Statistics Questions

Determine whether the study is an experiment or an study A sociologist has created a brief survey to be given to 2000 adults randomly selected from the U S population Here are her first two questions 1 Have you ever been the victim of a felony crime 2 Have you ever been convicted of a felony This is an observational study experiment ww because the researcher the individuals

Types of Observational Studies flow chart interactive to answer the following question A sociologist conducts a study of the number of single parents between the years of 2000 and 2015 What type of study is that Choose the correct answer below A Retrospective B Cross sectional OC Prospective OD None of the above

described below A woman experienced a tax audit The tax department claimed that the woman was audited because she was randomly selected from all women in her age group Which type of sampling did the tax department use Stratified sampling O Systematic sampling Random sampling Cluster sampling Convenience sampling

A television station asks its viewers to call in their opinion regarding the desirability of programs in high definition TV Which type of sampling is used A Cluster B Convenience OC Stratified OD Random OE Systematic

In a study designed to test the effectiveness of a medication as a treatment for lower back pain 1643 patients were randomly assigned to one of three groups 1 the 547 subjects in the placebo group were given pills containing no medication 2 550 subjects were in a group given pills with the medication taken at regular intervals 3 546 subjects were in a group given pills with the medication to be taken when needed for pain relief In what specific way was replication applied in the study Choose the correct answer below OA The group sample sizes are all large so the researchers could see the effects of the treatment B The subjects were divided into three groups so the researchers could compare the effects of the different treatments OC The researchers did not know whether the subjects in the study were taking a placebo or the medication OD The subjects in the study did not know whether they were taking a placebo or the medication

K The diameter of a brand of tennis balls is approximately normally distributed with a mean of 2 72 inches and a standard deviation of 0 04 inch A random sample of 11 tennis balls is selected Ca through d below a What is the sampling distribution of the mean A Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will also be approximately normal B Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 cannot be found C Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will be the uniform distribution OD Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will not be approximately normal b What is the probability that the sample mean is less than 2 71 inches P X 2 71 0 2035 Round to four decimal places as needed c What is the probability that the sample mean is between 2 70 and 2 74 inches P 2 70 x 2 74 0 9027 Round to four decimal places as needed d The probability is 68 that the sample mean will be between what two values symmetrically distributed around the population mean The lower bound is 2 71 inches The upper bound is Round to two decimal places as needed inches

A study of an association between which ear is used for cell phone calls and whether the subject is left hande or right handed began with a survey e mailed to 5000 people belonging to an otology online group and 717 were returned Otology relates to the ear and hearing What percentage of the 5000 surveys were returned that response rate appear to be low In general what is a problem with a very low response rate CHOCOH Of the 5000 surveys were returned This response rate Round to the nearest whole number as needed to be low

Identify which of these types of sampling is used random systematic convenience stratified or cluster To determine her water pressure Keri divides up her day into three parts morning afternoon and evening She then measures her water pressure at 4 randomly selected times during each part of the day What type of sampling is used O Random Cluster Systematic Convenience Stratified

Identify the type of sampling used random systematic convenience stratified or cluster sampling in the situation described below A woman is selected by a marketing company to participate in a paid focus group The company says that the woman was selected because every 2500th person in the phone number listings was being selected Which type of sampling did the marketing company use Stratified sampling Cluster sampling Systematic sampling Random sampling Convenience sampling

times and her results had a standard deviation of 1 33 The same sample was analyzed five times by Student B and his results had a standard deviation of 3 42 To determine if their standard deviations are similar they perform an F The calculated F value is and the table value of Fis Refer to Table for critical values of F Degrees of freedom for 2 2 AWN 3 4 5678 c 10 2 3 39 00 39 17 16 04 15 44 15 10 10 65 9 98 9 60 8 43 7 76 7 39 7 26 6 60 6 54 5 89 6 06 5 42 5 71 5 08 5 46 4 5 39 25 39 30 39 33 14 88 14 73 9 36 9 20 Fcalculated 6 61 Ftable 9 36 Fcalculated 6 61 Ftable 5 19 Fcalculated 2 57 Ftable 7 39 6 23 5 52 5 05 4 72 4 83 4 47 6 7 39 36 14 62 9 07 7 15 6 98 6 85 5 99 5 82 5 70 5 29 5 12 4 99 4 82 4 65 4 53 4 32 Degrees of freedom for s 8 9 39 37 39 39 14 54 14 47 8 98 8 90 6 76 6 68 5 60 5 52 4 90 4 82 4 43 4 36 4 10 4 03 4 48 4 20 4 24 4 07 3 95 3 85 3 78 10 39 40 14 42 8 84 6 62 5 46 4 76 4 30 3 96 3 72

In a study of a sample of babies born at hospitals in one state it was found that 49 of the babies were girls Identify whether this value is a statistic or a par Choose the correct answer below OA The value is a statistic because it describes some characteristic of a population OB The value is a parameter because it describes some characteristic of a population OC The value is a parameter because it describes some characteristic of a sample OD The value is a statistic because it describes some characteristic of a sample

A simple random sample of n subjects is selected in such a way that every possible sample of the same size n has the same chance of being chosen A simple random sample is often called a random sample K but strictly speaking a random sample has the weaker requirement that all members of the population have the same chance of being selected Determine whether each of the following is a simple random sample and a random sample Complete parts a through c C a In Major League Baseball there are 30 teams each with an active roster of 25 players The names of the teams are printed on 30 separate index cards the cards are shuffled and one card is drawn The sample consists of the 25 players on the active roster of the selected team This sample a simple random sample It a random sample

Variable a Customer satisfaction rating very satisfied somewhat satisfied somewhat dissatisfied or very dissatisfied b Temperature in degrees Fahrenheit c Closing price in dollars of a stock variable O Quantitative O categorical O Quantitative O categorical O Quantitative O Categorical measuremen O Nominal Oordinal O Interval O Ratio O Nominal O Ordinal O Interval O Ratio O Nominal O Ordinal O Interval O Ratio

In a double blind experiment designed to test the effectiveness of a new medication as a treatment for lower back pain 1643 patients were randomly assigned to one of three groups 1 the 547 subjects in the placebo group were given pills containing no medication 2 550 subjects were in a group given pills with the new medication taken at regular intervals 3 546 subjects were in a group given pills with the new medication to be taken when needed for pain relief What does it mean to say that the experiment was double blind Choose the correct answer below OA Both the subjects in the study and those who administered the pills knew whether they were taking a placebo or the new medication OB The subjects in the study did not know whether they were taking a placebo or the new medication but those who administered the pills did know OC The subjects in the study did not know whether they were taking a placebo or the new medication and those who administered the pills also did not know COD The subjects in the study were given the choice of taking the placebo or the new medication but those who administered the pills did not know which group the subjects were in

When a car is randomly selected it is found to have an engine with 4 cylinders Choose the correct answer below OA A continuous data set because there are infinitely many possible values and those values can be counted OB The data set is neither continuous nor discrete OC A discrete data set because there are a finite number of possible values OD A continuous data set because there are infinitely many possible values and those values cannot be counted

Identify the level of measurement of the data and explain what is wrong with the given calculation In a set of data mood levels are represented as 1 for bad 2 for OK and 3 for good The average mean of the 619 mood levels is 1 3 The data are at the level of measurement nominal ratio ordinal interval COD

Identify the level of measurement of the data and explain what is wrong with the given calculation In a survey the favorite foods of respondents are identified as 0 for italian food 1 for mexican food 2 for chinese food and 3 for anything else The average mean is calculated for 551 respondents and the is 1 1 The data are at the level of measurement nominal ratio ordinal interval

the four levels of measurement nominal ordinal interval ratio is most appropriate for the data below Years in which a war was started Choose the correct answer below OA The ratio level of measurement is most appropriate because the data can be ordered differences obtained by subtraction can be found and are meaningful and there is a natural startr OB The nominal level of measurement is most appropriate because the data cannot be ordered OC The interval level of measurement is most appropriate because the data can be ordered differences can be found and are meaningful and there is no natural starting zero point OD The ordinal level of measurement is most appropriate because the data can be ordered but differences obtained by subtraction cannot be found or are meaningless

Determine which of the four levels of measurement nominal ordinal interval ratio is most appropriate for the data below Diameter of circles in inches Choose the correct answer below OA The nominal level of measurement is most appropriate because the data cannot be ordered OB The ratio level of measurement is most appropriate because the data can be ordered differences obtained by subtraction can be found and are meaningful and there is a natural starting zero p OC The ordinal level of measurement is most appropriate because the data can be ordered but differences obtained by subtraction cannot be found or are meaningless OD The interval level of measurement is most appropriate because the data can be ordered differences obtained by subtraction can be found and are meaningful and there is no natural starting po

Determine which of the four levels of measurement nominal ordinal interval ratio is most ap Positions of persons in a line Choose the correct level of measurement OA Ordinal OB Interval OC Nominal OD Ratio

Experimental Inquiry What Can You Learn About the Process of Science from Investigating a Cricket s Chirp 15 20 25 30 Snowy tree cricket 09 100 117 128 No elements selected Common field cricket mte 130 73 Plot the data for the two species of crickets one graph for the snowy tree cricket and a separate graph for the common field cricket To plot the data for each species click Add Element and then select the appropriate graph t View Available Hint s 120 115 110 82 90 100

In a study conducted by a human resource management organization 277 human resource professionals were surveyed Of those surveyed 73 said that their companies conduct criminal backgro all job applicants Complete parts a through d a What is the exact value that is 73 of 277 survey subjects The exact value is 202 21 Type an integer or a decimal Do not round b Could the result from part a be the actual number of survey subjects who said that their companies conduct criminal background checks on all job applicants Why or why not A Yes the result from part a could be the actual number of survey subjects who said this because the polling numbers are accurate OB No the result from part a could not be the actual number of survey subjects who said this because that is a very rare outcome OC Yes the result from part a could be the actual number of survey subjects who said this because the result is statistically significant D No the result from part a could not be the actual number of survey subjects who said this because a count of people must result in a whole number c What is the actual number of survey subjects who said that their companies conduct criminal background checks on all job applicants The actual number of survey subjects with this response is Round to the nearest whole number as needed

users this question was posted online Have you ever been hit by a computer virus Among the 170 063 responses 63 answered yes What term is used to describe this t of survey in which the people surveyed consist of those who chose to respond What is wrong with this type of sampling method What term is used to describe this type of survey Select all that apply A The respondents are a voluntary response sample B The respondents are a self selected sample C The respondents are a population D The respondents are a census What is wrong with this type of sampling method Select all that apply A The survey question is loaded or intentionally worded to elicit a desired response B It is too time consuming C Many people may choose not to respond to the survey D Responses may not reflect the opinions of the general population E It is too expensive CIB

Determine whether the given source has the potential to create a bias in a statistical study A data set includes brain volumes from 10 pairs of monozygotic identical twins The data were collected by researchers at Harvard University Massachusetts General Hospital Dartmouth College and the University of California at Davis Choose the correct answer below OA There does not appear to be a potential to create a bias The organizations collecting the data are reputable OB There does appear to be a potential to create a bias There is an incentive to produce data showing identical brain volumes OC There does appear to be a potential to create a bias There is an incentive to make the results statistically insignificant OD There does not appear to be a potential to create a bias The organizations would not gain from putting spin on the data

In a study conducted by a human resource management organization 277 human resource professionals were surveyed Of those surveyed 73 said that their companies conduct criminal background checks on all job applicants Complete parts a through d a What is the exact value that is 73 of 277 survey subjects The exact value is 202 21 Type an integer or a decimal Do not round b Could the result from part a be the actual number of survey subjects who said that their companies conduct criminal background checks on all job applicants Why or why not OA Yes the result from part a could be the actual number of survey subjects who said this because the polling numbers are accurate OB No the result from part a could not be the actual number of survey subjects who said this because that is a very rare outcome OC Yes the result from part a could be the actual number of survey subjects who said this because the result is statistically significant D No the result from part a could not be the actual number of survey subjects who said this because a count of people must result in a whole number

Determine whether the results appear to have statistical significance and also determine whether the results appear to have practical significance One of a botanist s hybridization experiments with peas yielded 528 offspring with 139 of those peas or 26 having yellow pods According to genetic theory 25 of the offspring peas should have yellow pods Does the result have statistical significance According to genetic theory of the 528 peas would have yellow pods The difference between the actual number of peas with yellow pods and the expected number of peas with yellow pods is pea s This difference to be statistically significant Type whole numbers

When testing a new treatment what is the difference between statistical significance and practical significance Can a treatment have statistical significance but not practical significance BEKL Choose the correct answer below OA Statistical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use Practical significance is achieved when the result is very unlikely to occur by chance It is not possible for a treatment to have statistical significance but not practical significance OB Statistical significance is achieved when the result is very unlikely to occur by chance Practical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use It is not possible for a treatment to have statistical significance but not practical significance OC Statistical significance is achieved when the result is very unlikely to occur by chance Practical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use It is possible for a treatment to have statistical significance but not practical significance OD Statistical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use Practical significance is achieved when the result is very unlikely to occur by chance It is possible for a treatment to have statistical significance but not practical significance

Determine whether the given source has the potential to create a bias in a statistical study The American Automobile Association AAA is a not for profit federation of motor clubs that provides automotive and travel services AAA conducts a survey of its members about their use of public transportation versus private automobiles Choose the correct answer below OA There does appear to be a potential to create a bias There is an incentive to make the results statistically insignificant OB There does not appear to be a potential to create a bias The organization would not gain from putting spin on the results OC There does not appear to be a potential to create a bias The organization is reputable and has many professional and credible members O D There does appear to be a potential to create a bias There is an incentive to produce results that are in line with preferring private automobiles

Determine whether the sampling method described below appears to be sound or is flawed In a survey of 616 subjects the following question was posted on a newspaper s website In your view are nucle plants safe The survey subjects were Internet users who chose to respond to the question posted on the electr edition of the newspaper Choose the correct answer below OA It is flawed because it is a voluntary response sample B It is flawed because it is not statistically significant C It is flawed because it is a census D It appears to be sound because the data are not biased in any way

Determine whether the given source has the potential to create a bias in a statistical study A data set includes brain volumes from 10 pairs of monozygotic identical twins The data were collected by researchers at Harvard University Massachusetts General Hospital Dartmouth College and the University of California at Davis Choose the correct answer below OA There does not appear to be a potential to create a bias The organizations would not gain from putting spin on the data B There does appear to be a potential to create a bias There is an incentive to make the results statistically insignificant OC There does not appear to be a potential to create a bias The organizations collecting the data are reputable OD There does appear to be a potential to create a bias There is an incentive to produce data showing identical brain volumes

Determine whether the results below appear to have statistical significance and also determine whether the results have practical significance In a study of a weight loss program 4 subjects lost an average of 46 lbs It is found that there is about a 35 chance of getting such results with a diet that has no effect Does the weight loss program have statistical significance OA No the program is not statistically significant because the results are likely to occur by chance B Yes the program is statistically significant because the results are likely to occur by chance C Yes the program is statistically significant because the results are unlikely to occur by chance D No the program is not statistically significant because the results are unlikely to occur by chance

Refer to the sample of body temperatures degrees Fahrenheit in the table below Given these temperatures what issue can be addressed by conducting a statistical analysis of the data Subject 8 AM 12 AM 1 2 3 4 5 97 0 98 5 97 6 97 7 98 7 97 6 97 8 98 0 98 4 98 4 Choose the correct answer below OA The data can be used to address the issue of whether there is a correlation between body temperatures at 8 AM and at 12 AM OB The data can be used to address the issue of whether there is a correlation between average body temperature and a person s exposure to sunlight OC The data can be used to address the issue of whether there is a difference between average body temperature for males and for females D The data can be used to find the percentage of people whose body temperature increases with illness

Choose the correct answer below O A A radio station asks for call in responses to a question concerning city recycling OB Quiz scores from a college level statistics course are analyzed to determine student progress OC A local dentist asks her patients to fill out a questionnaire and mail it back to determine the quality of received during an office visit OD A survey is taken at a mall by asking passersby if they will fill out the survey

In the data table the entries are for five different years and consist of the weights metric tons of lemons imported from another country and car crash fatality rates per 100 000 population The data are based on data from a scientific publication Given the context of the data in the table what issue can be addressed by conducting a statistical analysis of the values Lemon Imports Crash Fatality Rate What issue can be addressed 235 264 358 473 15 5 15 3 15 5 15 4 15 3 530 C A Does the car crash fatality rate cause the amount of lemon imports to increase or decrease OB Is there a correlation or an association between the amount of lemon imports and the car crash fatality rate OC Is there a correlation or an association between the mean amount of lemon imports and the mean car crash fatality rate O D Does an increase or decrease in the amount of lemon imports cause the car crash fatality rate to increase or decrease E No conclusion can be drawn from the data

ted by a hun management organizatio source professionals were surveyed Of those surveyed 73 said that their companies conduct criminal background checks on all job applicants Complete parts a through d www a What is the exact value that is 73 of 277 survey subjects The exact value is Type an integer or a decimal Do not round

Determine whether the given source has the potential to create a bias in a statistical study An article in Journal of Nutrition Vol 130 No 8 noted that chocolate is rich in flavonoids The article notes regular consumption of foods rich in flavonoids may reduce the risk of coronary heart disease The study received funding from Mars Inc the candy company and the Chocolate Manufacturers Association Choose the correct answer below OA The data do not seem to support the claims being made by the article The article s author should consult an expert to make sure that he or she is correctly interpreting the study s results OB The questions used in the study might have caused the respondents to give inaccurate or dishonest responses The question wording should be changed to be more neutral OC The researchers may have been more inclined to provide favorable results because funding was provided by a party with a definite interest The bias could have been avoided if the researchers were not paid by the candy company and the chocolate manufacturers D Since the sample is self selected there is a definite participation bias in this study The researchers should randomly select the subjects of the study

In recent years the numbers of cell phones and the numbers of pirates have both increased so there is a correlation or association between those two variables Therefore pirates cause increases in cell phones Choose the correct answer below www OA The data on the numbers of pirates and cell phones are sample data and cannot be used to make conclusions about the populations OB The direction of the causal relationship is reversed Cell phones cause increases in pirates OC The number of pirates cannot be accurately measured D The correlation or association between two variables does not mean that one of the variables is the cause of the other Correlation does not imply causation

In an AOL survey of Internet users this question was posted online Have you ever been hit by a computer virus Among the 170 063 responses 63 answered yes What term is used to describe this type of survey in which th people surveyed consist of those who chose to respond What is wrong with this type of sampling method What term is used to describe this type of survey Select all that apply A The respondents are a voluntary response sample B The respondents are a self selected sample C The respondents are a population D The respondents are a census

b Create a side by side bar chart with store location on the horizontal axis and side by side bars of the percentage of time spent on each task 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 0 11 Meetings Reports Customers Idle Meetings Reports Customers Idle 4 70 60 50 40 30 20 10 0 c Which type of bar chart stacked or side by side do you prefer for these data Why Meetings Reports Customers Idle G 70 60 50 40 30 20 10 0

Class A B C D a What is the relative frequency of class D 0 7 b The total sample size is 200 What is the frequency of class D 35 c Show the frequency distribution Class Relative Class Frequency A A B B C C D D Total 200 Total 0 22 0 08 d Show the percent frequency distribution 0 40 35 Frequency Percent Frequency

The London School of Economics and the Harvard Business School have conducted studies of how chief executive officers CEOS spend their time These studies have found that CEOS spend many hours per week in meetings include conference calls business meals and public events Suppose that the data below show the time spent per week in meetings hours for a sample of 25 CEOS 14 15 18 23 15 19 20 13 15 23 23 21 16 15 19 11 12 13 14 a What is the least amount of time spent per week on meetings h 15 16 What is the highest amount of time spent per week on meetings h 17 18 b Use a class width of two hours to prepare a frequency distribution and a percent frequency distribution for the data Hours in Meetings per Week 19 20 5 18 18 18 21 22 15 20 21 22 23 24 Total 18 19 23 21 12 Frequency Percent Frequency

Olivia Sophia Emma Charlotte Emma Amelia Ava Sophia Sophia Ava Charlotte Olivia Emma Amelia Emma Ava Emma Sophia Amelia Ava Emma Ava Charlotte Charlotte Emma Amelia Charlotte Sophia Amelia 7 Summarize the data by constructing the following a Relative and percent frequency distributions Baby Name Relative Frequency 12 5 Sophia Sophia Olivia 6 8 Olivia Sophia Emma Ava Ava Sophia Emma Charlotte Emma Ava Olivia Olivia Emma Ava 0 1 0 2 0 1 0 1 Emmal 0 1 Ava Olivia Ava Olivia Percer Frequency Charlotte Amelia Emma Amelia 4

Consider the following frequency distribution Class Frequency Frequency 10 19 20 29 30 39 40 49 50 59 10 15 18 6 1 Construct a histogram for the data above 18 10 19 20 29 30 39 40 49 50 59 Frequency 18 16 ONACON 10 19 20 29 30 39 40 49 50 59 Frequency 16 10 19 20 29 30 39 40 49 50 59 Frequency 18 16 ONO 10 19 20 29 30 39 4

d A pie chart Olivia Charlotte Emma Emma Amelia Ava Sophia Charlotte Amelia Sophia sorted bar chart Ava Olivia Charlotte Which type of chart makes this most apparent obar chart Sophia Emma Amelia e Based on these data what are the three most common last names Select all that apply Olivia Ava Olivia Charlotte Emma Sophia Amelia Ava

6 Consider a game where you have five distinct coins in a row and you are allowed to move them in pairs and insert them between other coins in the row without changing the order of the pair For example here is an allowable move As above you can slide the coins over horizontally to create space to insert the pair or to remove extra space as long as you do not change the order of the coins in the row a Suppose the coins are placed in order of increasing value is there a sequence of moves as described above that reverses the order so they are in order of decreasing value b Again suppose the coins are placed in order of increasing value is there a sequence of moves that switches the first coin with the last and where the middle three end up in their starting positions

3 a On a 3x3 grid draw a path that starts in the center of a square in the corner of the grid and repeatedly moves from the center of one square to the center of an adjacent square without visiting any squares twice and such that every square in the grid is included For example the following is a path that meets these conditions What are the possible ending positions for a path drawn according to these rules What if you started from an edge square or the center square of the grid Explain b Repeat the above problem for a 4 x 4 grid e Can you describe what happens in general for an nxn grid where n is a positive integer

A pair of distinct two digit integers n d is misleading if the first digit of d equals the second digit of n and if you write the numbers as a fraction and cancel i e cross out the common digits you get an equal fraction For example 16 64 is a misleading pair since 16 16 64 a Find all pairs of two digit misleading integers Explain how you know you have found all possible misleading two digit pairs b Are there any misleading pairs of three digit integers For a three digit pair to be mis leading when written as a fraction cancelling the first digit of d with either the second or third digit of n where the digits being canceled are equal results in a fraction equal in value to the original fraction Furthermore the two integers cannot both end in 0 otherwise the pair could be obtained directly from a pair of misleading two digit num bers

The graph illustrates a normal distribution for the prices paid for a particular model of HD television The mean price paid is 1000 and the standard deviation is 70 790 860 930 1000 1070 1140 1210 Distribution of Prices Q What is the approximate percentage of buyers who paid between 790 and 1000 What is the approximate percentage of buyers who paid between 1000 and 1070 What is the approximate percentage of buyers who paid between 930 and 1070 What is the approximate percentage of buyers who paid between 1000 and 1140 What is the approximate percentage of buyers who paid more than 1140 What is the approximate percentage of buyers who paid more than 1210

The annual per capita consumption of ice cream in pounds in the United States can be approximated by the normal distribution as shown in the figure to the right a What is the largest annual per capita consumption of ice cream that can be in the bottom 10 of consumptions b Between what two values does the middle 80 of the consumptions lie Click to view page 1 of the Standard Normal Table Click to view page 2 of the Standard Normal Table a The largest annual per captita consumption of ice cream that be in the bottom 10 of consumptions is 12 9 pounds Round to one decimal place as needed b The middle 80 of consumption lies between and pounds Round to one decimal place as needed IN H 19 8 Q 5 4 20 32 Consumption in lbs

Adult men have heights with a mean of 69 0 inches and a standard deviation of 2 8 inches Find the z score of a man 68 7 inches tall to 2 decimal places