Statistics Wk 1-3

Relative frequency table

a frequency table with an additional column giving the relative frequencies of the data Instead of a Relative frequency table, some sources may give this information in the form of a Relative frequency histogram

The event is

a grouping of the outcomes in the sample space

discrete quantitative

its possible values form a set of separate numbers such as 0,1,2,3

What is a particular result of an experiment?

outcome An outcome is defined as a particular result of an experiment

Third quartile (Q3)

the 75th percentile, or the 'middle number' of the upper half of the data

binomial distribution

The probability distribution of a binomial random variable

Observational study

a study in which no attempt is made to control any aspect of the individuals in the sample

Pie chart

a type of graph where a circle is divided into sectors, or slices, which each show the proportion of that subcategory compared to the whole A Pie chart may also be referred to as a Pie graph or a Circle graph

Which of the following best describes the term explanatory variable?

the independent variable in an experiment An explanatory variable is defined as the independent variable in an experiment.

Discrete random variable

variables whose possible values are a list of distinct values

Janice is investigating if grade level has any effect on time spent studying. What is the explanatory variable?

grade level The explanatory variable is the independent variable, which is the variable that is changed to determine its effect on a dependent variable. In this case the explanatory variable is grade level.

A computer randomly generates numbers 1 through 100 for a lottery game. Every lottery ticket has 7 numbers on it. Identify the correct experiment, trial, and outcome below:

The experiment is the computer randomly generating a number A trial is one number generated An outcome is the number 2 being generated

Stratified sampling

a variation of random sampling; the population is divided into subgroups and weighted based on demographic characteristics of the national population

a bar graph is appropriate for

categorical data where you want to see the absolute quantity and compare between subcategories

Frequency distribution

contains the number of times each possible outcome occurs within a study or experiment

Probability distribution

contains the probability of each possible outcome for a discrete random variable and always adds up to 1.0

Timothy is collecting data on the number of dental cavities. What type of data is this?

discrete quantitative The number of dental cavities is discrete quantitative data because it is obtained by counting.

cluster sampling

A probability sampling technique in which clusters of participants within the population of interest are selected at random, followed by data collection from all individuals in each cluster.

Dot plot

a graph that helps show the distribution of values in a set of data, consisting of a horizontal axis labeled with data values, dots drawn above the axis to represent single data values, and no vertical axis A Dot plot is sometimes written as Dotplot and in some cases can be referred to as a Dot chart

An energy drink manufacturer wants to find out the effect of different amounts of their product on the maximum anaerobic power level for rowers. Would an experimental or observational study design be more appropriate?

An experimental study should be used with the maximum anaerobic power as the controlled factor. It is not possible to directly control the maximum anaerobic power of the rowers

Let R be the event that a randomly chosen person has red hair. Let G be the event that a randomly chosen person has green eyes. Place the correct event in each response box below to show: Given that the person has red hair, the probability that a randomly chosen person has green eyes.

P(G|R)

Stem-and-leaf plot

a graph, especially good for small sets of data, that separates data points into a leaf consisting of the last significant digit and a stem that consists of any numbers to the left of that digit and can be arranged in ascending or descending order A Stem-and-leaf plot is also commonly referred to as a Stemplot

Categorical data

data separated into qualitative categories

A biologist wants to study the effect of vitamin D enriched food on the weight of adult mice. Would an experimental or observational study design be more appropriate?

An experimental study should be used where the food consumed is the controlled factor. An experimental study should be used because the goal involves a cause and effect relationship.

A new mother keeps track of the time when her baby wakes up each morning. What is the level of measurement of the data?

Interval This is interval data because time of day is a numerical scale where differences are meaningful. However, because the time of day does not have a true zero value, it is not ratio data.

conveinance sampling

is a non-probability sampling technique where subjects are selected because of their convenient accessibility and proximity to the researcher.

Interquartile range:

useful for identifying outliers, a number that indicates the spread of the middle half or the middle 50% of the data in a data set; the difference between the third quartile (Q3) and the first quartile (Q1)

Box-and-whisker plots

(also called box plots) give a good graphical image of the concentration of the data values in a data set. It gives a good, quick picture of the data. They also show how far the extreme values (minimum and maximum) are from most of the other values.

systematic sampling

A procedure in which the selected sampling units are spaced regularly throughout the population; that is, every n'th unit is selected.

A researcher is interested in the effects of watching videos just before bed on the quality of sleep. He has decided to test the claim "Watching 1 hour of video just before going to bed reduces the number of minutes of REM sleep by more than 10%. " Which of the following data collection processes would be appropriate? Select only one answer choice.

Choose a random sample of people. On multiple randomly selected nights, randomly assign each person to either watch one hour of video or not watch video and monitor their sleep. To perform an experiment, a control group that does not watch video and a test group that does should be used. It is also important for the process to account for possible confounding variables. In this case both the person and the night of the week could be confounding variables. By choosing random nights and having the same people either watch 1 hour of videos or not, variations due to the person and the night of the week should not interfere with the analysis.

In reference to different sampling methods, is the following statement true or false? Cluster sampling includes the steps: divide the population into groups; use simple random sampling to identify a proportionate number of individuals from each group.

FALSE Cluster sampling includes the steps: use simple random sampling to select a set of groups; every individual in the chosen groups is included in the sample. Stratified sampling includes the steps: divide the population into groups; use simple random sampling to identify a proportionate number of individuals from each group.

True or False? An explanatory variable is a value or component of the independent variable applied in an experiment.

False An explanatory variable is defined as the independent variable in an experiment. The value or component of the independent variable applied in an experiment is called the treatment.

A restaurant surveys its patrons to pick their favorite item on the menu. What is the level of measurement of the data?

Nominal This is nominal data because the menu items do not have an order to them. Even though a person may have a personal preference, there is no ranking that is universal.

A market researcher finds the price of several brands of fabric softener. What is the level of measurement of the data?

Ratio This is ratio data because price has a true zero, and ratios of prices are meaningful.

Is an online poll asking the preferred mobile phone type used by school children an observational study or experimental study? If it is an experiment, what is the controlled factor?

The study is an observational study. The samples are chosen using an appropriate process; however, no attempt is made to control any aspect of the sample even though the variables of interest are recorded for each group.

Line graph

a bar graph with the tops of the bars represented by points joined by lines (with rest of the bar not shown) and are only appropriate for ordered (rather than qualitative) variables that show how a quantity changes over time A Line graph may also be referred to as a Line chart or as a Time series graph

Bar graph

a graph to summarize and organize categorical data consisting of rectangular bars that are separated from each other and the length (or height) of the bar for each category is proportional to the number or percent of individuals in each category A Bar graph may also be referred to as a Bar chart

a statistical graph is

a tool that helps you learn about the shape or distribution of a sample or a population - a good choice when data sets are small - each data value will be separated into a "stem" and a "leaf" using its digits - the "leaf" consists of a final significant digit - the "stem" contains the remaining digits in front of the "leaf"

A statistic and a parameter are

both descriptions of groups, but they represent different groups

Patrick is collecting data on shoe size. What type of data is this?

discrete quantitative Shoe size is discrete quantitative data because it only takes on certain values, such as 5, 5.5, 6, 6.5, etc.

Jason is collecting data on the number of books students read each year. What type of data is this?

discrete quantitative The number of books students read each year is discrete quantitative data because it is obtained by counting.

Over the past 8 seasons, a baseball player has hit a home run in 6% of his at-bats

empirical probability because the probability is based on observations obtained from a probability experiment - this is because data had to be gathered

mutually exclusive events

events that cannot happen at the same time there can be no overlap you cannot belong to both groups

Michelle is investigating if gender has any effect on political party associations. What is the explanatory variable?

gender The explanatory variable is the independent variable, which is the variable that is changed to determine its effect on a dependent variable. In this case the explanatory variable is gender.

The construction of a box-and-whisker plot uses a...

horizontal or vertical number line and a rectangular box with extending lines (or "whiskers") It is important to start a box plot with a scaled number line. Otherwise the box plot may not be useful. The sample minimum and maximum should be the labels of the endpoints of the number line. The box is marked from the first quartile and the third quartile, with a line marking the median. The median or second quartile can be between the first and third quartiles, or it can be one, or the other, or both. The "whiskers" extend from the ends of the box to the smallest and largest data values.

Which of the following sets of data should not be displayed with a pie chart? Assume that only the two given categories will be included.

the percentage of students on a school sports team and the percentage of people on an intramural sports team A pie chart is not appropriate if the given data has percentages that sum to greater than 100%, resulting from overlap in categories, or less than 100%, resulting from omitting other categories. If an individual could be in each of the categories or neither of the categories, a pie chart will not be sufficient.

A relative frequency is

the ratio (fraction or proportion) of the number of times a certain data value occurs in the set of the total number of data values.

A box plot is constructed from the five-number summary:

the sample minimum, the first quartile, the median, the third quartile, and the sample maximum. We use these values to compare how close other data values are to them.

The sample space is

the set of all possible outcomes of an experiment

interquartile range is

the third quartile minus the first quartile

Expected value

the weighted average of all possible outcomes Expected value is also commonly referred to as Expectation

a pie chart is useful for

categorical data with several categories and where you are interested in the relative quantity Data must fall into one category (no overlap, and nothing excluded)

Given the following list of values, is the mean or the median likely to be a better measure of the center of the data set? 39, 41, 38, 39, 38, 41, 41, 39, 40

Mean Most of the values are close together in the range between 38 and 41. There are no very large or very small values in the list, so the mean is a good measure of the center because it takes into account all the values but will not be pulled up or down by any one value.

A track runner keeps track of how long it takes her to run the 200 meter dash. What is the level of measurement of the data?

Ratio This is ratio data because length of time has a true zero, and ratios are meaningful. If someone took twice as long to run the 200 meters, then that person ran half as fast.

A biologist samples and measures the length of the fish in a lake. What is the level of measurement of the data?

Ratio This is ratio data because length has a true zero, and ratios of lengths are meaningful.

subjective probability

Uses a probability value based on an educated guess or estimate, employing opinions and inexact information based off a persons expertise; its like a really good guess example would be a doctor telling a patient that having this surgery has a 90% success rate

A researcher is interested in the effects of watching videos just before bed on the quality of sleep. Which of the following claims would be appropriate for this situation? Select only one answer choice.

Watching 1 hour of video just before going to bed reduces the number of minutes of REM sleep by more than 10%. The claim or question needs to be testable in order to be analyzed using a statistical process. By using a specified number of hours of video and making a quantitative claim about the effect on REM sleep, the claim can be tested using an appropriate data set.

A parameter is

a number that is a property of the population. Sometimes, when the population is small enough, there is no need to take a sample, and so determining a parameter is easier and a more accurate reflection of the population. A statistic is a number that represents a property of the sample. The statistic is an estimate of a population parameter.

Five-number summary

a set of descriptive statistics that provides information about the five most important percentiles from the data set - the sample minimum, the first quartile (Q1), the second quartile (median, or Q2), the third quartile (Q3), and the sample maximum

Quantifying Statistical Significance

If the probability of an observed result occurring by chance is 0.05 or less, the difference is statistically significant at the 0.05 level. If the probability of an observed result occurring by chance is 0.01 or less, the difference is statistically significant at the 0.01 level ***The general rule is that the smaller number we use, the stronger the significance

Given the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Median Most of the values are close together in the range between 33 and 37, but because there is one number, 20, which is much smaller than the rest of the values, the mean would not be a good measure because that one small value would pull the mean down. Therefore, the median is probably a better measure of the center of this data set.

You want to find the average test score of the students in every math class in 11th grade. There are 15 math classes in 11th grade. You survey the scores of 7 math classes, and you calculate that the average test score is 83%. Which of the following represent the sample and the parameter in this scenario?

Parameter: There is no parameter in this scenario. The sample of 7 math classes is a portion of the total population (15 math classes). A parameter is a number that is representative of the population, but the 83% is the average test score determined from the sample. 83% is the statistic here, because it is a number that represents the sample.

In order to study the shoe sizes of people in his town, Billy samples the population by dividing the residents by age and randomly selecting a proportionate number of residents from each age group. Which type of sampling is used?

Stratified sampling This scenario demonstrates stratified sampling. Stratified sampling involves dividing the population into groups and randomly selecting a proportionate number of individuals from each group

A study was performed with a random sample of 513 bolts produced at one factory. What population would be appropriate for generalizing conclusions from the study, assuming the data collection methods used did not introduce biases?

The conclusions would apply to all bolts produced by that factory. Since the sample is fairly large and was randomly obtained, the results should apply to the bolts made at the factory in the study. However, there could be systematic differences between that factory and any other run by the same company, so the results should not be generalized any further.

A 2016 Pew Research poll estimated the proportion of people supporting each presidential candidate in the fall election. They based their estimates on telephone interviews of 2,010 adults living in all 50 U.S. states. They asked each participant which presidential and vice presidential candidate they support, and used that data to calculate the proportions. What is the statistic in this experiment?

The proportions of participants who support each presidential candidate. Remember that the statistic in an experiment is the numerical characteristic of the sample. Therefore, the statistic in this example is the proportions in the sample, the proportions of participants who support each candidate.

​​​​​​​Probability distribution

contains the probability of each possible outcome for a discrete random variable and always adds up to 1.0

Jacqueline will spin a fair spinner with the numbers 0, 1, 2, 3, and 4 a total of 3 times. If Event A = spinner lands on numbers all greater than 2 and Event B = total sum of 9, which of the following best describes events A and B?

dependent Events are independent if the knowledge of one event occurring does not affect the chance of the other event occurring. In this case, the chance of Event B occurring is greater if Event A occurs, so these events are dependent.

statistically significant:

if measurements or observations in a statistical experiment are unlikely to have occurred by chance

A histogram

is a graph that consists of contiguous (adjoining) boxes, and can show you the shape, the center and the spread of the data - One advantage of a histogram is that it can readily display large data sets - A histogram has both a horizontal axis and a vertical axis - the horizontal axis is labeled with what the data represents. The vertical axis is labeled with either the frequency or the relative frequency

a cluster sample is...

obtained by dividing the population into groups and selecting all individuals from within a random sample of groups.

Quartiles

special (more specific) percentiles that divide ordered data into quarters, and they may or may not be part of the data (depending on the number of data values)

First quartile (Q1)

the 25th percentile, or the 'middle number' of the lower half of the data

Probability

the proportion of times an outcome would occur if we observed the random process an infinite number of times, measured on a scale from 0 (impossible event) to 1 (certain event)

A poll is conducted to determine if political party has any association with whether a person is for or against a certain bill. In the poll, 214 out of 432 Democrats and 246 out of 421 Republicans are in favor of the bill. Assuming political party has no association, the probability of these results being by chance is calculated to be 0.01. Interpret the results of the calculation.

there is no significant evidence to support the claim that the results being by chance

Uniform dot plot

way to explain the general shape of a distribution with no distinct peaks, so the data is spread evenly

For which of the following sets of data is a pie chart appropriate? Assume that only the two given categories will be included.

the percentage of people that support a new speed limit law and the percentage of people who oppose the new speed limit law A pie chart is only appropriate when the data categories account for the entirety of the sample. A pie chart is not appropriate if the given data has percentages that sum to greater than 100%, resulting from overlap in categories, or less than 100%, resulting from omitting other categories. If an individual could be in each of the categories or neither of the categories, a pie chart will not be sufficient.

Unimodal dot plot

way to explain the general shape of a distribution with a single peak

Multimodal dot plot

way to explain the general shape of a distribution with more than one peak

Let B be the event that a randomly chosen person has low blood pressure. Let E be the event that a randomly chosen person exercises regularly. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person exercises regularly, given that the person has low blood pressure.

P(E|B)

Each side of a fair, six-sided die has a certain number of dots on it, 1,2,3,4,5, or 6. The die is rolled by a board game player as part of their turn. Identify the correct experiment, trial, and outcome below:

An experiment is a planned operation carried out under controlled conditions. Here, rolling the die is the experiment. A trial is one instance of a an experiment taking place. A trial here is one rolling of the die. An outcome is any of the possible results of the experiment. Here, the outcome is any of the numbers on the die.

A study is planned to research the effects of drinking coffee on hours of sleep. Would an experimental or observational study design be more appropriate?

An experimental study should be used where the consumption of coffee is the experimental factor. An experimental study should be used because the goal involves a cause and effect relationship.

Jeremy will roll a standard die 3 times and record the total sum of the three rolls. If Event A = roll an even sum, what is an outcome in Event A?

An outcome of an experiment in probability is any of the possible results. An event is some subset of the possible outcomes of an experiment. In this case, an outcome in Event A is a combination of three die rolls whose sum is even. In this case, the only option in Event A is rolling 1,2,1, which results in a total sum of 4, which is even.

Is the statement below true or false? Continuous is the type of quantitative data that is the result of counting.

False Continuous data is defined as the type of quantitative data that is the result of measuring.

Sample maximum

the greatest value in the ordered data set; together with the sample minimum are the two values that show the range of the data set

Sample minimum

the least value in the ordered data set; together with the sample maximum are the two values that show the range of the data set

Conditional probability:

the likelihood that an event will occur given that another event has already occurred

Experimental study

a study in which the researchers attempt to control one or more aspects of the individuals in the sample

Find the median of the following set of data. 7,26,7,9,11,4,15,22

It helps to put the numbers in order. 4,7,7,9,11,15,22,26 Now, because the list has length 8, which is even, we know the median number will be the average of the middle two numbers, 9 and 11. So the median is 10.

An opinion poll was conducted to know whether parents play a role in shaping a career for their children. Respondents were asked to vote "Yes" if they think parents have a role to play and "No" if they think parents have no role to play in shaping the career of their children. Is the study observational or experimental? If it is an experiment, what is the controlled factor?

The study is an observational study. The samples are chosen using an appropriate process; however, no attempt is made to control any aspect of the sample even though the variables of interest are recorded for each group.

Data

Facts and statistics collected together for reference or analysis

A biology student is interested in the relationship between temperature and germination time for soy beans. The student gets sample seeds from multiple sources and randomly selects 180 seeds from them. Then he sets up three growing stations, one at 20∘C, one at 15∘C, and one at 25∘C. He divides the seeds into three groups of 60 and plants the seeds from each group in a growing station. He records the germination time for each seed.

The study is an experiment. The controlled factor is the temperature. Since the seeds are divided into three groups and each group is subjected to a controlled temperature, the study is an experiment. The temperature is the controlled factor.

In a study to add a new feature to a software program, the programmer introduced two categories, men and women, in the survey she conducted. Is the study observational or experimental? If it is an experiment, what is the controlled factor?

The study is an observational study. The samples are chosen using an appropriate process; however, no attempt is made to control any aspect of the sample even though the variables of interest are recorded for each group.

statistically significant

an observed effect so large that it would rarely occur by chance

T or F: Data is a subset of the population studied

false A sample is defined as a subset of the population studied. Data is defined as a set of observations or possible outcomes.

Which of the following events seem like they would be unlikely to occur by chance? Select all that apply.

flipping a coin 100 times and having it land on heads 15 times (If the coin was fair, one would expect to see 50 heads. It would be very unlikely to observe only 15 heads by chance) picking 10 people out of a crowd and having them all have brown eyes guessing somebody's name correctly without having previously met the person

If the difference between the results and the expectation of an experiment is unlikely to be due to chance, the difference is

statistically significant

Conditional probability

the likelihood that an event will occur given that another event has already occurred

Deborah conducted a survey in which she collected data on the percentage of workers with college degrees and the percentage of workers without college degrees. Which of the following could sufficiently display the data if only the two given categories are to be included?

either a pie chart or a bar graph Each person surveyed must fit in exactly one of given data categories. This implies that the percentages must account for 100% of the individuals in the sample. Therefore, either a bar graph or a pie chart will sufficiently display the data.

Donald is studying the eating habits of all students attending his school. He samples the population by dividing the students into groups by grade level and randomly selecting a proportionate number of students from each group. He then collects data from the sample. Which type of sampling is used?

stratified sampling Stratified sampling involves dividing the population into groups and randomly selecting a proportionate number of individuals from each group

When considering different sampling methods, cluster sampling includes the steps: _______.

use simple random sampling to select a set of groups; every individual in the chosen groups is included in the sample Cluster sampling includes the steps: use simple random sampling to select a set of groups; every individual in the chosen groups is included in the sample.

A zoologist measures the birthweight of each cub in a litter of lions. What is the level of measurement of the data?

Ratio This is ratio data because weight has a true zero, and ratios of weights are meaningful.

There are 6000 people who attended a political debate at a convention center. There are three candidates who are participating in the debate - Mr. Smith, Mr. Jones, or Ms. Roberts. You want to find out which candidate is most popular amongst the people attending. You ask 500 people as they walk into the convention center.

The 6000 people attending the debate The three candidates should not be surveyed in order to determine which candidate is most popular amongst the people attending. The population is the total number of people who are attending the political debate.

Data are information about....

populations or samples. Generally, there are unknown characteristics, or variables, about a population that we wish to know. To obtain this information, we collect data by sampling some members of a population. Once collected, the data are used as the actual values of the variables. Data may be numbers (quantitative data) or they may be words (qualitative data)

Second quartile (Q2)

the 50th percentile, and it is a number that separates ordered data such that half the values are the same number or smaller than the median, and half the values are the same number or larger The Second quartile (Q2) is the same as the Median of a data set

Which of the following sets of data should not be displayed with a pie chart? Assume that only the two given categories will be included.

the percentage of people who own cell phones and the percentage of people who own tablets A pie chart is not appropriate if the given data has percentages that sum to greater than 100%, resulting from overlap in categories, or less than 100%, resulting from omitting other categories. If an individual could be in each of the categories or neither of the categories, a pie chart will not be sufficient.

Let C be the event that a randomly chosen cancer patient has received chemotherapy. Let E be the event that a randomly chosen cancer patient has received elective surgery. Identify the answer which expresses the following with correct notation: Of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery.

Remember that in general, P(A|B) is read as "The probability of A given B," or equivalently, as "Of all the times B occurs, the probability that A occurs also." So in this case, the phrase "Of all the cancer patients who have received chemotherapy" can be rephrased to mean "Given that a cancer patient has received chemotherapy," so the correct answer is P(E|C).