Statistics Questions and Answers

11 Consider the system 3 in Example 1 of the text Recall that this system has an asymptot ically stable critical point at 0 5 0 5 corresponding to the stable coexistence of the two population species Now suppose that immigration or emigration occurs at the constant rates of 8a and b for the species x and y respectively In this case Eqs 3 are replaced by dx dt x 1 x y a i dy dt y 0 75 y 0 5x Sb The question is what effect this has on the location of the stable equilibrium point a To find the new critical point we must solve the equations x 1 x y da 0 y 0 75 y 0 5x 8b 0 One way to proceed is to assume that x and y are given by power series in the parameter 5 thus iii 11 x x0 x 8 y yo y o Substitute Eqs iii into Eqs ii and collect terms according to powers of d b From the constant terms the terms not involving 8 show that xo 0 5 and yo 0 5 thus confirming that in the absence of immigration or emigration the critical point is 0 5 0 5 c From the terms that are linear in 8 show that x 4a 4b y 2a 4b iv d Suppose that a 0 and b 0 so that immigration occurs for both species Show that the resulting equilibrium solution may represent an increase in both populations or an increase in one but a decrease in the other Explain intuitively why this is a reasonable result

Criterion for the Galois Group of an Irreducible Cubic over an Arbitrary Field Suppose K is a field and f x x ax bx c K x is irreducible so the Galois group of f x over K is either S3 or A3 a Show that the Galois group of f x is A3 if and only if the resultant quadratic poly nomial g x x ab 3c x b a c 6abc 9c has a root in K If a B y are the roots of f x show that the Galois group is A3 if and only if the element 0 a By ya is an element of K and that is a root of g x Show that the discriminant of g x is the same as the discriminant of f x b ch K 2 If K has characteristic different from 2 show either from a or directly from the definition of the discriminant that the Galois group of f x is A3 if and only if the discriminant of f x is a square in K c ch K 2 If K has characteristic 2 show that the discriminant of f x is always a square Show that f x can be taken to be of the form x px q and that the Galois group of f x is A3 if and only if the quadratic x qx p q has a root in K equivalently if p3 q2 q2 K is in the image of the Artin Schreier map xx x mapping K to K d If K F 1 where is transcendental over F Prove that the polynomials x 1 x 1 x 1 1 1 x 1 1 and x3 1 1 x 1 1 1 have A3 as Galois group while x3 x t and x x t have S3 as Galois group

d Joachim a third student noted that the confidence interval in part a gives plausible values of the parameter as an interval between two values He suggested that they develop a one sided confidence interval because they were only concerned with whether the number 6 was landing face up more often than expected not less often The one sided interval will determine a value I such that all plausible values of pare greater than L The formula for L is L p z V p 1 p n 1 Determine the values of z needed to create the one sided 95 percent confidence interval Then calculate the value of L ii Do the results of Joachim s one sided confidence interval agree with results of Aurelia s significance test in part b Explain your reasoning

c Two standard normal curves are shown below one for the confidence interval calculated in part a and one for the significance test conducted in part b 1 For the confidence interval curve label the critical values for the 95 confidence level and shade the area that represents values in the outer 5 ii For the significance test curve label the critical value for the 5 significance level and shade the area representing the values of z that would lead to a rejection of the null hypothesis in part b Confidence Interval Hypothesis Test

Julie knows that the adult population gets on average eight hours of sleep each night A hypothesis test can help her see if college students are different from the adult population Julie tabulated that her sample of 101 students got an average of 7 1 hours of sleep each night with a standard deviation of 2 48 Using the data provided and the formula below what is the t statistic that Julie calculates Answer choices are rounded to the hundredths place x S n a 2 61 b 6 78 c 0 24 d 3 64

What happens to a t distribution as the degrees of freedom increase A B C D E The center increases and the area in the tails increases The center increases and the area in the tails decreases The center increases and the area in the tails remains constant The center remains constant and the area in the tails increases The center remains constant and the area in the tails decreases

C explanatioNIS A recent survey collected information on television viewing habits from a random sample of 1 000 people in the United States Of those sampled 37 percent indicated that their favorite sport to watch on television was American tootball a Construct and interpret a 95 percent confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football Use the 4 step process state plan do check procedures conclude b Based on your answer to part a is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football Justify your answer

A fair die with its faces numbered from 1 to 6 is one in which each number is equally likely to land face up when the die is rolled On a fair die the probability that the number 6 will land face up is A group of students wanted to investigate a claim about manipulating a fair die so that it favors one outcome The claim states that if a fair die is put into an oven and baked at 200 F for 10 minutes the inside of the die will begin to melt When the die cools the inside will be solid again but with more weight toward the bottom This shift in weight will cause the face that was up when the die cooled to land up more often that the other faces The students obtained a fair die and baked it according to the preceding directions The die cooled with the number 6 face up After the die cooled they rolled the die 200 times and the number 6 landed face up 43 times Let p represent the population proportion of times the number 6 will land face up on the baked die if the die could be rolled an infinite number of times a Clarke one of the students constructed a 95 percent confidence interval for p as 0 215 0 057 Does the interval provide convincing statistical evidence that the number 6 will land face up more often on the baked die than on a fair die Explain your reasoning BIU X X C2E E BI U X X 3 C 2 a b Aurelia another student suggested they conduct a significance test to investigate the claim She tested the hypotheses Ho p versus H p at the significance level of a 0 05 She obtained a test statistic of 1 83 with a p value of 0 033 Do the results of the significance test agree with the results of Clarke s confidence interval in part a Explain your reasoning O E Pa 0 10000 Word Limit

Jackson sampled 101 students and calculated an average of 6 5 hours of sleep each night with a standard deviation of 2 14 Using a 95 confidence level he also found that t 1 984 confidence interval X t s n A 95 confidence interval calculates that the average number of hours of sleep for working college students is between hours Answer choices are rounded to the hundredths place a 6 15 and 6 94 O b 6 46 and 6 54 c 6 08 and 6 92 d1646 and 6 92

A consulting firm records its employees income against the number of hours worked in the scatterplot shown below Income in dollars 1000 900 800 700 600 500 400 300 200 100 O y 5 10 Employee s Income vs Hours Worked 15 Using the best fit line which of the following predictions is TRUE 20 Hours of Work a An employee would earn 730 if they work for 27 hours on a project b An employee would earn 310 if they work for 7 hours on a project c An employee would earn about 470 if they work for 15 hours on a project 25 30 35 X

If n 490 and p p hat 0 79 find the margin of error at a 95 confidence level Give your answer to three decimals

Which of the following data sets would most likely have a positive association and a correlation coefficient between 0 and 1 a Speed of Frank s vehicle Time it takes Frank to arrive at destination b Number of school absences Test scores in school c Level of education Head circumference d Number of months Lela owns a car Number of scratches on Lela s car

The average number of bridge construction projects that take place at any one time in a certain state is 7 ke k P X k k is the given number of event occurrences A is the average rate of event occurrences Using the Poisson distribution formula what is the probability of exactly 4 bridge construction projects taking place at one time in this state Answer choices are rounded to the hundredths place a 0 19 O b 0 05 c 0 06 d 0 09

Theresa really loves orange flavored Fruity Tooty candies but there always seems to be a lot of cherry flavored candies in each bag To determine whether this is because cherry candies are so popular or because each bag contains fewer orange candies Theresa randomly picks a Fruity Tooty candy from her bag records its flavor and places it back in the bag Each bag contains a mixture of cherry grape apple lemon and orange flavors Which statement about Theresa s distribution of sample proportions is true a The distribution of lemon candy can be modeled as normal if Theresa picks candies 12 times b The sample proportion of cherry candies has a binomial distribution c The distribution of the count of picking an apple candy cannot be modeled as approximately normal if Theresa picks candies over 300 times O d The count of drawing an orange candy is not a binomial distribution

An article claims that 12 of trees are infested by a bark beetle A random sample of 1 00 trees were tested for traces of the infestation and found that 127 trees were affected Z p pq n Using the formula and data provided what is the value of the z test statistic Answer choices are rounded to the hundredths place a 0 25 b 0 68 O c 0 50 d 0 35

5 4 Unexpected expense In a random sample 765 adults in the United States 322 say they could not cover a 400 unexpected expense without borrowing money or going into debt a What population is under consideration in the data set b What parameter is being estimated c What is the point estimate for the parameter d What is the name of the statistic we use to measure the uncertainty of the point estimate e Compute the value from part d for this context f A cable news pundit thinks the value is actually 50 Should she be surprised by the data g Suppose the true population value was found to be 40 If we use this proportion to recompute the value in part e using p 0 4 instead of p does the resulting value change much

O a The critical z score for a right tailed test at a 17 significance level is 0 57 Ob The critical z score for a left tailed test at a 25 significance level is 0 40 Oc The critical z score for a two sided test at a 3 significance level is 2 17 O d The critical z score for a two sided test at a 5 significance level is 1 65

Jeanette really loves apple flavored Fruity Tooty candies but there always seems to be a lot of cherry flavored candies in each bag To determine whether this is because cherry candies are so popular or because each bag contains fewer apple candies Jeanette randomly picks a Fruity Tooty candy from her bag records its flavor and places it back in the bag Each bag contains a mixture of cherry grape apple lemon and orange flavors Which statement about Jeanette s distribution of sample proportions is true a The distribution of the count of picking a lemon candy can be modeled as approximately normal if Jeanette picks candies 15 times O b The count of drawing an apple candy is not a binomial distribution c The distribution of the count of picking a cherry candy cannot be modeled as approximately normal if Jeanette picks candies over 100 times O d The count of drawing an orange candy has a binomial distribution

5 26 Which is higher In each part below there is a value of interest and two scenarios I and II For each part report if the value of interest is larger under scenario I scenario II or whether the value is equal under the scenarios a The standard error of p when I n 125 or II n 500 b The margin of error of a confidence interval when the confidence level is 1 90 or II 80 c The p value for a Z statistic of 2 5 calculated based on a I sample with n 500 or based on a II sample with n 1000 d The probability of making a Type 2 Error when the alternative hypothesis is true and the significance level is I 0 05 or II 0 10

A school counselor claims that the average number of sleep students get each night is 6 hours A researcher has taken a well designed survey and his sample mean is 8 5 hours and sample standard deviation is 0 25 The sample size is 400 Which statement is correct O a The sample size should be much more O b The result of the survey is not statistically significant Oc The difference exists due to chance since the test statistic is small d The result of the survey is statistically significant

Jackson wants to test an assumption on the number of red candies in a bag of Fruity Toot candy The company claims that five flavors are given an equal distribution in each bag meaning that the proportion of each color should be 20 Jackson decides to focus his efforts on examining whether there is a difference in the proportion of red candies than what the company claims What should Jackson state for the null hypothesis and alternative hypothesis a Null hypothesis p 0 20 Alternative hypothesis p 0 20 b Null hypothesis p 0 20 Alternative hypothesis p 0 20 c Null hypothesis p 0 20 Alternative hypothesis p 0 20 d Null hypothesis p 0 20 Alternative hypothesis p 0 20

5 22 Getting enough sleep 400 students were randomly sampled from a large university and 289 said they did not get enough sleep Conduct a hypothesis test to check whether this represents a statistically significant difference from 50 and use a significance level of 0 01

Find the confidence interval specified Assume that the population is normally distributed The football coach randomly selected ten players and timed how long each player took to perform a certain drill The times in minutes were 6 4 12 8 11 4 8 0 7 4 12 8 7 0 7 1 10 2 5 8 Determine a 95 confidence interval for the mean time for all players O 10 90 to 6 88 minutes O 10 80 to 6 98 minutes O 6 88 to 10 90 minutes 10 90 X S

What value of z divides the standard normal distribu decimal places

A random sample of 861 births in a state included 421 boys Construct a 95 confidence interval estimate of the proportion of boys in all births It is believed that among all births the proportion of boys is 0 506 Do these sample results provide strong evidence against that belief

There are 84 students in a history class The instructo Academ Fresh Sophor Juni Seni

The diameters of ball bearings are distribu

The following is a stem leaf plot of sample temperatures in New York throughout the year Stem Leaf 4679 4 5 6 7 2256 8 148 9 10 6 34688 a List out all the data values from this dataset separated by commas b What is the 5 number summary of this dataset c Find the sample mean and sample standard deviation of this dataset mean rounded to nearest tenths place standard deviation rounded to nearest hundredths place d What is the approximate shape of the data e Can we use the empirical rule for this type of dataset Why or why not

An honor society on campus only accepts the top 4 of students according to GPA If the GPAs of all students on campus are known to be a normally distributed with a mean of 2 99 and a standard deviation of 0 51 determine the minimum GPA a student must achieve in order to be admitted to the honor society Round the solution to two decimal places if necessary A minimum GPA of is required for admission to the honor society

A newspaper company classifies its customers by ge sample of 1736 customers The data is summarized So

The diameters of ball bearings are distribut diameter of a selected bearing is between 8

A single serving coffee machine is programmed to dispense 12 ounces of coffee per serving Occasionally the machine will start to overfill or underfill cups of coffee and require recalibration According to the company the coffee machine requires recalibration if it underfills or overfills cups of coffee by 1 5 or more ounces Determine the probability that a randomly selected cup of coffee was underfilled or overfilled by more than 1 5 ounces Assume the distribution of the amounts of all coffee dispensed by the machine are normally distributed with a mean of 12 ounces and a standard deviation of 1 8 ounces The probability that a randomly selected cup of coffee is underfilled or overfilled by 1 5 ounces is

Given a standard normal distribution determine the following probability Round the solution to four decimal places if necessary P 12 68 z 0 88

The speeds of all pitches thrown by the IRSC softball team are normally distributed with a mean of 68 25 miles per hour and a standard deviation of 4 miles per hour Determine the probability that a randomly selected pitch has a speed of more than 78 miles per hour Round the solution to four decimal places if necessary P x 78

Assume X is a normally distributed continuous variable with 121 and o 13 Determine the following probability Round the solution to four decimal places if necessary P x 120

Find the area under the standard normal curve within one standard deviation of the mean Round the solution to four decimal places if necessary Area

Find the area under the standard normal curve to the right of z 1 39 Round the solution to four decimal places if necessary Area

Classify the following data Indicate whether the data is qualitative or q indicate the level of measurement for the data A company s international marketing group asked the following qu Americ Answer 5 Points 1 Below Average 2 Average Are these data qu

Determine whether the statement Answer 5 Points T

Calculate the sample mean and sample variance for the following frequenc necessary round to one more decimal place than the largest number of de Hourly Wages Class 6 51 8 50 8 51 10 50 10 51 12 50 12 51 14 50 14 51 16 50

Question 19 2 points The tort of invasion of privacy

In a student s calculus class the final grade is based on four categories The categories grades in percentages and weights are shown in this table Category Grade Weight Tests 91 23 Labs 63 5 Assignments 65 17 Final Exam 77 55 Compute a weighted average to determine the student s overall final grade in the course Convert the weig percentages to decimals before multiplying them times the grades Record the grade below as a percentage Round accurately to two decimal places Overall Grade

Annual Home Expenses ater Sewer Charges ectricity one Service ater Heater Replacement ndow Repairs w Furnace AC ernet Phone Bundle tter Replacements w Lawn Mower cal 352 66 1 194 00 655 92 590 34 580 10 6 458 90 1 198 80 2 560 00 579 20 7 David and Helen Voss have a combined monthly gross income of 5 750 00 Their records show that for last year they paid 10 789 20 in mortgage payments 581 00 for insurance premiums and 2 685 00 in real estate taxes In addition they had the annual home expenses listed to the left What was their average monthly housing cost for last year Was it within the 35 recommendation

Calculate x bar for the data shown to two decimal places X 9 25 9 12 5 28 8 27 7 15 7 4 4 2 1 18 2 14 8

This graph shows the maximum towing capacity in pounds of five full size pickup trucks 17000 16800 16600 16400 16200 L 16000 A B C D E Q Which of the following is the best indication of what is misleading about the graph shown Towing Capacity Truck B has a bar that is too short Truck C appears to have a much higher capacity Truck E should have been made shorter There is nothing misleading about this graph Which of the following is the best indication as to why the graph is misleading The bottom of the graph has been cut off The bars were made longer The axes are mislabeled There is nothing misleading about this graph

Two dice are rolled and the faces are added together Give your answer as a decimal out to at least 4 places a What is the probability of getting a sum of eight b What is the probability of getting doubles c What is the probability of getting a sum more than ten

AD of 34 v4 4 20 4 2o 29 24 24 4 29 34 34 IMP Me 2 A Club or Heart 34 ew 44 CA of so e 4 4 CH 54 54 59 of 20 above As shown a classic deck of cards is made up of 52 cards 26 are black 26 are red Each color is split into two suits of 13 cards each clubs and spades are black and hearts and diamonds are red Each suit is split into 13 individual cards Ace 2 10 Jack Queen and King If you select a card at random what is the probability of getting 1 A n 2 of Spade s 3 A number smaller than 9 counting the ace as a 1 I

Here is a dataset containing 11 GPAs of students at a university answers can be rounded to the nearest hundredths place 2 45 3 33 3 49 2 0 3 45 4 0 2 96 2 77 3 89 3 0 3 1 a What are the mean median and mode b Is this discrete or continuous data c Which is an appropriate graphical display for this type of data stem leaf plot or a pie graph d Create a histogram for this dataset and describe the shape of the data as best as possible you do not need to copy or draw a histogram in your answer just the shape description

Co What is the margin or erro this interval in the format 14 31 Deer mice continued Exerc interval for the mean body length habitat a What is the standard devia mean body lengths b What critical value was use

Complete the following diagram by choosing the best answer choice below Tariff O Trade barriers Import Quota O Protective measures Economic Sanctions Embargo