Statistics Questions

What is the probability of finding a score lower than a z score of 1 81 Write your answer below as a percentage to two decimal places do not include the symbol

31 Sensitive questions The 2013 Youth Risk Behavior Survey found that 349 individuals in its random sample of 1367 Ohio high school students said that they had texted or emailed while driving in the previous 30 days That s 25 5 of the sample Why is this estimate likely to be biased Do you think it is biased high or low Does the margin of error of a 95 confidence inter val for the proportion of all Ohio high school students who texted or emailed while driving in the previous 30 days allows for this bias

8 13 Number Skills of Eighth Graders Suppose that scores on the mathemat ics part of the National Assessment of Educational Progress NAEP test for eighth grade students follow a Normal distribution with standard deviation 110 You want to estimate the mean score within 10 with 90 confi dence How large an SRS of scores must you choose

33 An outlier strikes You have data on an SRS of freshmen from your college that shows how long each student spends studying and working on homework The data contain one high outlier Will this outlier have a greater effect on a confidence interval for mean completion time if your sample is small or if it is large Why

3 29 Attractive instructors A psychologist claims that stu dents are more attentive in classes taught by instruc tors rated as attractive by students than in classes with instructors rated as unattractive He surveys students in classes taught by instructors rated as attractive and in classes taught by instructors rated as unattractive The proportion of students claiming to be highly at tentive in class is significantly higher P 0 05 among students with attractive instructors What additional in formation would you want to help you decide whether you believe the psychologist s claim

7 Do You Drink Pop A July 2015 Gallup poll asked a national sample of 1009 adults aged 18 and over if they actively avoided drinking soda or pop Of those sampled 61 indicated they do so Gallup announced the poll s margin of error for 95 confidence as 4 percentage points Which of the following sources of error are included in this margin of error a Gallup dialed landline telephone numbers at random and so missed all people without landline phones including people whose only phone is a cell phone b Some people whose numbers were chosen never answered the phone in several calls or answered but refused to participate in the poll c There is chance variation in the random selection of telephone numbers

3 Sampling Students A college administrator questions the first 50 stu dents he meets on campus the day after final exams are over He asks them whether they had positive neutral or negative overall feelings about the term that had just ended Suggest some reasons it may be risky to act as if the first 50 students at this particular time are an SRS of all students at this college

1 Rate This Product An online shopping site asks customers to rate the products they buy on a scale from 1 strongly dislike to 5 strongly like The invitation to rate a recent purchase is sent by email to customers one week after they purchase a product and customers can choose to ignore the invitation Which of the following is the most important reason a confidence interval based on the data from such ratings is of little use for the mean rating by all customers who purchase a particular product Comment briefly on each reason to explain your answer a For some products the number of customers who purchase the product is small so the margin of error will be large b Many of the customers may not read their email or have a spam filter that wrongly identifies the email requesting a review as spam c The customers who provide ratings can t be considered a random sample from the population of all customers who purchase a particular product

4 14 pts The life expectancy of a microwave oven is normally distributed with an average of 16 7 years and a standard deviation of 3 1 years Round all probabilities to 4 decimal places a Sketch and label a complete normal distribution for this situation include values of 30 30 16 7 3 3 1 16 7 9 3 7 4 years years H 20 16 7 2 3 1 16 7 6 2 10 2 H 0 16 7 3 1 13 6 years H 16 7 years H 0 16 7 3 1 19 8 8 years H 20 16 7 2 3 1 16 7 6 2 22 9 years H 30 16 7 3 3 1 16 7 9 3 26 0 years 7 4 center of distribution 10 5 13 6 16 7 22 9 b Find the probability of buying a microwave that lasts for more than 24 years Label this value shade in this area in the distribution above to show the connection to the probability you calculate P x 24 P z 2 35 1 P Z 2 35 0 0094 c Find the probability of buying a microwave that lasts for less than 13 years

4 14 pts The life expectancy of a microwave oven is normally distributed with an average of 16 7 and a standard deviation of 3 1 years Round all probabilities to 4 decimal places a Sketch and label a complete normal distribution for this situation include values of 30 I 7 4 years 10 2 years I center of distribution H 30 16 7 3 3 1 16 7 9 3 H 20 16 7 2 3 1 16 7 6 2 H o 16 7 3 1 13 6 years H 16 7 years H 0 16 7 3 1 19 8 years 1 H 20 16 7 2 3 1 16 7 6 2 22 9 years 1 H 30 16 7 3 3 1 16 7 9 3 26 0 17 4 10 5 13 6 years 16 7 b Find the probability of buying a microwave that lasts for more than 24 years Label this value shade in this area in the distribution above to show the connection to the probability you calculate years 19 8 22 9

A random sample of n 10 regions in New England gave the following violent crime rates per million population x New England Crime Rate 3 3 3 7 4 2 4 1 3 3 4 1 1 8 4 8 Another random sample of n 12 regions in the Rocky Mountain states gave the following violent crime rates per million population X Rocky Mountain Crime Rate IX 3 5 3 7 S 1 0 84 X 3 83 20 94 4 3 4 7 5 3 3 3 4 8 3 5 USE SALT Assume that the crime rate distribution is approximately normal in both regions Use a calculator to calculate x S X and S Round your answers to four decimal place X X X 2 9 3 1 1 What is the level of significance 0 01 2 4 3 1 3 5 5 2 2 8 a Do the data indicate that the violent crime rate in the Rocky Mountain region is higher than in New England Use 0 01 State the null and alternate hypotheses H i Hi Hi Hi Hi 12 H H H i H H H

eye gre 4ster Does eye grease work In one study 16 student subjects took a test of sensitivity to contrast after three hours fac ing into bright sun both with and without eye grease This is a matched pairs design If eye grease is effective subjects will be more sensitive to contrast when they use eye grease Here are the differences in sensitivity with eye grease minus without eye grease 2 EYEGRS 0 07 0 64 0 12 0 05 0 18 0 14 0 16 0 03 0 05 0 02 0 43 0 24 0 11 0 28 0 05 0 29 Kathy Willens AP Images W SUTO We want to know whether eye grease increases sensitiv ity on the average a What are the null and alternative hypotheses Say in words what mean u your hypotheses concern b Suppose that the subjects are an SRS of all young people with normal vision that contrast differences follow a Normal distribution in this population and that the standard deviation of differences is

35 5 versus 1 Sketch the standard Normal curve for the z test statistic and mark off areas under the curve to show why a value of z that is statistically significant at the 1 level in a one sided test is always statistically significant at the 5 level If z is statistically significant at the 5 level what can you say about its significance at the 1 level

7 31 Is this what P means A randomized comparative experiment examined the effect of the attractiveness of an instructor on the performance of students on a quiz given by the instructor The researchers found a statistically significant difference in quiz scores between students in a class with an instructor rated as attrac tive and students in a class with an instructor rated as unattractive P 0 005 When asked to explain the meaning of P 0 005 a student says This means there is only probability of 0 005 that the null hypoth esis is true Explain what P 0 005 really means in a way that makes it clear that the student s explanation is wrong

7 17 Significance from a Table A test of Ho 0 against Ha 0 has test statistic g 1 65 Is this test statistically significant at the 5 level a 0 05 Is it statistically significant at the 1 level a 0 01 is

11 The GMAT continued Exercise 17 1 describes a study of the performance on the GMAT of college seniors in the Philippines You stated the null and alternative hypotheses in Exercise 17 3 542 Enter this a One sample of 250 students had mean GMAT score x x along with the other required information into the P Value of a Test of Significance applet What is the P value Is this outcome statistically signifi cant at the a 0 05 level At the a 0 01 level gid b Another sample of 250 students had x 532 Use the applet to find the P value for this outcome Is it statistically significant at the a 0 05 level At the a 0 01 level favel yogan spagpilinga sean c Explain briefly why these P values tell us that one outcome is strong gia evidence against the null hypothesis and that the other outcome is not bla

Job Satisfaction Find the P Value The P value in Example 17 6 is the probability taking the null hypothesis 0 to be true that x takes a value at least as far from 0 as 17 a What is the sampling distribution of x when 0 This distribution is p shown in Figure 17 3 Res b Do a Normal probability calculation to find the P value Recall that the alter native hypothesis is two sided Your result should agree with Example 17 6 up to roundoff error

Stating Hypotheses In planning a study on the number of days in the last 30 that high school students texted while driving sometime during the day a researcher states the hypotheses as What s wrong with this Ho x 15 days H x 15 days Rot 6 al airit ast

5 Too Early The examinations in a large multisection statistics class are scaled after grading so that the mean score is 75 The professor thinks that students in the 8 00 A M class have trouble paying attention because they are sleepy and suspects that these students have a lower mean score than the class as a whole The students in the 8 00 A M class this semester can be considered a sample from the population of all students in the course so the professor compares their mean score with 75 State the hypotheses Ho and Ha Agrote

80 Consider the system of components connected as in the accompanying picture Components 1 and 2 are connected in parallel so that subsystem works iff either 1 or 2 works since 3 and 4 are connected in series that subsystem works iff both 3 and 4 work If components work independently of one another and P component i works 9 for i 1 2 and 8 for i 3 4 calculate P system works 3 2

20 A certain factory operates three different shifts Over the last year 200 accidents have occurred at the factory Some of these can be attributed at least in part to unsafe working conditions whereas the others are unrelated to working conditions The accompanying table gives the percentage of accidents falling in each type of accident shift category Shift Day Swing Night Unsafe Conditions 10 Unrelated to Conditions 35 20 22 Suppose one of the 200 accident reports is randomly selected from a file of reports and the shift and type of accident are determined a What are the simple events b What is the probability that the selected accident was attributed to unsafe conditions What is the probability that the selected accident did

6 In October 1994 a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division The manu facturer initially claimed that the chance of any particular division being incorrect was only 1 in 9 billion so that it would take thousands of years before a typical user encountered a mistake However statisticians are not typical users some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility Assuming that the 1 in 9 billion figure is correct and that results of different divisions are independent of one another what is the probability that at least one error occurs in one billion divisions with this chip

4 pts The amount of time in minutes it will take students to answer a homework question follows the probability distribution given below Use the graph to answer the following questions P x 15 x time min to complete a homework question a What is the probability a student will take exactly 1 minute to complete the homework assignment P x 1 0 5 The probability that a student will take exactly 1 minute to complete the homework assignment is 50 b What s the probability a student will take between 1 and 2 minutes to complete the homework question P 14x2 5 5 0 25 12 5 The probability that a student will take between 1 and 2 minutes to complete the homework question is 25 c What s the probability a student will take less than 60 seconds to complete the question

8 pts The height of a certain plant after 1 week ranges from 1 cm to 5 cm with a uniform probability distribution The graph is given below Using this distribution answer the following questions 0 25 0 0 5 15 0 50 25 3 5 4 5 a What s the probability that after 1 week the plant will be between 2 and 4 cm tall P 2x34 0 25 0 25 4 2 The probability that the plant will be between 2cm and 4cm tall after 1 week is 50 0 25 2 1 the plant will be at least 3 5 cm tall 1 The probability that the plant will be at least 3 5cm tall is 75 b What s the probability P x23 5 1 P x 3 5 1 4 3 5 1 0 25 0 75 c What s the probability the plant will be under 1 5 cm or over 4 5 cm

Given x and x distributions that are normal or approximately normal with unknown and 2 the value of t corresponding to x x has a distribution that is approximated by a Student s t distribution We use the convention that the degrees of freedom is approximately the smaller of n 1 and 1 However a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite s formula 2 3 d f where s S n and n are the respective sample standard deviations and sample sizes of independent random samples from the x and x distributions This is the approximation used by most statistical software When both n and n are 5 or larger it is quite accurate The degrees of freedom computed from this formula are either truncated or not rounded a We tested whether the population average crime rate in the Rocky Mountain region is higher than that in New England The data were n Satterthwalte s formula to compute the degrees of freedom for the Student s t distribution Round your answer to two decimal places d f 4 58 16 3 51 S 1 09 n 19 X 3 87 and s 1 04 Use b Previously you followed the convention that degrees of freedom d f smaller of n 1 and n 1 Compare this value of d f with that found with Satterthwaite s formula O The previous method leads to a d f that is always less than or equal to that computed by Satterthwaite s formula O The previous method leads to a d f that could be less than or greater than that computed by Satterthwaite s formula The previous method leads to a d f that is always greater than or equal to that computed by Satterthwaite s formula O The previous method leads to a d f that is always equal to that computed by Satterthwaite s formula

of employed adults in the United States held multiple jobs A random sample of 61 employed adults is chosen Use the TI 84 Plus calculator as needed B Part 0 5 Part 1 of 5 a Is it appropriate to use the normal approximation to find the probability that less than 8 5 of the individuals in the sample hold multiple jobs If so find the probability If not explain why not It Choose one appropriate to use the normal curve since np 10 Part 1 5 Part 2 of 5 Choose one b A new sample of 125 employed adults is chosen Find the probability that less than 8 5 of the individuals in this sample hold multiple jobs Round the answer to at least four decimal places The probability that less than 8 5 of the individuals in this sample hold multiple jobs is X 5 B d

A survey of 1721 people found that 15 9 of individuals purchase their religious books primarily at a Christian bookstore a What is the sample proportion of success p 0 159 b What is 0 841 c Which type of distribution is this O Depends on the sample size Binomial O Student T t score O Normal z score d Which type of score will be used in finding the margin of error Ot score Z score c What upper critical value either z score or t score is associated with a 95 confidence level Round to 4 decimal places 1 9600 d What is the margin of error of the 95 confidence interval Round to 3 decimal places 3 92 e Find the 95 confidence interval of the true population proportion of individuals who purchase their religious books primarily at a Christinn

Below is the data for the exam scores of a sample of 9 students The population is normally distributed 85 70 66 95 77 94 65 79 84 a Find the margin of error for a 95 confidence interval for the mean Round to 3 decimal places To find the standard deviation of this sample use formulas from Chapter 2 or technology Enter an integer or decimal number more b Find the confidence interval for the mean Round to 3 decimal places To find the mean of this sample use the formulas in Chapter 2 or technology

Suppose you have sample data where 20 and n 25 You want to create a 99 confidence interval for the mean Also the sample standard deviation is 4 a What is the point estimate for the mean 20 b What is the critical value for the confidence interval Round to 4 decimal places 2 7969 c What is the margin of error for the confidence interval Round to 3 decimal places d What is the confidence interval Round to 3 decimal places I

In this problem assume that the distribution of differences is approximately normal Note For degrees of freedom d f not in the Student s t table use the closest d f that is smaller In some situations this choice of d f may increase the P value by a small a therefore produce a slightly more conservative answer In the following data pairs A represents birth rate and B represents death rate per 1000 resident population The data are paired by counties in the Midwest A random sample of 16 counties gave the following information A 12 5 13 2 12 6 12 1 B 9 8 14 3 10 5 11 6 14 2 15 1 13 2 12 9 10 9 10 0 A 12 5 12 3 13 1 B 14 1 13 6 9 1 15 8 10 3 12 7 11 1 10 2 17 9 11 8 7 0 Do the data indicate a difference elther way between population average birth rate and death rate in this region Use a 0 01 Let d A B a What is the level of significance 0 01 15 7 9 2 State the null and alterate hypotheses Will you use a left tailed right tailed or two tailed test Hoi Hd 0 H Ha 0 right tailed OHR 0 H 0 left talled H H 0 H 0 two tailled Ho H0 H H 0 two talled b What sampling distribution will you use What assumptions are you making The standard normal We assume that d has an approximately uniform distribution O The standard normal We assume that d has an approximately normal distribution The Student s t We assume that d has an approximately uniform distribution O The Student s t We assume that d has an approximately normal distribution What is the value of the sample test statistic Round your answer to three decimal places 1 385 X c Find or estimate the P value OP value 0 500 O 0 250 P value 0 500 0 100 P value 0 250 0 050 P value 0 100 O 0 010 P value 0 050

based on n 23 observations The original datasets had the following proportions of successes p 0 13 p 0 22 p 0 30 p 0 43 A Count C Count 250 200 150 100 50 0 150 100 50 0 0 0 0 1 0 2 0 3 0 4 Bootstrapped proportion B E 0 0 0 2 0 4 0 6 Bootstrapped proportion Histogram A Histogram B Histogram C B a 0 13 b 0 22 c 0 43 D Count Count 150 100 50 0 200 150 100 50 0 Match each histogram with the original data proportion of success 0 2 0 4 0 6 Bootstrapped proportion 0 0 1 0 4 0 2 Bootstrapped proportion

as the standard normal distribution but it reflects the greater variability that is expected with small samples 4 t distributions are used with small samples or when is not known Example of degrees of freedom If 10 test scores have the restriction that their mean is 80 then their sum must be 800 and we can freely assign values to the first 9 scores but the 10th score would then be determined so in this case there are 9 degrees of freedom How many degrees of freedom are there for a sample size of 34 Use the critical values of t table to find the critical value of t for a sample size of 34 when building a 90 confidence interval To find critical values of t on Excel use the following function T INV probability to left of critical value degrees of freedom

poll conducted in 2013 found that 52 of all US adult Twitter users get at least some news on witter However this value was based on a sample so it may not be a perfect estimate for the population parameter of interest on its own The study was based on a sample of 736 adults Below s a distribution of 1000 bootstrapped sample proportions from the Pew dataset Pew Research Center 2013b 0 46 0 47 0 48 0 49 0 50 0 51 0 52 0 53 0 54 0 55 0 56 0 57 a Using the distribution of 1000 bootstrapped proportions approximate a 98 confidence interval for the true proportion of US adult Twitter users in 2013 who get at least some of their news from Twitter b Interpret the interval in the context of the problem With 98 confidence the true proportion of all US adults who live with one or more chronic conditions is between and

Let s say that you want to estimate the proportion of YouTube videos which take place outside define outside to be if any part of the video takes place outdoors You take a random sample of 187 YouTube videos and determine that 64 of them take place outside You d like to estimate the proportion of all YouTube videos which take place outside so you decide to create a bootstrap interval from the original sample of 187 videos Describe how you could use a simulation method similar to using cards or marbles method to find one Bootstrap sample and compute its Bootstrap sample proportion

Researchers have been studying acorns produced by trees in a certain region The acorns are used as a food source by various animals The researchers have recorded the mass of a random sample of acorns displayed below Acorn mass grams 4 25 4 14 4 03 4 26 3 97 4 12 4 19 4 15 3 73 Use Statkey to resample the sample bootstrap with 10 000 iterations to create a 90 confidence interval for the mean mass of all acorns Result The researchers can be 90 confident that the mean mass of all acorns in this region and grams is between

4 pts IQ scores for PhD students is normally distributed with a mean of 110 pts and standard deviation of 17 pts Find the Z score range corresponding to P 100 X 120 on the standard normal distribution Hint your answer will look like P a Z b where you have given values for a and b

8 pts The height of a certain plant after 1 week ranges from 1 cm to 5 cm with a uniform probability distribution The graph is given below Using this distribution answer the following questions 0 25 0 5 1 5 25 45

14 pts The life expectancy of a microwave oven is normally distributed with an average of 16 7 years and a standard deviation of 3 1 years Round all probabilities to 4 decimal places Sketch and label a complete normal distribution for this situation include values of 30

Be sure to fully 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 4 pts The amount of time in minutes it will take students to answer a homework question follows the probability distribution given below Use the graph to answer the following questions P x 0 5 0 5 1 5 x time min to complete a homework question a What is the probability a student will take exactly 1 minute to complete the homework assignment

Unal Benford s Law claims that numbers chosen from very large data files tend to have 1 as the first nonzero digit disproportionately often In fact research has shown that if you randomly draw a number from a very large data file the prob of getting a number with 1 as the leading digit is about 0 301 Now suppose you are the auditor for a very large corporation The revenue file contains millions of numbers in a large computer data bank You draw a random sample of n 229 numb from this file and r 88 have a first nonzero digit of 1 Let p represent the population proportion of all numbers in the computer file that have a leading digit of 1 1 Test the claim that p is more than 0 301 Use a 0 10 a What is the level of significance 0 10 State the null and alternate hypotheses Ho P 0 301 H p 0 301 O Ho P 0 301 H p 0 301 Ho p 0 301 H p 0 301 O Ho P 0 301 H p 0 301 b What sampling distribution will you use The standard normal since np 5 and nq 5 O The Student s t since np 5 and nq 5 The Student s t since np 5 and nq 5 The standard normal since np 5 and ng 5 What is the value of the sample test statistic Round your answer to two decimal places 2 76 c Find the P value of the test statistic Round your answer to four decimal places 0 0056

Using Fig1a png Fig1b png and Fig1c png answer the following question For large values of p the binomial distribution is skewed to the 1 left II right III symmetric IV none of the above

Using Question 11 answer the following question Use your calculator to calculate the standard deviation of the number of minority students selected Use the formula o np 1 p to find the standard deviation Note round your answer to TWO DECIMAL places 3 1

According to the CDC alcohol impaired drivers are responsible for 32 of all crash deaths in the US Suppose we take a random sample of 100 car accident deaths and let X the number that are alcohol related Find the probability that fewer than 25 were alcohol related Note please round your answer to TWO DECIMAL places 0 05 QUESTION 5 Using Question 4 find the probability that between 23 and 40 inclusive were alcohol related Note please round your answer to TWO DECIMAL places 0 95 QUESTION 6 Using Question 4 find the probability that at least 38 were alcohol related Note please round your answer to TWO DECIMAL places

Using Question 11 answer the following question Assuming that the sentate represents the entire student body how many minority students would you expect to be selected for the sentate Use the formula n p to find expected value 12

Now see what happens when you hold p constant and vary n Let X be a binomial random variable with p 0 24 Obtain the probability distribution and graph of the distribution for both n 10 and 30 and save the graphs as Fig2a png and Fig2b png respectively in your preferred local folder Using Fig2a png and Fig2b png answer the following question As n increases the probability distribution becomes more Ol left skewed II right skewed III symmetric O IV none of the above in shape

Using Question 11 calculate the probability of the Senate being made up of so few minority students purely by chance That is what is the value of P X S 2 for this binom problem 1 0 941 OII 0 0001941 III 0 41 IV 9 1941

A Student Sentate at a large university is made up of 60 students randomly selected to represent the entire student body 20 of the students at this university are minoritie When the members of the senate are selected only two members are minority students The Minority Caucus claims that this Senate is racially biased They decide to use statistics to prove their claim Let X be the number of minority students who are selected for the senate Assume that X Binomial n p then answer the following questions The value of n is 60 and the value of p is 0 2 round your answer to TWO DECIMAL places

Using the results found in Question 2 calculate by hand the probability that more than 15 students in the sample graduate in 4 years That is find P X 15 Note please round your answer to TWO DECIMAL places 0 07

It has been reported that 39 of college students graduate in 4 years Consider a random sample of thirty students and let the random variable X be the number who graduate in 4 years Find the probability that exactly 13 students in the sample of 30 graduate in 4 years Note please round your answer to TWO DECIMAL places 0 13

According to the CDC alcohol impaired drivers are responsible for 32 of all crash deaths in the US Suppose we take a random sample of 100 car accident deaths and let X be the number that are alcohol related Find the probability that fewer than 25 were alcohol related Note please round your answer to TWO DECIMAL places 0 05