Questions and Answers
What is the probability of getting fewer than 2 days when the surf is at least 6 feet?
0.087
What is the expected number of days when the surf will be at least 6 feet?
3 days
What is the standard deviation of the r-probability distribution?
1.095 days
Can you be fairly confident that the surf will be at least 6 feet high on one of your days off?
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What is the expected number of defective syringes the inspector will find?
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What is the probability that the batch will be accepted?
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What is the expected number of friends for whom addresses will be found?
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What is the probability none of the tsunamis are nine meters or higher?
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What is the probability at least one tsunami is nine meters or higher?
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What is the expected number of tsunamis nine meters or higher?
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What is the standard deviation of the r-probability distribution of tsunamis?
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What is the expected number of vehicles out of 7 that will tailgate?
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What is the standard deviation of this distribution of tailgating?
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What percentage of the area under the normal curve lies to the left of μ?
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What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?
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Is the distribution valid? (a) P(x) 0.20 0.61 0.19, (b) P(x) 0.20 0.61 0.29
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What does a standard score measure?
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The expected value of the distribution is necessarily one of the possible values of x.
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Sketch the area under the standard normal curve over the indicated interval and find the specified area.
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Compute the expected value of the distribution for P(x) 0.25 0.70 0.05.
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The expected age of a super shopper is ____.
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What is the standard deviation for ages of super shoppers?
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What is the total expected cost to Big Rock Insurance for Jim's policy?
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What is the probability that Kevin will win the cruise?
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The outcome of one trial affects the probability of success on any other trial.
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Which outcomes are possible in a binomial experiment?
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What does the expected value of a binomial distribution with n trials tell you?
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In a binomial experiment, the probability of success can change from one trial to the next.
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Which of the following is NOT a characteristic of a binomial experiment?
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What is the area to the left of z = -1.32?
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What is the area to the left of z = -0.40?
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What is the area to the left of z = 0.54?
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What is the area to the left of z = 0.78?
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What is the area to the right of z = 1.63?
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What is the area to the right of z = -1.14?
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What is the area to the right of z = -2.12?
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What is the area between z = 0 and z = 2.46?
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What is the area between z = 0 and z = -1.97?
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What is the area between z = -2.29 and z = 1.33?
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What is the area between z = -1.46 and z = 1.93?
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What is the area between z = 0.31 and z = 1.84?
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What is the area between z = 1.32 and z = 2.15?
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What is the probability that P(z ≤ 0) for a standard normal variable?
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What is the probability that P(z ≤ -0.12) for a standard normal variable?
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What is the probability that P(z ≤ -2.04) for a standard normal variable?
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What is the probability that P(z ≤ 1.24) for a standard normal variable?
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What is the probability that P(z ≥ 1.44) for a standard normal variable?
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What is the probability that P(z ≥ 2.09) for a standard normal variable?
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What is the probability that P(z ≥ -1.27) for a standard normal variable?
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What is the probability that P(-1.10 ≤ z ≤ 2.64) for a standard normal variable?
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What is the probability that P(-2.02 ≤ z ≤ 1.08) for a standard normal variable?
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What is the probability that P(-2.13 ≤ z ≤ -0.36) for a standard normal variable?
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What is the z-score corresponding to x = 19 for a normal distribution with mean 𝜇 = 24 and standard deviation 𝜎 = 5?
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What is the z-score corresponding to x = 37 for a normal distribution with mean 𝜇 = 24 and standard deviation 𝜎 = 5?
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What is the final exam score for Robert if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17 with a z-score of 1.24?
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What is the final exam score for Juan if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17 with a z-score of 1.64?
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What is the final exam score for Haley if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17 with a z-score of -1.82?
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What is the final exam score for Joel if the mean score is 𝜇 = 148?
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What is the final exam score for Jan if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17?
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What is the final exam score for Linda if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17?
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Convert the x interval x < 30 to a z interval for a fawn weight distribution with mean 𝜇 = 29.0 and standard deviation 𝜎 = 3.4.
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Convert the x interval 19 < x to a z interval for a fawn weight distribution with mean 𝜇 = 29.0 and standard deviation 𝜎 = 3.4.
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Convert the x interval 32 < x < 35 to a z interval for a fawn weight distribution with mean 𝜇 = 29.0 and standard deviation 𝜎 = 3.4.
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If a fawn weighs 14 kilograms, would you say it is an unusually small animal?
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If a fawn is unusually large, what kind of z value would you expect?
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What is the probability of selecting a value greater than 26 from a normal distribution with mean 26 and standard deviation 7?
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If 90% of the area under the standard normal curve lies to the right of z, then z is positive.
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If 25% of the area under the standard normal curve lies to the left of z, then z is positive.
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For a normal distribution with mean 𝜇 = 8 and standard deviation 𝜎 = 2, what is P(7 ≤ x ≤ 10)?
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For a normal distribution with mean 𝜇 = 16.0 and standard deviation 𝜎 = 4.5, what is P(10 ≤ x ≤ 26)?
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For a normal distribution with mean 𝜇 = 40 and standard deviation 𝜎 = 15, what is P(50 ≤ x ≤ 70)?
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For a normal distribution with mean 𝜇 = 5.1 and standard deviation 𝜎 = 1.7, what is P(7 ≤ x ≤ 9)?
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For a normal distribution with mean 𝜇 = 28 and standard deviation 𝜎 = 4.2, what is P(x ≥ 30)?
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For a normal distribution with mean 𝜇 = 104 and standard deviation 𝜎 = 11, what is P(x ≥ 120)?
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For a normal distribution with mean 𝜇 = 100 and standard deviation 𝜎 = 19, what is P(x ≥ 90)?
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For a normal distribution with mean 𝜇 = 2.1 and standard deviation 𝜎 = 0.33, what is P(x ≥ 2)?
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Find z such that 70% of the standard normal curve lies to the left of z.
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Study Notes
Probability Distributions
- A valid probability distribution requires that the sum of probabilities equals 1.
- Example valid distribution: P(x) = {0.20, 0.61, 0.19} sums to 1.
- Example invalid distribution: P(x) = {0.20, 0.61, 0.29} sums to 1.10.
Expected Value and Standard Deviation
- The expected value is not necessarily one of the possible values in the distribution and can be a different value.
- Example of expected value calculation: For P(x) = {0.25, 0.70, 0.05}, expected value = 0.8, standard deviation is approximately 0.5099.
Age and Income Distributions of Super Shoppers
- Age distribution: 10% (18-28), 41% (29-39), 25% (40-50), 9% (51-61), 15% (62+); expected age = 42.58 years.
- Income distribution: 20% (5-15k), 15% (15-25k), 20% (25-35k), 16% (35-45k), 19% (45-55k), 10% (55k+); expected income = $32.9k.
Historical Context of Nurses' Age Distribution
- In 1851, 25,466 nurses in Great Britain were aged as follows: 5.7% (20-29), 9.1% (30-39), etc.
- Probability of a nurse being 60+ years was calculated at 36.6%.
- Expected age of nurses = 53.99 years, with a standard deviation of 13.65 years.
Parole Repeat Offenders
- Probability of one or more offenders among five parolees was calculated at 78%.
- Expected number of repeat offenders = 1.423; standard deviation = 1.1.
Continuous vs Discrete Variables
- Discrete variables count distinct outcomes (e.g., number of traffic fatalities).
- Continuous variables measure quantities (e.g., distance a golf ball travels).
Binomial Experiments
- Binomial experiments measure the number of successes over n trials, with each trial having only two outcomes.
- Trials are independent, and the probability of success remains constant across trials.
Sampling Without Replacement
- Sampling without replacement can affect the independence of trials; a hypergeometric model is more suitable when the population size is small.
Life Insurance Expected Costs
- For Jim, the total expected cost over 5 years was calculated at $4,234, requiring a premium over that to ensure profit.
- Sara's total expected cost was $2,184.
Probability of Hurricane or Tsunami Events
- Probability modeling is employed in assessing disaster risks, considering historical data and environmental factors.
Conclusion on Expectations and Standard Deviations
- The expected value gives the average outcome for a probability distribution, while standard deviation measures variability.### Tsunamis and Earthquakes
- Approximately 30% of recorded tsunamis reach a height of nine meters or more.
- Statistical profile of tsunamis in Hawaii can be inferred from recent Pacific Rim earthquakes.
- For 6 randomly selected earthquakes:
- Probability of no tsunamis reaching nine meters: 0.118
- Probability of at least one tsunami of nine meters or higher: 0.882
- Expected number of tsunamis nine meters or higher: 1.8
- Standard deviation of the probability distribution: 1.122
Illiteracy in the U.S.
- About 20% of the U.S. population is reported to be illiterate.
- For a sample of 8 randomly selected individuals:
- Mean number of illiterate individuals: 1.6
- Standard deviation: 1.131
- Expected number of illiterate individuals: 1.6
Parolee Recidivism
- Roughly 50% of all prison parolees are repeat offenders.
- For a group of 4 parolees:
- Probability of 0 successes (not reoffending): 0.0623
- Probability for 1, 2, 3 successes: 0.25, 0.375, 0.25, respectively
- Expected number of parolees not reoffending: 2
- Standard deviation: 1
Binomial Distributions
- A binomial distribution of 200 trials shows an expected value of 80 and standard deviation of 6.9.
- More than 120 successes is unusual (above 2.5 SD).
- Fewer than 40 successes is also unusual (below 2.5 SD).
- 70 to 90 successes is typical (within 2.5 SD).
Symmetry in Binomial Distributions
- A binomial distribution is symmetric when p = 0.50.
- The expected value for p = 0.50 in 10 trials: 5
- Small p leads to a right-skewed distribution; large p leads to a left-skew.
Days with Surf in Hawaii
- January surf conditions in Hawaii: 60% of days have at least 6 feet of surf.
- For 5 randomly chosen days:
- Probability of 3 or more days with surf: 0.683
- Probability of fewer than 2 days with surf: 0.087
- Standard deviation: 1.095 days
- High confidence that at least one day will have surf at least 6 feet.
Normal Distribution Properties
- Percentages of areas under the normal curve:
- 50% lies to the left of the mean.
- 68% lies within one standard deviation.
- 99.7% lies within three standard deviations.
Measurement Comparisons
- Standard scores measure the distance of a value from the mean in terms of standard deviations.
- Raw scores below mean result in negative standard scores, while those above result in positive scores.
Normal Distribution of RBC Count
- Female RBC count follows a normal distribution: mean = 4.3 million, standard deviation = 0.7 million.
- Understanding z-scores helps in comparing RBC counts to the healthy population.
Empirical Rule
- About 68% of a normally distributed data set falls within one standard deviation from the mean.
- About 95% falls within two standard deviations; nearly all (99.7%) falls within three standard deviations.### Z Interval Conversions
- For the interval 4.5 < x, the converted z interval is 0.29 < z.
- For the interval x < 4.2, the converted z interval is z < -0.14.
- For the interval 4.0 < x < 5.5, the converted z interval is -0.43 < z < 1.71.
- For the z interval z < -1.44, the converted x interval is x < 3.3.
- For the z interval 1.28 < z, the converted x interval is 5.2 < x.
- For the z interval -2.25 < z < -1.00, the converted x interval is 2.7 < x < 3.6.
RBC Count Analysis
- An RBC count of 5.9 or higher is not considered unusually high; a z score of 2.29 indicates normality.
Tree-Ring Dating at Burnt Mesa Pueblo
- Two archaeological sites produced tree-ring dates with distinct means and standard deviations.
- Site 1: Mean date (μ1) = year 1294, Standard deviation (σ1) = 30 years.
- Site 2: Mean date (μ2) = year 1149, Standard deviation (σ2) = 43 years.
- Object from Site 1 dated x1 = year 1175; calculated z1 = -3.97.
- Object from Site 2 dated x2 = year 1200; calculated z2 = 1.19.
- x1 (Site 1) is deemed more unusual due to its further z value from zero.
Normal Distribution Probability Insights
- In a normal distribution where the mean is 26 and standard deviation is 7, the probability of selecting a value greater than 26 is 0.5.
- If 90% of the area under the normal curve lies to the right of z, then z is negative.
- If 25% of the area lies to the left of z, then z is also negative.
Probability Calculations for Various Normal Distributions
- P(7 ≤ x ≤ 10) given μ = 8 and σ = 2 is 0.5328.
- P(10 ≤ x ≤ 26) given μ = 16.0 and σ = 4.5 is 0.895.
- P(50 ≤ x ≤ 70) given μ = 40 and σ = 15 is 0.2286.
- P(7 ≤ x ≤ 9) given μ = 5.1 and σ = 1.7 is 0.1204.
- P(x ≥ 30) given μ = 28 and σ = 4.2 is 0.3156.
- P(x ≥ 120) given μ = 104 and σ = 11 is 0.0735.
- P(x ≥ 90) given μ = 100 and σ = 19 is 0.7019.
- P(x ≥ 2) given μ = 2.1 and σ = 0.33 is 0.6179.
Finding z Values
- The z value such that 70% of the standard normal curve lies to the left is z = 0.52.
Studying That Suits You
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Test your understanding of probability distributions with these flashcards covering Chapters 6 to 8 of your statistics course. Determine the validity of given distributions and explore expected values for random variables. Perfect for final exam preparation!