Intro to Statistical Methods

pie chart

classes of the qualitative variable are represented by slices of a pie, circle. The size of each slice is proportional to the class relative frequency.

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution​ below: Price: ​$10.00, ​$12.50, ​$15.00, Probability: 0.4, 0.2, 0.4 How much should the lab budget for next​ year's frog orders assuming this distribution does not​ change? (Hint: Find the expected price and assume 52 weeks per​ year.) A. $1250 B. $13 C. $3,380,000 D. $650

$650 ( 52* [10*0.4+12.5*0.2+15*0.4] = 650)

Decide whether the following statement is true or false. In any experiment with exactly four sample points in the sample​ space, the probability of each sample point is 0.25. True False

False

Determine whether the following statement is true or false. The expected value of a discrete random variable must be one of the values in which the random variable can result. True False

False

stem and leaf display

The numerical value of the quantitative variable is partitioned into a "stem" and a "leaf." The possible stems are listed in order in a column. The leaf for each quantitative measurement in the data set is placed in the corresponding stem row. Leaves for observations with the same stem value are listed in increasing order horizontally.

histogram

The possible numerical values of the quantitative variable are partitioned into class intervals, each of which has the same width. These intervals from the scale of the horizontal axis. The frequency or relative frequency of observations in each class interval is determined. A vertical bar is placed over each class interval, with the height of the bar equal to either the class frequency or class relative frequency.

Decide whether the following statement is true or false. The probability of a sample point is usually taken to be the relative frequency of the occurrence of the sample point in a very long series of repetitions of the experiment. True False

True

pareto diagram

a bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged by height in descending order from left to right

Classify the following random variable according to whether it is discrete or continuous. The height of a player on a basketball team discrete continuous

continuous

Classify the following random variable according to whether it is discrete or continuous. The number of goals scored in a soccer game discrete continuous

discrete

class

one of the categories into which qualitative data can be classified

class relative frequency

the class frequency divided by the total number of observations in the data set (crf = class frequency/n)

class precentage

the class relative frequency multiplied by 100 (class percentage = crf x 100)

class frequency

the number of observations in the data set that fall into a particular calss

dot plot

the numerical value of each quantitative measurement in the data set is represented by a dot on a horizontal scale. When data values repeat, the dots are placed above one another vertically.

The random variable x represents the number of boys in a family with three children. Assuming that births of boys and girls are equally​ likely, find the mean and standard deviation for the random variable x. A. ​mean: 1.50; standard​ deviation: .87 B. ​mean: 2.25; standard​ deviation: .87 C. ​mean: 2.25; standard​ deviation: .75 D. ​mean: 1.50; standard​ deviation: .75

​mean: 1.50; standard​ deviation: .87 ( outcomes = {BBB, BBG, BGB, GBB, BGG, GGB, GBG, GGG}, odds = 1/8, mean = 0* 1/8 + 1 *1/8 + 2 * 1/8 + 3 * 1/8 = 1.5, sd = sqrt([0 *1/8 + 1 * 3/8 + 4 * 3/8 + 9 * 1/8] - [12/8]^2) = sqrt(3/4) = 0.87)

A human gene carries a certain disease from a mother to her child with a probability rate of 0.45. That​ is, there is a 45​% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has four children. Assume that the​ infections, or lack​ thereof, are independent of one another. Find the probability that all four of the children get the disease from their mother. Round to three decimal places. A. 0.959 B. 0.092 C. 0.075 D. 0.041

0.041 ( 4_C_4 * 0.45^4 * 0.55^0)

Transportation officials tell us that 60​% of drivers wear seat belts while driving. Find the probability that more than 562 drivers in a sample of 900 drivers wear seat belts. A. 0.937 B. 0.063 C. 0.6 D. 0.4

0.063 ( P[X>562] = ?, mean = 540, sd = 4.6476, P[X>562] = 1- P:[562.5-540]/sqrt[216] = 1 -P(z<1.53) = 1 - 0.937 = 0.063)

A​ one-week study revealed that​ 60% of a warehouse​ store's customers are women and that​ 30% of women customers spend at least​ $250 on a single visit to the store. Find the probability that a randomly chosen customer will be a woman who spends at least​ $250. A. 0.90 B. 0.18 C. 0.50 D. 0.36

0.18 (0.6*0.3 = 0.18)

Suppose that for a certain experiment P​(A​)=0.6 and P​(B​)=0.3. If A and B are independent​ events, find P(A∩B). A. 0.18 B. 0.30 C. 0.90 D. 0.50

0.18 (0.6*0.3 =0.18)

A recent article in the paper claims that business ethics are at an​ all-time low. Reporting on a recent​ sample, the paper claims that 36​% of all employees believe their company president possesses low ethical standards. Suppose 20 of a​ company's employees are randomly and independently sampled. Assuming the​ paper's claim is​ correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the​ company's president possesses low ethical standards. A. 0.243570 B. 0.165691 C. 0.335832 D. 0.427953

0.243570

Consider the given discrete probability distribution. Find the probability that x equals 4. x: 2, 4, 6, 8, P(x): 0.19, ?, 0.24, 0.08 A. 0.51 B. 0.49 C. 2.04 D. 1.96

0.49 (1 - 0.19+0.24+0.08 = 0.49)

In a box of 50​ markers, 30 markers are either red or black and 20 are missing their caps. If 12 markers are either red or black and are missing their​ caps, find the probability that a randomly selected marker is red or black or is missing its cap. A. 0.24 B. 0.38 C. 1 D. 0.76

0.76 ([30+20-12]/50 = 0.76)

Probabilities of different types of​ vehicle-to-vehicle accidents are shown below. Accident: Car to Car, Car to Truck, Truck to Truck Probability: 0.68, 0.1, 0.22 Find the probability that an accident involves a car. A. 0.68 B. 0.1 C. 0.78 D. 0.22

0.78

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 60°F to 94°F. What is the probability that the high temperature on a day in August exceeds 65°F? A. 0.8529 B. 0.4221 C. 0.1471 D. 0.0294

0.8529 ([94-65]/[94-60] = 0.8529)

Consider the given discrete probability distribution. Find P​(x≤​4). x: 0, 1, 2, 3, 4, 5 ​p(x): 0.30, 0.25, 0.20, 0.15, 0.05, 0.05 A. 0.90 B. 0.05 C. 0.95 D. 0.10

0.95

An experiment consists of rolling two dice and summing the resulting values. Which of the following is not a sample point for this​ experiment? A. 2 B. 1 C. 7 D. 6

1

The carbon dioxide emissions of a group of nations had a mean of 9.5 and standard deviation of 3.1. a. One​ country's observation was 14.9. Find and interpret its​ z-score relative to the distribution of values for the group of nations. b. Another​ country's observation was 4.8. Find and interpret its​ z-score. a. Find the​ z-score for the observation of 14.9. z= ? ​(Round to two decimal places as​ needed.) What does this​ z-score imply? A. The observation 14.9 is an outlier because it is greater than 3 standard deviations from the mean. B. The observation 14.9 is an outlier because its​ z-score is negative. C. The observation 14.9 is not an outlier because its​ z-score is positive. D. The observation 14.9 is not an outlier because it is less than 3 standard deviations from the mean. b. Find the​ z-score for the observation of 4.8. z= (Round to two decimal places as​ needed.) What does this​ z-score imply? A. The observation 4.8 is an outlier because its​ z-score is negative. B. The observation 4.8 is not an outlier because it is less than 3 standard deviations from the mean. C. The observation 4.8 is not an outlier because its​ z-score is positive. D. The observation 4.8 is an outlier because it is greater than 3 standard deviations from the mean.

1.74, The observation 14.9 is not an outlier because it is less than 3 standard deviations from the mean., -1.52, The observation 4.8 is not an outlier because it is less than 3 standard deviations from the mean.

Suppose x is a uniform random variable with c=40 and d=80. Find the standard deviation of x. A. σ = 11.547 B. σ = 1.826 C. σ = 34.641 D. σ = 3.162

11.547 (sd = [80-40]/sqrt[12] = 11.547)

Evaluate 8_C_2 A. 56 B. 28 C. 4 D. 16

28

Calculate the mean and median of the following grade point averages. 3.1, 3.3, 3.3, 3.7, 3.4, 2.6 Mean = ? Median = ?

3.23, 3.3

Compute the number of ways you can select 3 elements from 7 elements. A. 21 B. 10 C. 343 D. 35

35 (7_C_3)

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing​ process? A. 7 millimeters B. 8 millimeters C. 7.5 millimeters D. 8.5 millimeters

7.5 millimeters ([8.5-6.5]/2 = 7.5)

If sample points​ A, B,​ C, and D are the only possible outcomes of an​ experiment, find the probability of D using the table below. Sample Point A, B, C, D Probability 1/10, 1/10, 1/10, ? A. 3/10 B. 1/10 C. 7/10 D. 1/4

7/10

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 80°F to 110°F. Find the temperature which is exceeded by the high temperatures on​ 90% of the days in August. A. 90°F B. 107°F C. 83°F D. 110°F

83°F ([110-X]/[110-80] = 0.9, X = 110 -27 = 83)

A number between 1 and​ 10, inclusive, is randomly chosen. Events A​, B​, C​, and D are defined as follows. A​: ​{The number is​ even} B​: ​{The number is less than​ 7} C​: ​{The number is less than or equal to​ 7} D​: ​{The number is​ 5} Identify one pair of independent events. A. A and D B. B and D C. A and B D. A and C

A and B

Explain how populations and variables differ. Choose the correct answer below. A. A population is a set of units of interest to a study. A variable is a subset of the units of a population. B. A variable is a set of units of interest to a study. A population is a characteristic or property of the units being studied. C. A population is a set of units of interest to a study. A variable is a characteristic or property of the units being studied. D. A population is a set of units of interest to a study. A variable is an object upon which data is collected.

A population is a set of units of interest to a study. A variable is a characteristic or property of the units being studied.

What is a representative​ sample? Choose the correct answer below. A. A representative sample is a subset of the units of a population. B. A representative sample of n experimental units is a sample selected from the population in such a way that every different sample of size n has an equal chance of selection. C. A representative sample is a sample that exhibits characteristics typical of those possessed by the population of interest. D. A representative sample is a set of units of interest to a study.

A representative sample is a sample that exhibits characteristics typical of those possessed by the population of interest.

A company surveyed a random sample of 9,500 employees in the region. One question they asked​ was, "If your employer provides you with mentoring opportunities are you likely to remain in your job for the next five​ years?" They found that 630 members of the sample said yes. a. Identify the population of interest to the company. A. All people in the region B. The 630 employees who answered yes C. All employees in the region D. The 9,500 employees surveyed b. Based on the question posed by the company what is the variable of​ interest? A. If the employees are provided with mentoring opportunities Your answer is not correct. B. How long the employees are likely to remain in their job C. The number of employees surveyed D. If they answered yes to the survey question c. Is the variable quantitative or​ qualitative? A. The variable is a qualitative variable. Its values cannot be expressed on a naturally occurring numerical scale. B. The variable is a quantitative variable. Its values can be expressed on a naturally occurring numerical scale. C. The variable is a qualitative variable. Its values can be expressed on a naturally occurring numerical scale. D. The variable is a quantitative variable. Its values cannot be expressed on a naturally occurring numerical scale.

All employees in the region, If they answered yes to the survey question, The variable is a qualitative variable. Its values cannot be expressed on a naturally occurring numerical scale.

Which of the following is not a method used for determining whether data are from an approximately normal​ distribution? A. Find the interquartile​ range, IQR, and standard​ deviation, s, for the sample. Then IQRs≈1.3. B. Compute the intervals x±​s, x±​2s, and x±3s. The percentages of measurements falling in each should be approximately​ 68%, 95%, and​ 100% respectively. C. Construct a histogram or​ stem-and-leaf display. The shape of the graph or display should be uniform​ (evenly distributed). D. Construct a normal probability plot. The points should fall approximately on a straight line.

Construct a histogram or​ stem-and-leaf display. The shape of the graph or display should be uniform​ (evenly distributed)

Explain the difference between descriptive and inferential statistics. Choose the correct answer below. A. Descriptive statistics draws conclusions about the sets of data based on sampling. Inferential statistics summarizes the information revealed in data sets. B. Descriptive statistics is a characteristic or property of an individual experimental unit. Inferential statistics is the process used to assign numbers to variables of individual population units. C. Descriptive statistics describes sets of data. Inferential statistics draws conclusions about the sets of data based on sampling. D. Descriptive statistics are measurements that are recorded on a naturally occurring numerical scale. Inferential statistics are measurements that cannot be measured on a natural number​ scale; they can only be classified into one of a group of categories.

Descriptive statistics describes sets of data. Inferential statistics draws conclusions about the sets of data based on sampling.

A company tracked some credit card purchases during the year 2005 and measured two​ variables: (1) the purchase location (online, over the phone, or at a business)​, and (2) the amount (in dollars)of the monthly minimum payment. a. Identify the type​ (quantitative or​ qualitative) of each variable measured. b. Does the data set collected represent a population or a​ sample? a. Is the variable​ (1) qualitative or​ quantitative? Quantitative Qualitative Is the variable​ (2) qualitative or​ quantitative? Qualitative Quantitative b. Does the data set collected represent a population or a​ sample? Sample Population

Qualitative, Quantitative, Sample

Suppose​ you're given a data set that classifies each sample unit into one of four​ categories: A,​ B, C, or D. You plan to create a computer database consisting of these​ data, and you decide to code the data as A=​1, B=​2, C=​3, and D=4. Are the data consisting of the classifications​ A, B,​ C, and D qualitiative or​ quantitative? After the data are input as​ 1, 2,​ 3, or​ 4, are they qualitative or​ quantitative? Are the data consisting of the classifications​ A, B,​ C, and D qualitative or​ quantitative? A. ​Qualitative, because they are measured on a naturally occurring numerical scale. B. ​Quantitative, because they can only be classified into categories. C. ​Qualitative, because they can only be classified into categories. D. ​Quantitative, because they are measured on a naturally occurring numerical scale. After the data are input as​ 1, 2,​ 3, or​ 4, are they qualitative or​ quantitative? A. ​Qualitative, because they are measured on a naturally occurring numerical scale. B. ​Quantitative, because they are measured on a naturally occurring numerical scale. C. ​Qualitative, because they cannot be meaningfully​ added, subtracted,​ multiplied, or divided. D. ​Quantitative, because they cannot be meaningfully​ added, subtracted,​ multiplied, or divided.

Qualitative, because they can only be classified into categories. ​Qualitative, because they cannot be meaningfully​ added, subtracted,​ multiplied, or divided.

Identify each of the following variables as qualitative or quantitative. a. Number of sick days taken in a year b. Number of children in a family c. Eye color d. Natural hair color a. Is number of sick days taken in a year qualitative or​ quantitative? Quantitative Qualitative b. Is number of children in a family qualitative or​ quantitative? Qualitative Quantitative c. Is eye color qualitative or​ quantitative? Quantitative Qualitative d. Is natural hair color qualitative or​ quantitative? Quantitative Qualitative

Quantitative, Quantitative, Qualitative, Qualitative

bar graph

classes of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency, or class percentage

The probability that an individual is​ left-handed is 0.16. In a class of 30 ​students, what is the mean and standard deviation of the number of​ left-handed students? A. ​mean: 4.8​; standard​ deviation: 2.19 B. ​mean: 4.8​; standard​ deviation: 2.01 C. ​mean: 30​; standard​ deviation: 2.01 D. ​mean: 30​; standard​ deviation:

mean: 4.8​; standard​ deviation: 2.01 (mean = 30 * 0.16 = 4.8, sd = sqrt(4.8 * 0.16 * 0.84) = 2.01)