Statistics Questions

17 A 100 kg uniform rectangular plate is supported in the position showr by hinges A and B and by cable DCE that passes over a frictionless hook at C Assuming that the tension is the same in both parts of the cable determine a the tension in the cable b the reactions at A and B Assume that the hinge at B does not exert any axial thrust Z y D 450 mm 90 mm 690 mm A C 960 mm 270 mm 90 mm B E 675 mm X

c Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance To investigate this issue a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months At the end of the study 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults Nine of these variables were significantly better in the sense of statistical significance at the a 0 05 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole and one variable was significantly better at the a 0 01 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole It would be correct to conclude O A that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the a 0 01 level We should be somewhat cautious about making claims for the variables that were significant at the a 0 05 level OB that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance C that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured OD none of the above

You and your friend are engaging in a classic sports argument which famous athlete is better at their respective sport In other words we re going to explore the question in the year 2022 was Jimmy Butler better at basketball than Tom Brady was at football Because we re comparing to totally different sports with different statistics in order to answer this question we need to look at standardized scores z scores For Tom Brady we ll look at passing yards Passing yards are a good way to measure a quarterback s ability to move the ball in short higher average passing yards per game is a good thing For Jimmy Butler we can look at points per game In the 2022 season Tom Brady had an average of 276 1 passing yards per game The total league average for the quarterbacks in the NFL was 226 19 with a standard deviation of 37 80 Using this information tell me who was better Tom Brady or Jimmy Butler and why In your answer include the following information a The z scores for Tom Brady s passing yards and Jimmy Butler s points per game Hint you already calculated the z score for Jimmy Butler in the questions above b A comparison of the two z scores an explanation of your conclusion based on those z scores What is your final conclusion who is better relative to others in their respective sports Why

b The nicotine content in cigarettes of a certain brand is normally distributed with mean in milligrams and standard deviation 0 1 The brand advertises that the mean nicotine content of their cigarettes is 1 5 but you believe that the mean nicotine content is actually higher than advertised To explore this you test the hypotheses Hou 1 5 H 1 5 and you obtain a P value of 0 052 Which of the following is true A You have failed to obtain any evidence for Ha B There is some evidence against Ho and a study using a larger sample size may be worthwhile 0 C This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the a 0 05 significance level D At the a 0 05 significance level you have proven that Ho is true

The city of Raleigh has 11 800 registered voters There are two candidates for City Council in an upcoming election Brown and Feliz The day before the election a telephone poll of 500 randomly selected registered voters was conducted 181 said they d vote for Brown 282 said they d vote for Feliz and 37 were undecided a Give the sample statistic for the percentage of voters surveyed who said they would vote for Brown Round your answer to the nearest tenth of a percent 32 2 Preview 32 2 b Using the sample statistic from above how many of the 11 800 registered voters might we expect to vote for Brown Round your answer to the nearest whole number

If you randomly select an NBA player what percentage of the time would the player s points per game for the season be less than Tyler Herro s To answer this question you need to use the table in the appendix of the text to figure out the area below this score Show your work on this To get full credit you must demonstrate all calculations values you used from the appendix and steps you took to get your answer that is your answer must include the z score the area under the curve that you found in the appendix and how you came to your final answer

If you randomly select a professional basketball player what is the probability that the player s points per game for the season is between Cody Zeller and Udonis Haslim To answer this question you need to use the table in the appendix of the text to figure out the area between those two scores on the normal curve Show your work on this To get full credit you must demonstrate all calculations values you used from the appendix and steps you took to get your answer that is your answer must include the z score the area under the curve that you found in the appendix and how you came to your final answer

Problem 3 Let X the time between two successive arrivals at the drive up window of a local bank If X has an exponential distribution with X 1 which is identical to a standard gamma distribution with x 1 compute the following 1 The expected time between two successive arrivals 2 The standard deviation of the time between successive arrivals Blank 1 Blank 2

Problem 2 The error involved in making a certain measurement is a continuous rv X with the following pdf 0 0 f x x 2 x 2 otherwise 0 09375 4 x 0 Compute P X 0 5 or X 0 5 Round your answer to four decimal places

Problem 1 Let Z be a standard normal random variable and calculate the following probabilities 1 P Z 2 50 Round your answers to four decimal places 2 Find the 39th percentile Round your answers to two decimal places

Distribution Normal Exponential Gamma Weibull Applications Examples include heights weights and other physical characteristics Measurement errors in scientific experiments anthropometric Measurements on fossils LIN 1 E IN reaction times in psychological experiments measurements of intelligence and aptitude scores on various tests numerous economic measures and indicators

Let X be the total medical expenses in 1000s of dollars incurred by a particular individual during a given year Although X is a discrete random variable suppose its distribution is quite well approximated by a continuous distribution with pdf f x k 1 x 2 5 7 for x 0 a What is the value of k b Graph the pdf of X c What are the expected value and standard deviation of total medical expenses d This individual is covered by an insurance plan that entails a 500 deductible provision so the first 500 worth of expenses are paid by the individual Then the plan will pay 80 of any additional expenses exceeding 500 and the maximum payment by the individual including the deductible amount is 2500 Let Y denote the amount of this individual s medical expenses paid by the insurance company What is the expected value of Y Hint First figure out what value of X corresponds to the maximum out of pocket expense of 2500 Then write an expression for Y as a function of X which involves several different pieces and calculate the expected value of this function

22 The weekly demand for propane gas in 1000s of gallons from a particular facility is an rv X with pdf 2 f x 15x52 otherwise a Compute the cdf of X b Obtain an expression for the 100p th percentile What is the value of e Compute E X and VIX d Z If 1 5 thousand gallons are in stock at the beginning of the week and no new supply is due in during the week how much of the 15 thousand gallons is expected to be left at the end of the week Hint Let h x

92 The article The Load Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls Lubrication Engr 1984 153 159 reports the accom panying data on bearing load life million revs for bearings tested at a 6 45 kN load 47 1 126 0 289 0 68 1 68 1 90 8 103 6 106 0 115 0 146 6 229 0 240 0 240 0 278 0 278 0 289 0 367 0 385 9 392 0 505 0 a Construct a normal probability plot Is normality plausible b Construct a Weibull probability plot Is the Weibull distribution family plausible

81 Sales delay is the elapsed time between the manufacture of a product and its sale According to the article Warranty Claims Data Analysis Considering Sales Delay Quality and Reliability Engr Intl 2013 113 123 it is quite common for investigators to model sales delay using a lognormal distribution For a particu lar product the cited article proposes this distribution with parameter values 2 05 and 06 here the unit for delay is months a What are the variance and standard deviation of delay time b What is the probability that delay time exceeds 12 months C d e What is the probability that delay time is within one standard deviation of its mean value What is the median of the delay time distribution What is the 99th percentile of the delay time distribution f Among 10 randomly selected such items how many would you expect to have a delay time exceeding 8 months

Select True or False from each pull down menu depending on whether the corresponding statement is true or false False 1 In a sample of 500 students at a university 12 of them are accounting majors The 12 is an example of statistical inference True 2 A summary measure that is computed from a population is called a parameter True 3 Estimating characteristics of a population is the main goal of descriptive statistics True 4 Statistical inference is the process of making an estimate prediction or decision about a population based on sample data Note In order to get credit for this problem all answers must be correct

A random sample of 20 U S weddings yielded the following data on wedding costs in dollars 22187 29616 22000 18073 19688 19909 24431 27763 14843 22918 17886 23020 24295 23394 7448 6717 7278 19348 12701 25043 a Use the data to obtain a point estimate for the population mean wedding cost u of all recent U S weddings Note The sum of the data is 388558 b Is your point estimate in part a likely equal to exactly yes or no

b The nicotine content in cigarettes of a certain brand is normally distributed with mean in milligrams u and standard deviation 0 1 The brand advertises that the mean nicotine content of their cigarettes is 1 5 but you believe that the mean nicotine content is actually higher than advertised To explore this you test the hypotheses Ho 1 5 Ha 1 5 and you obtain a P value of 0 052 Which of the following is true A You have failed to obtain any evidence for Ha OB There is some evidence against Ho and a study using a larger sample size may be worthwhile O c This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the a 0 05 significance level D At the a 0 05 significance level you have proven that Ho is true

c Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance To investigate this issue a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months At the end of the study 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults Nine of these variables were significantly better in the sense of statistical significance at the a 0 05 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole and one variable was significantly better at the 0 01 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole It would be correct to conclude O A that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the a 0 01 level We should be somewhat cautious about making claims for the variables that were significant at the 0 05 level O B that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance C that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured D none of the above

blem select the best response a An engineer designs an improved light bulb The previous design had an average lifetime of 1200 hours The new bulb had a lifetime of 1200 2 hours using a sample of 40 000 bulbs Although the difference is quite small the effect was statistically significant The explanation is A that the sample size is very large B that new designs typically have more variability than standard designs C that the mean of 1200 is large D all of the above

A confidence interval for a population mean has a margin of error of 3 2 a Determine the length of the confidence interval b If the sample mean is 57 7 obtain the confidence interval Confidence interval

For each situation state the null and alternative hypotheses Type mu for the symbol e g mu 1 for the mean is greater than 1 mu 1 for the mean is less than 1 mu not 1 for the mean is not equal to 1 Please do not include units such as mm or in your answer a The diameter of a spindle in a small motor is supposed to be 2 6 millimeters mm with a standard deviation of 0 18 mm If the spindle is either too small or too large the motor will not work properly The manufacturer measures the diameter in a sample of 8 spindles to determine whether the mean diameter has moved away from the required measurement Suppose the sample has an average diameter of 2 51 mm Ho mu 2 6 Ha mu 2 6

c According to the Merck Veterinary Manual the average resting heart rate for a certain type of sheep dog is 115 beats per minute bpm A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing He measures her heart rate on 15 occasions and finds a sample mean heart rate of 118 2 bpm a A Ho 115 H 115 B Ho p 118 2 Ha 118 2 C Ho 115 H 115 O D Ho 115 H 115 E Ho 118 2 Ha OF Ho 115 Ha 118 2 115

For each statement select the correct null hypothesis Ho and alternative hypothesis Ha in symbolic form a A certain type of hummingbird is known to have an average weight of 4 55 grams A researcher wonders if hummingbirds of this same type living in the Grand Canyon differ in weight from the population as a whole The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3 75 grams A Ho 3 75 B Ho 4 55 c Ho 4 55 D Ho 4 55 E Ho 3 75 F Ho 4 55 Ho 4 55 Ha 3 75 Ha 4 55 Ho p 4 55 H 4 55 H 3 75

b The mean height of all adult American males is 69 inches 5 ft 9 in A researcher wonders if young American males between the ages of 18 and 21 tend to be taller than 69 inches A random sample of 100 young American males ages 18 to 21 yielded a sample mean of 71 inches A H 69 H 69 B H 71 H 71 Ha 69 Ha 71 69 c Ho 69 D H 71 O E Ho 69 OF Hou 69 Ha Ha 69

Be sure to jully justify your responses when appropriate Show where all your numbers came from if not explicitly written Don t forget Units and use complete sentences when appropriate asked for 1 10 pts We bought a pack of 10 lightbulbs from a garage sale and 6 of them don t work If we were to collect a random sample of 2 lightbulbs no replacement and count the number of lightbulbs that are defective create a probability distribution for this discrete random variable a First define your random variable X The random variable X will be defined as the number of defective lightbulbs in a random sample of 2 lightbulbs chosen without replacement from the pack of 10 bulbs X can take from values 0 to 2 where X 0 means both bulbs are working X 1 means one bulbs is working and X 2 means both bulbs are defective b Next showing work for all of your probabilities in the space below fill in the table below round probabilities to 4 decimal places X P X

9 pts According to a study 67 of cars are some shade of grey white black grey or silver If we take a random sample of 25 cars find the following probabilities a None of them are a shade of gray

2 8 pts Scott has 8 class periods per week If we let our random variable C the number of classes Scott is on time to in a week the corresponding probabilities are included in the table below Use the table to answer the following questions 0 0 02 1 0 08 2 0 14 3 0 16 4 0 22 5 0 18 6 0 10 8 0 04 C 0 06 a In a random week what is the probability that Scott is on time to exactly 5 classes

If a person doesn t feel refreshed by sleep that is otherwise unremarkable it is referred to as Restricted to nonrestorative sleep True False

You are measuring the concentration of stress hormone cortisol in saliva samples from volunteers and want to determine whether those who are parents have higher cortisol levels than non parents Which of these if true would reduce the statistical power of your study You can choose more than one Parents tend to be older and older people are less stressed There is no difference in stress levels between parents and nonparents You have measured only a very small number of parents You have measured only a very small number of non parents Parents tend to be older and older people are more stressed

In a controversial paper published in 2016 Dong et al proposed that the age of the longest living human has already reached a plateau indicating a limit to human lifespan of 115 years What criticisms did this analysis face O The conclusions of the analysis rest on a single data point O Only 33 data points are included in the analysis Dong et al used the same dataset to generate their hypothesis and to test it which tends to generate false findings O All of the above

You are measuring the concentration of stress hormone cortisol in saliva samples from volunteers and want to determine whether those who are parents have higher cortisol levels than non parents Which of these if true would be confounding factors in your study You can choose more than one Parents tend to be older and older people are less stressed O There is no difference in stress levels between parents and nonparents You have measured only a very small number of parents You have measured only a very small number of non parents Parents tend to be older and older people are more stressed

Your test gives you a p value of 0 03 so you reject the null hypothesis that parents and non parents have the same mean of cortisol levels But what does your p value mean O There is still a 3 chance that the null hypothesis is correct O 97 of parents are more stressed than any of the non parents and vice versa If you repeated this study with a new set of samples from each population you would find a difference like this between parents and non parents 97 of the time You would find a difference like this between parents and non parents 3 of the time even if the two groups have the same cortisol levels There is only a 0 03 chance that the null hypothesis is correct

Dr Lee gives volunteers a list of 50 foods and a questionnaire about which they like and dislike For each food separately she tests whether men or women are more likely to enjoy it Where the p value is less than 0 05 she rejects the null hypothesis She concludes women are more likely to enjoy salmon and men are more likely to enjoy grapes What s the problem with this statistical analyses cheating Too small a sample size O Confounding variables O Non independent replicates O Multiple comparisons

As part of her final year project Emma did a survey on religious attitudes She asked respondents to enter a number in a box corresponding to their religious belief I identify as 1 Christian 2 Atheist 3 Jewish 4 Hindu 5 Agnostic 6 Other The mean of the numbers she received was 2 08 She concluded the average respondent is atheist What is wrong with this statistical analysis O Treating a categorical variable as a continuous variable O Cheating O Undisclosed flexibility O Confounding variables O Multiple comparisons

As part of Chris s final year project she sends a SurveyMonkey link to her peers in her class asking what they think about fox hunting 80 think it should be banned She concludes that 80 of the public supports the ban on fox hunting What is wrong with this statistical analysis O non random sampling O Cheating O Undisclosed flexibility O Multiple comparisons O Non independent replicates

Dr Smith is trying to detect whether there is a correlation between soil acidity and the number of flowers on plants She grows a treatment group of plants on acidified soil and a control group on normal soil She measures 40 plants in each group and finds that there is a weak correlation but it is not quite significant p 0 054 so she measures another 10 plants in each group and recalculates the p value a 0 046 She rejects the null hypothesis In her paper she writes only that the sample size was 50 and the correlation was significant What is the problem with this statistical analysis O Cheating O Undisclosed flexibility O Confounding variables O Non independent replicates O Multiple comparisons

Dr Jones wants to know whether animals that can fly are more intelligent He measures the brain size of 20 species of mammals that can fly all of them bats and 20 that cannot fly all of them rodents The bats have on average bigger brains He concludes that yes flying animals have bigger brains What is wrong whith this statistical analysis O Undisclosed flexibility p hacking O Non independent replicates Confounding variables O Multiple comparisons cheating

Dr Khan wants to determine whether temperature affects sleep cycles in sheep He monitors a herd of sheep for 24 hours in December after a period of 10 days in which the average daytime temperature was 5 C He does the same again the following July after a period of 10 days in which the average Virtua daytime temperature was 25 C The sheep slept less in summer He concludes that heat makes sheep sleep less What is the problem with this statistical analysis O Non independent replicates O Multiple comparisons O Confounding variables O cheating O undisclosed flexibility

Fill the blanks in the statement below by choosing from the following options You may use these more than once Options range wider standard deviation average mean You are measuring the concentration of stress hormone cortisol in saliva samples from volunteers As a measure of how closely the data are grouped around the mean you might simply state the highest and lowest measurements of the data but this is not ideal because the more measurements you take the i e the becomes so it is better to describe this using the the

Suppose a random variable x arises from a binomial experiment If n 14 and p 0 2 find the following probabilities using the binomial formula Round to four decimal places if necessary P x 1 0 1539 P x 7 0 0092 P x 6 0 0322 P x 11 1 0000 P x 9 P x 4

A fair coin is flipped 14 times Find the probability that more than 6 of the flips turn up tails 0 662034 0 604736 O0 545104 0 633746 O0 575141

The television show September Road has been successful for many years That show recently had a share of 21 meaning that among the TV sets in use 21 were tuned to September Road Assume that an advertiser wants to verify that 21 share value by conducting its own survey and a pilot survey begins with 12 households have TV sets in use at the time of a September Road broadcast Please show your answer to 4 decimal places Find the probability that none of the households are tuned to September Road P none 0 0591 Find the probability that at least one household is tuned to September Road P at least one

According to the American Red Cross 8 9 of all Connecticut residents have Type B blood A random sample 18 Connecticut residents is taken X the number of CT residents that have Type B blood of the 18 sampled What is the standard deviation of the random variable X O 1 39995 O 1 83195 O 1 459422 O 1 414872 1 873368

According to the American Red Cross 8 1 of all Connecticut residents have Type B blood A random sample of 17 Connecticut residents is taken X the number of CT residents that have Type B blood of the 17 sampled What is the expected value of the random variable X 1 564 1 717 1 377 1 411 1 649 1 819 O O O

An insurance company prices its Tornado Insurance using the following assumptions In any calendar year there can be at most one tornado In any calendar year the probability of a tornado is 0 02 The number of tornadoes in any calendar year is independent of the number of tornados in any other calendar year Using the insurance company s assumptions calculate the probability that there are fewer than 2 tornadoes in a 12 year period Round your answer to four decimal places

A binomial experiment consists of 17 trials The probability of success on trial 10 is 0 42 What is the probability of success on trial 14 00 3 0 42 0 19 0 87 00 26 0 43

A baseball player has a batting average of 0 175 What is the probability that he has exactly 5 hits in his next 7 at bats Round your answer to four decimal places The probability the baseball player has exactly 5 hits in his next 7 bats is

6 5 In an effort to link cold environments with hypertension in humans a preliminary experi ment was conducted to investigate the effect of cold on hypertension in rats Two random samples of 6 rats each were exposed to different environments One sample of rats was held in a normal environment at 26 C The other sample was held in a cold 5 C environment Blood pressures and heart rates were measured for rats for both groups The blood pressures for the 12 rats are shown in the accompanying table Rat 1 34563 a Do the data provide sufficient evidence that rats exposed to a 5 C environment have a higher mean blood pressure than rats exposed to a 26 C environment Use a 05 b Evaluate the three conditions required for the test used in part a c Provide a 95 confidence interval on the difference in the two population means 26 C Blood Pressure 152 157 179 182 176 149 Rat 789062 10 11 12 5 C Blood Pressure 384 369 354 375 366 423

126 Approximately 8 percent of students at a local high school participate in after school sports all four years of high school A group of 60 seniors is randomly chosen Of interest is the number who participated in after school sports all four years of high school a In words define the random variable X b List the values that X may take on c Give the distribution of X X d How many seniors are expected to have participated in after school sports all four years of high school e Based on numerical values would you be surprised if none of the seniors participated in after school sports all four years of high school Justify your answer numerically f Based on numerical values is it more likely that four or that five of the seniors participated in after school sports all four years of high school Justify your answer numerically