard deviation of differences and n D is number of pairs Under the null it has standard normal distribution 3 Rejection Region The rejection region is given by 0 zaoo or o 00 Za or o 00 Z U 2oo Assumptions The differences are randomly selected from the population of differences Paired Difference Experiment Small Samples n 30 1 Setup Null Hypothesis H0 IuD D0 Alternative Hypothesis 0 Ha uD gtD0 or 0 Ha uD ltD0 or 0 Ha uD D0 2 Test Statistic The test statistic is given by YD D0 3 D In D a where X D is the sample mean difference 3 D is the sample standard deviation of t differences and n D is number of pairs Under the null it has t distribution with n D 1 degrees of freedom 3 Rejection region The rejection region is given by tamrl w or o oo t 1 or mm o wa tagwl U Pumped Assumptions The differences are randomly selected from the population of differences The difference population distribution is approximately normal Example 2 A tasting panel of 15 people is asked to rate two new kinds of tea on a scale ranging from 0 to 100 25 means I would try to nish it only to be polite 50 means I would drink it but not buy it 75 means It s about as good as any tea I know and 100 means It s superb I would drink nothing else The difference in rating is recorded for each person The mean of differences is 7 and the standard deviation is 1608 Does these data indicate the difference in ratings at a 5 Solution 1 Setup H0 yD 0 Ha yD 0 2 Test Statistic The test statistic is equal to YD D0 7 0 7 169 SD ME 16085 3 Rejection region 00 00 2 145 U 2145oo RR oo IYKD71UIYKD71 4 Conclusion The test statistic does not fall into the rejection region Therefore we cannot reject the null at significance level of 5 That is our data does not support the research hypothesis that there is difference in ratings of these two kinds of tea Exercises p 420 7147715 7197720 p 434 735 738 References 1 Chase and Bown General Statistics 2 Hildebrand and Ott Statistical Thinking for Managers 3 Keller and Warrack Statistics for Management and Economics 4 McClave Benson and Sincich A First Course In Business Statistics Exercises 1 Business schools A and B reported the following summary of GMAT Graduate Management Aptitude Test verbal scores n X s2 A 201 3475 4859 B 115 3374 3068 At a 5 level of signi cance is there sufficient evidence to believe there is a difference in the population means a Use the classical approach b Use the Pvalue approach 2 Assume independent samples from approximately normal populations with equal variances The data are given here n X s2 A 10 74 60 B 13 81 40 Is there sufficient evidence to conclude that the mean of A is smaller than the mean of B Use the 5 significance level 3 Twentyfour males age 2529 were selected from the Framingham Heart Study Twelve were smokers and 12 were nonsmokers The subjects were paired with one being a smoker and the other a nonsmoker Otherwise each pair was similar with regard to age and physical characteristics Systolic blood pressure readings were as follows Smokers Nonsmokers 122 114 146 134 120 114 114 116 124 138 126 110 118 112 128 116 130 132 134 126 116 108 130 116 List the differences A B and verify that YD 6 and SD 840 Use a 5 level of signi cance to determine whether the data indicate a difference in mean systolic blood pressure levels for the populations from which the two groups were selected You may assume that the population of differences is approximately normal 4 A salesman for a shoe company claimed that runners would record quicker times on the average with the company39s brand of sneaker A track coach decided to test the claim The coach selected eight runners Each runner ran two 100yard dashes on different days In one 100yard dash the runners wore the sneakers supplied by the school in the other they wore the sneakers supplied by the salesman Each runner was randomly assigned the sneakers to wear for the first run Their times measured in seconds were as follows A B With shoe company s sneakers With school s sneakers 108 114 123 125 107 108 120 117 106 109 115 118 121 122 112 117 Note For the differences YD 225 and SD 276 Assume the population of differences is approximately normal 1 Data Graphical Descriptive Techniques Introduction Descriptive statistics involves the arrangement summary and presentation of data to enable meaningful interpretation and to support decision making Descriptive statistics methods make use of o graphical techniques 0 numerical descriptive measures The methods presented apply to both 0 the entire population 0 the population sample Types of data A variable is a characteristic of population or sample that is of interest for us for instance 0 Cereal choice 0 Capital expenditure 0 The waiting time for medical services Data the actual values of variables 0 Quantitative data are numerical observations 0 Qualitative data are categorical observations Quantitative data Age income 55 75000 42 68000 Weight gain 10 5 Qualitative data Person Marriedunmarried 1 yes 2 no 3 no Professor Rank 1 Lecturer 2 Full 3 Assistant With qualitative data all we can do is to calculate the proportion of data that falls into each category Lecturers Assistant Associate Full Total 15 25 5 15 60 25 4167 833 25 100 Knowing the type of data is necessary to properly select the technique to be used Type of analysis allowed for each type of data 0 Quantitative data arithmetic calculations 0 Qualitative data counting the number of observation in each category Cross sectional and Time Series Data Cross sectional data is collected at a certain point in time for example 0 Marketing survey observe preferences by gender age 0 Test score in a statistics course 0 Starting salaries of an MBA program graduates Time series data is collected over successive points in time for instance 0 Weekly closing price of gold 0 Amount of crude oil imported monthly Graphical Techniques for Quantitative Data Histogram Example Providing information concerning the monthly bills of new subscribers in the rst month after signing on with a telephone company 1 Collect data 2 Prepare a frequency distribution 3 Draw a histogram Collect data Prepare a frequency distribution Draw a Histogram Frequency 5 cn o o How many classes to use of observations of classes Less then 50 57 50 200 79 200 500 910 500 1000 1011 Class width Range of classes Range Largest Observation Smallest Observation What information can we extract from this histogram 0 About half of all the bills are small 0 A few bills are in the middle range 0 Relatively large number of large bills Relative Frequency It is o en preferable to show the relative frequency proportion of observations falling into each class rather than the frequency itself Class relative frequency Class frequency Total number of observations Relative frequencies should be used when 0 the population relative frequencies are studied 0 comparing two or more histograms 0 the number of observations of the samples studied are different Class width It is generally best to use equal class Width but sometimes unequal class width are called for Unequal class width is used when the frequency associated with some classes is too low Then 0 several classes are combined together to form a wider and more populated class 0 It is possible to form an open ended class at the higher end or lower end of the histogram Shapes ofhistograms There are four typical shape characteristics Symmetry Frequency 3 C1 skewness 0 Positively skewed 6 Frequency A 0 Negatively skewed 9 Frequency A m i 10 9 8 7 e 5 4 3 2 1 0 CS Number of modal classes A modal class is the one with the largest number of observations 0 unimodal histogram 257 207 Frequency C4 0 bimodal histogram 20 7 gt 0 3 10 7 039 9 LL 0 7 l l l l l 5 0 5 10 15 Bell shaped histogram 257 Frequency 0 Many statistical techniques require that the population be bell shaped 0 Drawing the histogram helps verify the shape of the population in question Stem and Leaf Display This is an interval scaled display most useful in preliminary analysis Stem and leaf diagram shows the value of the original observations whereas the histogram loses them Creating a stem and leaf display Observe the data in the table below 191 198 180 192 195 173 200 203 196 185 181 197 184 176 212 206 222 191 211 193 208 212 210 187 199 187 221 172 184 214 Determine what constitutes a stem and a leaf there is more than one way For example 0 the digits to the left of the decimal point is the stem 0 the digits to the right of the decimal point is the leaf List the stems in a column from smallest to largest Place each leaf at the same row as its stem The complete display is Stem Leaf 17 623 18 4705147 19 1983627571 20 038 21 12204 22 12 Conclusions from the stem and leaf display 0 The observations range from 172 to 222 0 Most of the observations fall between 180 and 200 0 The shape of the distribution is not symmetrical 0 Half the observations are below 195 and half above it Pie Charts Bar Charts Line Charts 0 The graphical presentations shown here are used primarily for qualitative data 0 These graphical tools are most appropriate when the raw data can be naturally categorized in a meaningful manner Pie Charts 0 Pie chart is a very popular tool used to represent the proportions of appearance for nominal data 0 The pie chart is a circle subdivided into a number of slices that represent the various categories 0 The size of each slice is proportional to the percentage corresponding to the category it represents Bar Charts 0 Bar charts provide an alternative to pie charts 0 The frequency or relative frequency of each category is represented by a vertical 0 Use bar charts also when the order in which qualitative data are presented is meaningful Line charts 0 Plot the frequency of a category above the point on the horizontal aXis representing that category 0 Use line charts when the categories are points in time References 1 Chase and Bown General Statistics 2 Hildebrand and Ott Statistical Thinking for Managers 3 Keller and Warrack Statistics for Management and Economics 4 McClave Benson and Sincich A First Course In Business Statistics 5 Conditional Probability and Independence Conditional Probability o The probability of an event when partial knowledge about the outcome of an experiment is known is called conditional probability 0 We use the notation PA l B for the conditional probability that event A occurs given that event B has occurred PA and B PAlB PB Example 1 The following data are characteristics of the votingage population regarding the 1992 presidential election in the United States Number of persons is measured in thousands Voted Did not vote Total Males 53312 35245 88557 Females 60554 36573 97127 Total 113866 71818 185684 For a randomly selected person from the population let A be the event that the person selected voted and Bbe the event that the person selected is a male Find each of the following 1 PA 2 PA 3 PAlB Solution 1 PA1138685684613 2 PA 1 PA 1 613 387 53312 3 PA BFW M wz 88557 PB A85684 Independent and Dependent Events 0 Two events A and B are said to be independent ifPAlBPA or PB l A PB Otherwise the events are dependent 0 Note that if the occurrence of one event does not change the likelihood of occurrence of the other event the two events are independent Note that independent events and mutually exclusive events are not the same Example 1 7 cont d Are eventsA and B independent Solution Since PA 613 at 602 PA B the events A and B are not independent Probability Rules Again Complement rule Each simple event must belong to either A or A Since the sum of the probabilities assigned to simple events is one we have for any event A PA 1 PA Addition rule For any two events A and B PA or B PA PB PA and B Multiplication rule For any two events A and B PA and B PAPB A PBPA B When A and B are independent PA and B PAPB Example 2 Let A and Bbe independent events with PA 3 and PB 4 What is PA or B Solution Let us use the addition law PA or B PA PB PA and B 3 4 PA and B Because of the independence of these events PA and B PAPB 3 X 4 12 Therefore PA or B 7 12 58 Probability Trees This is a useful device to build a sample space and to calculate probabilities of simple events and events Rules for constructing a probability tree 1 Events forming the rst set of branches must have known marginal probabilities must be mutually exclusive and should exhaust all possibilities so that the sum of branch probabilities is 1 2 Events forming the second set of branches must be entered at the tip of each of the sets of rst branches Conditional probabilities given the relevant rst branch must be entered unless assumed independence allows the use of unconditional probabilities Again the branches must be mutually exclusive and exhaustive 3 If there are additional sets of branches the probabilities must be conditional on all preceding events As always the branches must be mutually exclusive and exhaustive 4 The sum of path probabilities must be taken over all paths included in the relevant event Example 3 In a certain television game show a valuable prize is hidden behind one of the three doors You the contestant pick one of the three doors Before opening it the announcer opens one of the other two doors and you see that the prize isn t behind that door The announcer offers you the chance to switch the remaining door Should you switch or it does not matter Solution Call the door that you select A the others B and C Assuming that the prize is distributed randomly among the doors the probability that it s behind each of the doors is 13 If you picked a wrong door in A the announcer has no choice If B contains the prize the announcer must open C if C has the prize he must open B But if you picked correctly and A has the prize the announcer does have a choice Let us assume that the announcer picks B or C randomly each with probability 12 in this situation We can construct the following tree B 12 16 A 13 C 12 16 B 13 C 1 13 C 13 B 1 13 Door containing prize Announcer s choice Path probability Suppose that the announcer has chosen B and you chose A initially What is the probability that the prize behind C Pmehmd C chose B Pbehina C and chose B 13 Pchose B 1613 so Pbehina A l chose B 1 23 13 You have a better chance of winning if you switch to door C 23 Recommendedexercises 3867394 311073111 3113 311573116 311873121 References 1 Chase and Bown General Statistics Hildebrand and Ott Statistical Thinking for Managers Keller and Warrack Statistics for Management and Economics McClave Benson and Sincich A First Course In Business Statistics 59 Exercises 1 N E 4 Considered the tossing of two fair dice Consider the following events A sum is 7 or more B sum is even C sum is 7 andD sum is less than 11 a Verify that the only pair of mutually exclusive events isB C b Use the Addition Rule to nd PA or C c Let E sum is less than 4 and F 3 3 Find PA or E or F A card is to be randomly selected from an ordinary deck of 52 cards Consider the following events A ace B face card and C club a Verify that the only pair of mutually exclusive events is A B b Use the Addition Rule to find PA or B Assume that the mother is a carrier for colorblindness and the father is normal Assume also that when a parent contributes a gene from a gene pair either gene is equally likely to be contributed Let event A carries gene for colorblindness and event B is colorblind a Assign probabilities to each outcome b Are the events A B mutually exclusive c Find the probability that an offspring will either carry the colorblind gene or be colorblind The following data are characteristics of the votingage population regarding the 1992 presidential election in the United States Number of persons is measured in thousands A A Voted Did not vote B Males 53312 35245 E Females 60554 36573 For a randomly selected person from the population let A be the event that the person selected voted and B be the event that the person selected is a male Find each of the following 20133 bPA cPA and B There are 2000 voters in a town Consider the experiment of randomly selecting a voter to be interviewed The voters in the town are the possible outcomes of the experiment The event A consists of being in favor of more stringent building 0 gt1 9 0 codes the event B consists of having lived in the town less than 10 years The following table gives the numbers of voters in various categories A A B 100 700 E 1000 200 Find each of the following a PA b PB c PA and B Two cards are to be selected from an ordinary deck of 52 cards Assume the first card is replaced before the second card is selected Consider the following events A first card is an ace B second card is an ace C second card is a king a Find PA and B b Find PB or C c Now suppose a third card is selected after replacement of the first and second cards Let D third card is not an ace Find PA and B and D A sporting goods store has a large batch of cans of tennis balls on hand Ten percent of the cans are unacceptable that is contain at least one defective ball a A customer decides to purchase one can What is the probability that the customer will be satisfied b A customer is to purchase two cans Find the probability that 1 Both cans will be satisfactory ii Exactly one can will be satisfactory iii At least one can will be satisfactory Suppose that the probability that a child produced by a couple will have a particular disease is 1 If they plan to have four children what is the probability that one or more children will have the disease A large shipment contains 2 defective items Five items are to be selected What is the probability of getting one or more defective items Assume that we tossed two fair dice Consider the following events A sum is 7 or more B sum is even C sum is 7 and D sum is less than 11 Find a PA or B b PA or C c PA or D Two cards are to be selected from an ordinary deck of 52 cards Assume that the first card is not replaced before the second one is drawn Consider the following events A first card is an ace B second card is an ace a Find PA and B b Find PB Are the events A B independent c Find PA or B 12 If the mother is a carrier for colorblindness and the father is normal a Find the probability that a child will be colorblind if it is male b Find the probability that a child will be colorblind if it is female 13 A committee of seven consists of two males and five females Two members are to be chosen randomly to look into a specific problem What is the probability that both males will be chosen Hint Imagine the selection as a twostage processselect one member then another without replacement 14 A business employs 600 men and 400 women Five percent of the men and 10 of the women have been working there for more than 20 years If an employee is selected by chance what is the probability that the employee is male given that the length of employment is more tan 20 the years 15 A change was proposed in the mathematics curriculum at a college The mathematics majors were asked whether they approved of the proposed change The results of the survey follow Approved No opinion Did not approve Female 21 6 12 Male 14 10 7 Suppose that a mathematics major is selected by chance Find the probability that a The student is female given no opinion b The student approves of the proposed change given the student is male c The student is male given the student does not approve of the proposed change d The student is male and approves of the proposed change 5 Conditional Probability and Independence Conditional Probability o The probability of an event when partial knowledge about the outcome of an experiment is known is called conditional probability 0 We use the notation PA l B for the conditional probability that event A occurs given that event B has occurred PA and B PAlB PB Example 1 The following data are characteristics of the votingage population regarding the 1992 presidential election in the United States Number of persons is measured in thousands Voted Did not vote Total Males 53312 35245 88557 Females 60554 36573 97127 Total 113866 71818 185684 For a randomly selected person from the population let A be the event that the person selected voted and Bbe the event that the person selected is a male Find each of the following 1 PA 2 PA 3 PAlB Solution 1 PA1138685684613 2 PA 1 PA 1 613 387 53312 3 PA BFW M wz 88557 PB A85684 Independent and Dependent Events 0 Two events A and B are said to be independent ifPAlBPA or PB l A PB Otherwise the events are dependent 0 Note that if the occurrence of one event does not change the likelihood of occurrence of the other event the two events are independent Note that independent events and mutually exclusive events are not the same Example 1 7 cont d Are eventsA and B independent Solution Since PA 613 at 602 PA B the events A and B are not independent Probability Rules Again Complement rule Each simple event must belong to either A or A Since the sum of the probabilities assigned to simple events is one we have for any event A PA 1 PA Addition rule For any two events A and B PA or B PA PB PA and B Multiplication rule For any two events A and B PA and B PAPB A PBPA B When A and B are independent PA and B PAPB Example 2 Let A and Bbe independent events with PA 3 and PB 4 What is PA or B Solution Let us use the addition law PA or B PA PB PA and B 3 4 PA and B Because of the independence of these events PA and B PAPB 3 X 4 12 Therefore PA or B 7 12 58 Probability Trees This is a useful device to build a sample space and to calculate probabilities of simple events and events Rules for constructing a probability tree 1 Events forming the rst set of branches must have known marginal probabilities must be mutually exclusive and should exhaust all possibilities so that the sum of branch probabilities is 1 2 Events forming the second set of branches must be entered at the tip of each of the sets of rst branches Conditional probabilities given the relevant rst branch must be entered unless assumed independence allows the use of unconditional probabilities Again the branches must be mutually exclusive and exhaustive 3 If there are additional sets of branches the probabilities must be conditional on all preceding events As always the branches must be mutually exclusive and exhaustive 4 The sum of path probabilities must be taken over all paths included in the relevant event Example 3 In a certain television game show a valuable prize is hidden behind one of the three doors You the contestant pick one of the three doors Before opening it the announcer opens one of the other two doors and you see that the prize isn t behind that door The announcer offers you the chance to switch the remaining door Should you switch or it does not matter Solution Call the door that you select A the others B and C Assuming that the prize is distributed randomly among the doors the probability that it s behind each of the doors is 13 If you picked a wrong door in A the announcer has no choice If B contains the prize the announcer must open C if C has the prize he must open B But if you picked correctly and A has the prize the announcer does have a choice Let us assume that the announcer picks B or C randomly each with probability 12 in this situation We can construct the following tree B 12 16 A 13 C 12 16 B 13 C 1 13 C 13 B 1 13 Door containing prize Announcer s choice Path probability Suppose that the announcer has chosen B and you chose A initially What is the probability that the prize behind C Pmehmd C chose B Pbehina C and chose B 13 Pchose B 1613 so Pbehina A l chose B 1 23 13 You have a better chance of winning if you switch to door C 23 Recommendedexercises 3867394 311073111 3113 311573116 311873121 References 1 Chase and Bown General Statistics Hildebrand and Ott Statistical Thinking for Managers Keller and Warrack Statistics for Management and Economics McClave Benson and Sincich A First Course In Business Statistics 59 Exercises 1 N E 4 Considered the tossing of two fair dice Consider the following events A sum is 7 or more B sum is even C sum is 7 andD sum is less than 11 a Verify that the only pair of mutually exclusive events isB C b Use the Addition Rule to nd PA or C c Let E sum is less than 4 and F 3 3 Find PA or E or F A card is to be randomly selected from an ordinary deck of 52 cards Consider the following events A ace B face card and C club a Verify that the only pair of mutually exclusive events is A B b Use the Addition Rule to find PA or B Assume that the mother is a carrier for colorblindness and the father is normal Assume also that when a parent contributes a gene from a gene pair either gene is equally likely to be contributed Let event A carries gene for colorblindness and event B is colorblind a Assign probabilities to each outcome b Are the events A B mutually exclusive c Find the probability that an offspring will either carry the colorblind gene or be colorblind The following data are characteristics of the votingage population regarding the 1992 presidential election in the United States Number of persons is measured in thousands A A Voted Did not vote B Males 53312 35245 E Females 60554 36573 For a randomly selected person from the population let A be the event that the person selected voted and B be the event that the person selected is a male Find each of the following 20133 bPA cPA and B There are 2000 voters in a town Consider the experiment of randomly selecting a voter to be interviewed The voters in the town are the possible outcomes of the experiment The event A consists of being in favor of more stringent building 0 gt1 9 0 codes the event B consists of having lived in the town less than 10 years The following table gives the numbers of voters in various categories A A B 100 700 E 1000 200 Find each of the following a PA b PB c PA and B Two cards are to be selected from an ordinary deck of 52 cards Assume the first card is replaced before the second card is selected Consider the following events A first card is an ace B second card is an ace C second card is a king a Find PA and B b Find PB or C c Now suppose a third card is selected after replacement of the first and second cards Let D third card is not an ace Find PA and B and D A sporting goods store has a large batch of cans of tennis balls on hand Ten percent of the cans are unacceptable that is contain at least one defective ball a A customer decides to purchase one can What is the probability that the customer will be satisfied b A customer is to purchase two cans Find the probability that 1 Both cans will be satisfactory ii Exactly one can will be satisfactory iii At least one can will be satisfactory Suppose that the probability that a child produced by a couple will have a particular disease is 1 If they plan to have four children what is the probability that one or more children will have the disease A large shipment contains 2 defective items Five items are to be selected What is the probability of getting one or more defective items Assume that we tossed two fair dice Consider the following events A sum is 7 or more B sum is even C sum is 7 and D sum is less than 11 Find a PA or B b PA or C c PA or D Two cards are to be selected from an ordinary deck of 52 cards Assume that the first card is not replaced before the second one is drawn Consider the following events A first card is an ace B second card is an ace a Find PA and B b Find PB Are the events A B independent c Find PA or B 12 If the mother is a carrier for colorblindness and the father is normal a Find the probability that a child will be colorblind if it is male b Find the probability that a child will be colorblind if it is female 13 A committee of seven consists of two males and five females Two members are to be chosen randomly to look into a specific problem What is the probability that both males will be chosen Hint Imagine the selection as a twostage processselect one member then another without replacement 14 A business employs 600 men and 400 women Five percent of the men and 10 of the women have been working there for more than 20 years If an employee is selected by chance what is the probability that the employee is male given that the length of employment is more tan 20 the years 15 A change was proposed in the mathematics curriculum at a college The mathematics majors were asked whether they approved of the proposed change The results of the survey follow Approved No opinion Did not approve Female 21 6 12 Male 14 10 7 Suppose that a mathematics major is selected by chance Find the probability that a The student is female given no opinion b The student approves of the proposed change given the student is male c The student is male given the student does not approve of the proposed change d The student is male and approves of the proposed change 11 Confidence Interval for a Population Proportion Sampling Distribution of f7 0 f7 where n is number of trials andX is the total number of successes is an unbiased consistent estimator of p X has the binomial distribution bn p A A l Ep p and Vamp g for large 71 when 7431 f7 is greater than 5 by the Central Limit Theorem p 7 is approximately normal N 01 x p1 p n Large Sample Con dence Interval for p i231 f7n Example 1 A retail lumberyard routinely inspects incoming shipments of lumber from suppliers For select grade 8foot 2by4 pine shipments the lumberyard supervisor chooses one gross 144 boards randomly from a shipment of several tens of thousands of boards In the sample 18 boards are not salable as select grade Calculate a 95 CI for the proportion in the entire shipment that is not salable as select grade Solution The sample proportionf is equal to 18144125 Since nfa1 f7 144 X 125 X 875 1575 gt 5 the con dence interval is given by pizyd l f7n 125i196gtlt 125i054 2 Small Sample Con dence Interval for p If the sample size is small or p is close to 0 or 1 statisticians used adjusted con dence N 1171 piz n4 interval where g X 2n4 Sample Size Determination The required sample size to produce an interval estimator p i W with l a con dence zwpa p 2 n T is Since the value of p is unknown it can be estimated by using the sample proportion from a prior sample or we can use the conservative choice of p 5 in which case Example 2 A manufacturer of boxes of candy is concerned about the proportion of imperfect boxes 7 those containing cracked broken or otherwise unappetizing candies How large a sample is needed to get 95 CI for this proportion with a width no greater than 02 Solution Since the value of p is unknown we will use the conservative substitution 2 21 2 n 2 A 13996 m 9604 2W 2 X 01 Exercises p 319 5417543 p 320 5467550 p 326 5637564 References 1 Chase and Bown General Statistics 2 Hildebrand and Ott Statistical Thinking for Managers 3 Keller and Warrack Statistics for Management anal Economics 4 McClave Benson and Sincich A First Course In Business Statistics Exercises 1 The Aid to Families with Dependent Children AFDC program has an overall error rate of 4 in determining eligibility The state of California uses sampling to monitor its counties to see whether they exceed the 4 error rate which can result in economic sanctions In one county 9 cases out of 150 were found to be in error a Find a 95 con dence interval for the error rate for the county proportion of all cases in error b In 1982 the California legislature mandated that a 95 confidence interval be used in studying error rate Based on your answer in part a would you conclude that the error rate for the county is above the 4 rate 2 To estimate the proportion p of passengers who had purchased tickets for more than 400 over a year39s time an airline official obtained a random sample of 75 The number of those purchasing tickets for more than 400 was 45 a What is a point estimate for p b Find a 90 confidence interval forp 3 A city council commissioned a statistician to estimate the proportion p of voters in favor of a proposal to build a new library The statistician obtained a random sample of 200 voters with 112 indicating approval of the proposal a What is a point estimate for p b Find a 90 confidence interval forp 4 A union official wanted to get an idea of whether a majority of workers at a large corporation would favor a contract proposal She surveyed 500 workers and found that 260 favored the proposal a Find a 95 confidence interval for the proportion of all the workers who favor the contract proposal b Find the maximum error of the estimate c Based on the results of part a can we conclude that the contract will be ratified by the membership 5 Candidates A and B are opponents for political office Candidate A39s pollster conducts a poll one week before the election and finds that 165 of 300 potential voters say they will vote for candidate A At the time of the poll can we be a 80 confident that candidate A will win b 98 confident that candidate A will win 6 A town official wants to estimate the proportion p of voters who favor the granting of a variance so that a builder can construct a health spa in a residential area How large a random sample is required to estimate p to within 4 percentage points with 95 confidence 7 To estimate the proportion p of voters favoring a nuclear freeze in your voting district how large a random sample is needed to estimate p to within 2 percentage points with a 90 confidence b 99 confidence 11 Con dence Interval for a Population Proportion Sampling Distribution of f7 0 f7 where 7 is number of trials and X is the total number of successes is an unbiased consistent estimator of P X has the binomial distribution Mquot P EP P and VarP y for large 7 when 7430 f7 is greater than 5 by the Central Limit Theorem p 7p1 p m is approximately normal N 01 Large Sample Con dence Interval for P t z l l f7n a 2 Example 1 A retail lumberyard routinely inspects incoming shipments of lumber from suppliers For select grade 8foot 2by4 pine shipments the lumberyard supervisor chooses one gross 144 boards randomly from a shipment of several tens of thousands of boards In the sample 18 boards are not salable as select grade Calculate a 95 CI for the proportion in the entire shipment that is not salable as select grade Solution The sample proportionf7 is equal to 18144 125 Since n l f7 144gtlt 125X 875 1575 gt 5 the con dence interval is given by fat zaJ jaw 125 196gtlt 1251 054 2 Small Sample Con dence Interval for P If the sample size is small or P is close to 0 or 1 statisticians used adjusted con dence interval N l 15 i z 7 p 1 71 4 Sample Size Determination The required sample size to produce an interval estimator P i W with l a con dence is 2 zwp p W Since the value of P is unknown it can be estimated by using the sample proportion from a prior sample or we can use the conservative choice of p 5 in which case Z 2 LA 2W Example 2 A manufacturer of boxes of candy is concerned about the proportion of imperfect boxes 7 those containing cracked broken or otherwise unappetizing candies How large a sample is needed to get 95 CI for this proportion with a width no greater than 02 Solution Since the value of P is unknown we will use the conservative substitution 2 2a 2 n2 A 13996 J z 9604 2W 2X 01 Exercises p 319 5417543 p 320 5467550 p 326 5637564 References 1 Chase and Bown General Statistics 2 Hildebrand and Ott Statistical Thinking forM anagers 3 Keller and Warrack Statistics for Management anal Economics 4 McClave Benson and Sincich A First Course In Business Statistics Exercises 1 The Aid to Families with Dependent Children AFDC program has an overall error rate of 4 in determining eligibility The state of California uses sampling to monitor its counties to see whether they exceed the 4 error rate which can result in economic sanctions In one county 9 cases out of 150 were found to be in error a Find a 95 confidence interval for the error rate for the county proportion of all cases in error b In 1982 the California legislature mandated that a 95 con dence interval be used in studying error rate Based on your answer in part a would you conclude that the error rate for the county is above the 4 rate 2 To estimate the proportion p of passengers who had purchased tickets for more than 400 over a year39s time an airline of cial obtained a random sample of 75 The number of those purchasing tickets for more than 400 was 45 a What is a point estimate for p b Find a 90 con dence interval for p 3 A city council commissioned a statistician to estimate the proportion p of voters in favor of a proposal to build a new library The statistician obtained a random sample of 200 voters with 112 indicating approval of the proposal a What is a point estimate for p b Find a 90 con dence interval for p 4 A union of cial wanted to get an idea of whether a majority of workers at a large corporation would favor a contract proposal She surveyed 500 workers and found that 260 favored the proposal a Find a 95 con dence interval for the proportion of all the workers who favor the contract proposal b Find the maximum error of the estimate c Based on the results of part a can we conclude that the contract will be ratified by the membership 5 Candidates A and B are opponents for political of ce Candidate A39s pollster conducts a poll one week before the election and nds that 165 of 300 potential voters say they will vote for candidate A At the time of the poll can we be a 80 con dent that candidate A will win b 98 con dent that candidate A will win 6 A town of cial wants to estimate the proportion p of voters who favor the granting of a variance so that a builder can construct a health spa in a residential area How large a random sample is required to estimate p to within 4 percentage points with 95 con dence 7 To estimate the proportion p of voters favoring a nuclear freeze in your voting district how large a random sample is needed to estimate p to within 2 percentage points with a 90 con dence b 99 con dence 11 Confidence Interval for a Population Proportion Sampling Distribution of f7 0 f7 where n is number of trials andX is the total number of successes is an unbiased consistent estimator of p X has the binomial distribution bn p A A l Ep p and Vamp g for large 71 when 7431 f7 is greater than 5 by the Central Limit Theorem p 7 is approximately normal N 01 x p1 p n Large Sample Con dence Interval for p i231 f7n Example 1 A retail lumberyard routinely inspects incoming shipments of lumber from suppliers For select grade 8foot 2by4 pine shipments the lumberyard supervisor chooses one gross 144 boards randomly from a shipment of several tens of thousands of boards In the sample 18 boards are not salable as select grade Calculate a 95 CI for the proportion in the entire shipment that is not salable as select grade Solution The sample proportionf is equal to 18144125 Since nfa1 f7 144 X 125 X 875 1575 gt 5 the con dence interval is given by pizyd l f7n 125i196gtlt 125i054 2 Small Sample Con dence Interval for p If the sample size is small or p is close to 0 or 1 statisticians used adjusted con dence N 1171 piz n4 interval where g X 2n4 Sample Size Determination The required sample size to produce an interval estimator p i W with l a con dence zwpa p 2 n T is Since the value of p is unknown it can be estimated by using the sample proportion from a prior sample or we can use the conservative choice of p 5 in which case Example 2 A manufacturer of boxes of candy is concerned about the proportion of imperfect boxes 7 those containing cracked broken or otherwise unappetizing candies How large a sample is needed to get 95 CI for this proportion with a width no greater than 02 Solution Since the value of p is unknown we will use the conservative substitution 2 21 2 n 2 A 13996 m 9604 2W 2 X 01 Exercises p 319 5417543 p 320 5467550 p 326 5637564 References 1 Chase and Bown General Statistics 2 Hildebrand and Ott Statistical Thinking for Managers 3 Keller and Warrack Statistics for Management anal Economics 4 McClave Benson and Sincich A First Course In Business Statistics Exercises 1 The Aid to Families with Dependent Children AFDC program has an overall error rate of 4 in determining eligibility The state of California uses sampling to monitor its counties to see whether they exceed the 4 error rate which can result in economic sanctions In one county 9 cases out of 150 were found to be in error a Find a 95 con dence interval for the error rate for the county proportion of all cases in error b In 1982 the California legislature mandated that a 95 confidence interval be used in studying error rate Based on your answer in part a would you conclude that the error rate for the county is above the 4 rate 2 To estimate the proportion p of passengers who had purchased tickets for more than 400 over a year39s time an airline official obtained a random sample of 75 The number of those purchasing tickets for more than 400 was 45 a What is a point estimate for p b Find a 90 confidence interval forp 3 A city council commissioned a statistician to estimate the proportion p of voters in favor of a proposal to build a new library The statistician obtained a random sample of 200 voters with 112 indicating approval of the proposal a What is a point estimate for p b Find a 90 confidence interval forp 4 A union official wanted to get an idea of whether a majority of workers at a large corporation would favor a contract proposal She surveyed 500 workers and found that 260 favored the proposal a Find a 95 confidence interval for the proportion of all the workers who favor the contract proposal b Find the maximum error of the estimate c Based on the results of part a can we conclude that the contract will be ratified by the membership 5 Candidates A and B are opponents for political office Candidate A39s pollster conducts a poll one week before the election and finds that 165 of 300 potential voters say they will vote for candidate A At the time of the poll can we be a 80 confident that candidate A will win b 98 confident that candidate A will win 6 A town official wants to estimate the proportion p of voters who favor the granting of a variance so that a builder can construct a health spa in a residential area How large a random sample is required to estimate p to within 4 percentage points with 95 confidence 7 To estimate the proportion p of voters favoring a nuclear freeze in your voting district how large a random sample is needed to estimate p to within 2 percentage points with a 90 confidence b 99 confidence

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