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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

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Social Statistics for a Diverse Society

Instructor resources, solutions to chapter exercises and spss exercises.

Solutions to all Chapter Exercises and SPSS Exercises are provided here for reference.

N ( 244 , 15 50 ) N ( 244 , 15 50 )

As the sample size increases, there will be less variability in the mean, so the interval size decreases.

X is the time in minutes it takes to complete the U.S. Census short form. X ¯ X ¯ is the mean time it took a sample of 200 people to complete the U.S. Census short form.

CI: (7.9441, 8.4559)

The level of confidence would decrease, because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases.

• x ¯ x ¯ = 2.2

X ¯ X ¯ is the mean weight of a sample of 20 heads of lettuce.

EBM = 0.07 CI: (2.1264, 2.2736)

The interval is greater, because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.

The confidence level would increase.

(24.52,36.28)

We are 95 percent confident that the true mean age for winter Foothill College students is between 24.52 and 36.28.

The error bound for the mean would decrease, because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

X is the number of hours a patient waits in the emergency room before being called back to be examined. X ¯ X ¯ is the mean wait time of 70 patients in the emergency room.

CI: (1.3808, 1.6192)

• x ¯ x ¯ = 151
• s x s x = 32
• n – 1 = 107

X ¯ X ¯ is the mean number of hours spent watching television per month from a sample of 108 Americans.

CI: (142.92, 159.08)

(2.93, 3.59)

We are 95 percent confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.

The error bound would become EBM = 0.245. This error bound decreases, because as sample sizes increase, variability decreases, and we need less interval length to capture the true mean.

It would decrease, because the z -score would decrease, which would reduce the numerator and lower the number.

X is the number of successes where the woman makes the majority of the purchasing decisions for the household. P ′ is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.

CI: (0.5321, 0.6679)

EBM : 0.0679

X is the number of successes where an executive prefers a truck. P ′ is the percentage of executives sampled who prefer a truck.

CI: (0.19432, 0.33068)

EBM : 0.0707

The sampling error means that the true mean can be 2 percent above or below the sample mean.

P ′ is the proportion of voters sampled who said the economy is the most important issue in the upcoming election.

CI: (0.62735, 0.67265);

EBM: 0.02265

the number of girls, ages 8 to 12, in the 5 p.m. Monday night beginning ice-skating class

P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 ) P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 )

CI = (0.72171, 0.87829).

(0.72; 0.88)

With 92 percent confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72 percent and 88 percent.

The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.

• X is the height of a Swedish male, and is the mean height from a sample of 48 Swedish males.
• Normal. We know the standard deviation for the population, and the sample size is greater than 30.
• CI: (70.151, 71.49)
• EBM = 0.849
• The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.
• x ¯ x ¯ = 23.6
• X is the time needed to complete an individual tax form. X ¯ X ¯ is the mean time to complete tax forms from a sample of 100 customers.
• N ( 23.6 , 7 100 ) N ( 23.6 , 7 100 ) because we know sigma.
• (22.228, 24.972)
• EBM = 1.372
• It will need to change the sample size. The firm needs to determine what the confidence level should be and then apply the error bound formula to determine the necessary sample size.
• The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.
• According to the error bound formula, the firm needs to survey 206 people. Because we increase the confidence level, we need to increase either our error bound or the sample size.
• X is the number of letters a single camper will send home. X ¯ X ¯ is the mean number of letters sent home from a sample of 20 campers.

N 7.9 ( 2.5 20 ) 7.9 ( 2.5 20 )

• CI: (6.98, 8.82)
• The error bound and confidence interval will decrease.
• x ¯ x ¯ = $568,873
• CL = 0.95, α = 1 – 0.95 = 0.05, z α 2 z α 2 = 1.96 EBM = z 0.025 σ n z 0.025 σ n = 1.96 909200 40 909200 40 = $281,764

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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Geometric Mean
• Quadratic Mean
• Probability
• Standard Deviation
• Lower Quartile
• Upper Quartile
• Interquartile Range
• Standard Normal Distribution
• Conversions

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Number line.

• arithmetic\:mean\:1,\:2,\:3,\:4,\:5,\:6
• geometric\:mean\:\left\{0.42,\:0.52,\:0.58,\:0.62\right\}
• quadratic\:mean\:-4,\:5,\:6,\:9
• median\:\:\left\{1,\:7,\:-3,\:4,\:9\right\}
• mode\:\left\{90,\:94,\:53,\:68,\:79,\:94,\:87,\:90,\:70,\:69,\:65,\:89,\:85\right\}
• minimum\:-4,\:5,\:6,\:9
• maximum\:\frac{31}{100},\:\frac{23}{105},\:\frac{31}{205},\:\frac{54}{205}
• mid\:range\:1,\:2,\:3,\:4,\:5,\:6
• range\:\:\left\{1,\:7,\:-3,\:4,\:9\right\}
• standard\:deviation\:\:\left\{1,\:7,\:-3,\:4,\:9\right\}
• variance\:1,\:2,\:3,\:4,\:5,\:6
• lower\:quartile\:-4,\:5,\:6,\:9
• upper\:quartile\:\left\{0.42,\:0.52,\:0.58,\:0.62\right\}
• interquartile\:range\:1,\:2,\:3,\:4,\:5,\:6
• midhinge\:\left\{90,\:94,\:53,\:68,\:79,\:84,\:87,\:72,\:70,\:69,\:65,\:89,\:85\right\}
• What is the best calculator for statistics?
• Symbolab offers an online calculator specifically for statistics that can perform a wide range of calculations, including standard deviation, variance, range and normal distribution. It also provides detailed step-by-step solutions.
• What is statistics?
• Statistics is the branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. There are two main branches of statistics: descriptive statistics, and inferential statistics.
• What is descriptive statistics?
• Descriptive statistics is a branch of statistics that deals with summarizing, organizing and describing data. Descriptive statistics uses measures such as central tendency (mean, median, and mode) and measures of variability (range, standard deviation, variance) to give an overview of the data.
• What is inferential statistics?
• Inferential statistics is a branch of statistics that deals with making predictions and inferences about a population based on a sample of data. Inferential statistics uses probability theory and statistical models to make predictions and inferences about a population.
• What is the difference between statistics and probability?
• Statistics is the branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data, while probability is the branch of mathematics dealing with the likelihood of occurrence of different events.

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• Lies, Damned Lies, and Statistics Statistics is about analyzing data, for instance the mean is commonly used to measure the “central tendency” of...

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