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**ALEKS Probability and Statistics Answer Key**

Yes, we will be discussing all ALEKS Probability and Statistics topics including all previously asked questions.

**Q.** population**Ans:** the entire group of objects or individuals under study

**Q.** data**Ans:** observations (such as measurements or survey responses)

**Q.** census**Ans:** the collection of data from every element in a population

**Q.** sample**Ans:** a subcollection of the population that is actually used to get information

**Q.** unit/subject**Ans:** an individual object or person in the population

**Q.** population size**Ans:** N

**Q.** sample size**Ans:** n

**Q.** parameter**Ans:** a number that describes some aspect of a population

**Q.** statistic**Ans:** a number that describes some aspect of a sample

**Q.** univariate data**Ans:** observations are made on a single variable

**Q.** bivariate data**Ans:** observations are made on 2 variables

**Q.** multivariate data**Ans:** observations are made on several variables

**Q.** descriptive statistics**Ans:** used to summarize and describe data sets many of the methods are graphical in nature

**Q.** inferential statistics**Ans:** the process of drawing conclusions about a population based on information from a sample from that population

**Q.** probability vs statistics**Ans:**

probability: given the information in the box, what is in your hand, deductive

statistics: given the information in your hand, what is in the box, inductive

**Q.** selection bias**Ans:** the systematic tendency on the part of the sampling procedure to exclude or include a certain type of unit

**Q.** nonresponse bias**Ans:** the distortion that can arise because a large number of units selected for the sample do not respond or refuse to respond, and these nonresponders have a tendency to be different from the responders

**Q.** self-selecting bias**Ans:** occurs when respondents themselves decide whether to be included in the study (people with strong opinions are more likely to participate)

**Q.** convenience sample

Ans: a sample that is not drawn by a well-defined random method

**Q.** probability sampling method**Ans:** gives each unit in the population a known, non-zero chance of being selected

**Q.** simple random sample**Ans:** units are selected in such a way that every possible sample of size n has the same chance of being chosen

**Q.** stratified random sampling**Ans:**

divide the population into subgroups, that share some characteristic

draw a SRS from each stratum (the samples are not required to be of the same size from each strata)

**Q.** systematic sampling**Ans:** 1-in-k

**Q.** cluster sampling**Ans:** the units of the population are grouped into clusters (no obvious shared characteristic) and then one of the clusters is chosen as the sample

**Q.** multistage sampling**Ans:** sampling in stages (each stage could be a different method)

**Q.** voluntary response sampling**Ans:** the respondents themselves decide whether to be included

**Q.** comparative study**Ans:** purpose is to compare two or more alternative methods or groups distinguished by some attribute

**Q. **descriptive study**Ans: **purpose is to learn about characteristics of group(s) but not necessarily to make comparisons

**Q. **response variable**Ans:** measures and outcome of the study

**Q.** explanatory variable/factor**Ans:** variable that is throught to explain or cause the observed outcomes, explains the changes in the response variables

**Q.** levels**Ans:** possible values of an explanatory variable

**Q.** treatment**Ans:** specific combination of the levels of the explanatory variables

**Q.** observational study**Ans:** records data without interfering with the course of events

**Q.** experimental study**Ans:** the researcher actively intervenes to control the study conditions and records the responses

**Q.** prospective cohort study**Ans:** researchers identify subjects which have various explanatory variables and follow them into the future and record the responses

**Q.** retrospective cohort study**Ans:** researchers identify subjects that have experienced certain responses and look back to see if the subjects also have various explanatory variables

**Q.** cross-sectional study**Ans: **researchers identify subjects that have experienced certain responses and look back to see if the subjects also have vairous explanatory variables

**Q.** lurking variable**Ans:** a variable that has an important effect on the relationship among the variables in the study, but is not one of the explanatory variables studied

**Q.** confounded**Ans:** two variables are confounded when their effects on a response variable cannot be distinguished from each other

**Q.** experimenter bias**Ans:** the distortion that can arise on the part of the experimenter due to how the subjects are assigned to the groups, which variables are measured and how they are measured, and how the results are interpreted (bias is generally in the direction of the researcher’s theory)

**Q.** categorical/qualitative variable**Ans:** classifies the unit into one of several categories

**Q. **nominal**Ans:** categorical/qualitative variable when the categories are simply labels

**Q.** ordinal**Ans:** categorical/qualitative variable then the categories can be ordered or ranked

**Q.** numerical/quantitative variable**Ans:** takes values from a set of numbers

**Q.** continuous**Ans:** numerical/quantitative variable can take on any values within a given interval

**Q.** discrete**Ans:** numerical/quantitative variable one for which you can list the possible values (whole numbers)

**Q.** selecting the number of classes (for histogram)**Ans:** √n

**Q.** skewed right/positively skewed**Ans:** tail goes towards the right, income distributions

**Q.** skewed left/negatively skewed**Ans:** tail goes to the left

**Q.** sample median**Ans:** x with squiggly line over it, resistant

**Q.** sample mean**Ans:** x bar

**Q.** symmetric**Ans:** mean=median=mode

**Q.** bimodal**Ans:** 2 modes, mean=median

**Q.** 5 number summary**Ans:** min, Q₁, median, Q₃, max

**Q.** null hypothesis H₀**Ans:** the status quo or prevailing viewpoint about a population

**Q.** alternative hypothesis H₁**Ans:** competing hypothesis about the population – it may represent the change in the population that the researcher believes is true

**Q. **left-tailed alternative hypothesis**Ans:** H₁ : X<x

**Q.** right-tailed alternative hypothesis**Ans:** H₁ : X>x

**Q.** two-tailed alternative hypothesis**Ans:** states that the parameter is less than or greater than the value specified by the null hypothesis, i.e. H₁ : X≠x

**Q.** statistically significant**Ans:** unlikely to be observed under the assumption that H₀ is true→reject H₀

**Q.** type I error**Ans:** rejecting H₀ when in fact it is true

**Q.** type II error**Ans:** accepting H₀ when in fact it is not true

**Q.** α**Ans:** level of significance (0.05), the chance of a type I error occurring

**Q.** β**Ans:** the chance of a type II error occurring

**Q.** decision rules**Ans:** formal rule that states, based on the data obtained, when to reject H₀, generally it specifies a set of values, called the critical region, which are contradictory to H₀ and that favor H₁

**Q.** direction of extreme**Ans:** corresponds to the position of the values that are more likely under H₁ than under H₀ (larger→right/smaller→left)

**Q.** critical region**Ans:** the set of values for which you would reject H₀

**Q.** critical value**Ans:** the value which marks the starting point of the set of values that comprise the critical/rejection region

**Q.** one-sided**Ans:** a critical region is called one-sided if its set of extreme values are all in one direction

**Q.** two-sided**Ans:** a critical region is called two-sided if its set of extreme values are in two directions

**Q.** p-value**Ans:** the probability, computed under the assumption that H₀ is true, of getting the observed value plus the chance of getting all of the more extreme values

smaller p-value→stronger evidence against H₀

**Q.** p-value ≤ α**Ans:** the data are statistically significant at the given level α and we reject H₀

**Q.** p-value > α**Ans:** the data are not statistically significant at the given level α and we do not reject H₀

**Q.** permutation**Ans:** order matters

**Q.** combination**Ans:** for groups, order doesn’t matter

### ALEKS Probability & Statistics Exam QnA

Below are few sample questions & answers asked in ALEKS previous exams:

**Q. **A carpet store allows customers to choose carpet based on color, thread type, and thickness. If a customer can choose among 5 colors, 3 thread types, and 3 thicknesses, how many different types of carpet could the customer choose?**Ans:** 45

**Q. **Horace owns 9 science books, 12 poetry books, and 6 novels. How many total books does he own?**Ans:** 27

**Q. **Evaluate the following expression. 7!4!3!**Ans:** 35

**Q. **There are 12 candidates running for any of 6 distinct positions in a school’s student council. In how many different ways could the 6 positions be filled?**Ans:** 12!/6! = 665280

**Q. **How many distinct rearrangements of the letters in “tweedledee” are there? Give your answer as an integer.**Ans:** 15120

**Q. **A track race has 6 participants. In how many orders could the runners possibly finish?**Ans:** 720

**Q. **Victor will choose 6 colors to use for an art project. If there are 12 colors to choose from, how many different color combinations are possible?**Ans:** 924

**Q. **A car dealership offers 10 different upgrades with each of the vehicles it sells. If a customer can choose any number of upgrades offered, how many upgrade combinations are possible?**Ans:** 1024

**Q. **A particular 3rd grade class has 20 students. When the teacher checks attendance for the day, 3 are sick and 1 spends the day in the principal’s office. Assuming this is the normal occurrence for each day, what is the empirical probability that a student is not in class? Write your answer as an exact fraction which is reduced as much as possible.**Ans:** 0.2

**Q. **A bag of marbles contains 6 green marbles, 10 purple marbles, and 8 yellow marbles. What is the probability that a marble picked at random is green? Give your answer as a simplified fraction.**Ans:** 1/4

**Q. **A pet store has the top 8 ranked dog toys. What are the chances that your dog chooses the top-ranked dog toy first and the second top-ranked dog toy second?

Write your answer as an exact fraction which is reduced as much as possible.**Ans:** 1/56

**Q. **An elementary school art class teacher plans to display artwork next to the door of each of the classrooms in the school. Each classroom door will only have one piece of artwork displayed, and the school has 37 such doors. If the teacher has 22 sculptures and 22 oil paintings, what is the probability that 20 sculptures and 17 oil paintings are chosen to be displayed?**Ans:** (22/20)(22/17)/(44/37)

**Q. **A spinner contains the numbers 1 through 50. What is the probability that the spinner will land on a number that is not a multiple of 6? Give your answer as a simplified fraction.**Ans:** 21/25

**Q. **A coin is tossed 6 times. Find the probability that a heads is flipped at least once. Round your answer to two decimal places.**Ans:** 0.98

**Q. **A person chooses a single state from all 50 states in the United States. Find the set representing the event E that state starts with an H.**Ans:** Hawaii

**Q. **A person chooses 1 odd number between 3 to 9 inclusive. What is the sample space in this experiment?**Ans:** S= { 3, 4, 5, 6, 7, 8, 9}

**Q. **If a card is picked at random from a standard 52-card deck, what is the probability of getting a black card or a Queen?**Ans:** 7/13

**Q. **Suppose you took a poll of kids that like cookies. Out of 10 kids, 7 like cookies. You put each of their names in a bag and randomly draw a name, then replace the name and draw again. If you pull 3 names from the bag and replace them after each draw, what is the probability each name drawn likes cookies? Round to the nearest thousandths.**Ans:** 0.343

**Q. **C and D are two events with P(C)=0.4,P(D)=0.5, and P(C|D)=0.4. Calculate P(D|C). Give your answer as a decimal to the tenths place.**Ans:** 0.5

**Q. **A survey of households owned by couples was done. It found the number of children couples had in each household. The following probability distribution of the random variable X was created from that survey. What is the probability of more than two children in a household?

X-valueP(X)-00.1310.420.3130.1140.05

Write your answer as a decimal.**Ans:** 0.16

**Q. **Let N be a random variable defined as the number of days of paid vacation for people in the U.S. Let’s suppose A statistician comes up with the following probability function for N:

fN(n)={0.15 for n=0,0.15 for n=5,0.35 for n=8,0.1 for n=10, and 0.25 for n=12}.

What is the expected value of N? Give your answer as a decimal.**Ans:** 7.55

**Q. **True or False? In reference to different sampling methods, systematic sampling includes the steps: divide the population into groups; use simple random sampling to identify a proportionate number of individuals from each group.**Ans:** False

**Q. **Karen wants to estimate the mean number of siblings for each student in her school. She records the number of siblings for each of 200 randomly selected students in the school. What is the sample?**Ans:** 200 randomly selected students in school.

**Q. **Select the scenario below that demonstrates sampling bias.**Ans:** Elizabeth wants to estimate the mean height of students at her school. She collects data by selecting a random group of students within her classroom.

**Q. **Karen collected data from a random sample of 1200 homeowners in her state asking whether or not they use electric heat. Based on the results, she reports that 50% of the homeowners in the nation use electric heat. Why is this statistic misleading?**Ans:** The sample is biased.

**Q. **ValueFrequency07152332435560798196104**Ans: **20

**Q. **Given the following histogram for a set of data, how many values in the data set are between 8.5 and 11.5?**Ans: **22

**Q. **Find the median of the numbers in the following list.

26,13,39,22,42,39,18,21,32,14**Ans: **24

**Q. **Alice sells boxes of candy at the baseball game and wants to know the mean number of boxes she sells. The numbers for the games so far are listed below.

16,14,14,21,15**Ans: **16

**Q. **Given the frequency table below for a list of data, find the mean of the data.

ValueFrequency111125133141152163174185**Ans: **15

**Q. **Find the mode of the following list of numbers.

8,8,13,13,15,2,15,8,8,2**Ans: **8

**Q. **The set {14,5,5,8,7,6,2,3,7,11} represents the weight in pounds for ten pigs. What is the midrange of this set? Round your answer to decimal to two places.**Ans: **3

**Q. **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?32, 33, 36, 36, 32, 31, 32, 32, 34, 34**Ans: **Mean

**Q. **Which of the data sets represented by the following box and whisker plots has the smallest standard deviation?**Ans: **C

**Q. **Find the sample variance of the following set of data:4, 12, 7, 5.**Ans: **12.7

**Q. **Two students, Stephanie and Maria, want to find out who has the higher GPA when compared to each of their schools. Stephanie has a GPA of 3.85, and her school has a mean GPA of 3.1 and a standard deviation of 0.4. Maria has a GPA of 3.8, and her school has a mean of 3.05 and a standard deviation of 0.2. Who has the higher GPA when compared to each of their schools?**Ans: **Maria

**Q. **Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.**Ans:**

B has larger mean.

A has the larger standard deviation.

**Q. **Given the following box-and-whisker plot, decide if the data is skewed or symmetrical.**Ans: **The data are skewed to the right.

**Q. **Suppose X∼N(8,1.5), and x=5. Find and interpret the z-score of the standardized normal random variable.**Ans: **This means that x=5 is two standard deviations (2σ) below or to the left of the mean, μ=8.

**Q. **Porter’s points per basketball game are normally distributed with a standard deviation of 6 points. If Porter scores 50 points, and the z-score of this value is 4, then what is his mean points in a game? Do not include the units in your answer. For example, if you found that the mean is 33 points, you would enter 33.**Ans: **26

**Q. **In studying the relationship between average daily temperature and time spent on a telephone, suppose you computed r=0.607 using n=31 data points. Using the critical values table below, determine if the value of r is significant or not.**Ans: **r is not significant because it is between the positive and negative critical values.

**Q. **The scatter plot below shows data relating IQ and the number of hours people spend watching TV each week. Which of the following patterns does the scatter plot show?**Ans: **Negative linear pattern with deviations

**Q. **The table shows data collected on the relationship between the average daily temperature and time spent watching television. The line of best fit for the data is y^=−0.68x+85.1. According to the line of best fit, what would be the predicted number of minutes spent watching television for an average daily temperature of 32 degrees? Is it reasonable to use this line of best fit to make this prediction?**Ans: **The predicted number of minutes spent watching television is 63.34, and the estimate is not reasonable.

**Q. **Find the linear regression line for the following table of values. You will need to use a calculator, spreadsheet, or statistical software.**Ans: **xy13.8226.0636.6246.5354.5169.36

*Note: Probability & Statistics questions in ALEKS are prepared by professional mathematicians whose areas of expertise give the program the ability to ask questions that are both challenging and beneficial to the learning process*.

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