Stats Finals Chapter 6-8 Flashcards

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?

True Signup and view all the answers

What is the expected number of defective syringes the inspector will find?

0.06 syringes Signup and view all the answers

What is the probability that the batch will be accepted?

0.999 Signup and view all the answers

What is the expected number of friends for whom addresses will be found?

4.8 friends Signup and view all the answers

What is the probability none of the tsunamis are nine meters or higher?

0.168 Signup and view all the answers

What is the probability at least one tsunami is nine meters or higher?

0.832 Signup and view all the answers

What is the expected number of tsunamis nine meters or higher?

1.5 Signup and view all the answers

What is the standard deviation of the r-probability distribution of tsunamis?

1.025 Signup and view all the answers

What is the expected number of vehicles out of 7 that will tailgate?

2.1 vehicles Signup and view all the answers

What is the standard deviation of this distribution of tailgating?

1.212 vehicles Signup and view all the answers

What percentage of the area under the normal curve lies to the left of μ?

50% Signup and view all the answers

What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?

99.7% Signup and view all the answers

Is the distribution valid? (a) P(x) 0.20 0.61 0.19, (b) P(x) 0.20 0.61 0.29

Yes. The probabilities sum to 1. Signup and view all the answers

What does a standard score measure?

the number of standard deviations a measurement is from the mean Signup and view all the answers

The expected value of the distribution is necessarily one of the possible values of x.

False Signup and view all the answers

Sketch the area under the standard normal curve over the indicated interval and find the specified area.

The area to the right of z = 0 is 0.5, The area to the left of z = 0 is 0.5 Signup and view all the answers

Compute the expected value of the distribution for P(x) 0.25 0.70 0.05.

4/5 Signup and view all the answers

The expected age of a super shopper is ____.

42.58 yr Signup and view all the answers

What is the standard deviation for ages of super shoppers?

13.25 Signup and view all the answers

What is the total expected cost to Big Rock Insurance for Jim's policy?

$4234 Signup and view all the answers

What is the probability that Kevin will win the cruise?

0.00811 Signup and view all the answers

The outcome of one trial affects the probability of success on any other trial.

False Signup and view all the answers

Which outcomes are possible in a binomial experiment?

Both A and B Signup and view all the answers

What does the expected value of a binomial distribution with n trials tell you?

The average number of successes Signup and view all the answers

In a binomial experiment, the probability of success can change from one trial to the next.

False Signup and view all the answers

Which of the following is NOT a characteristic of a binomial experiment?

Probability of success varies Signup and view all the answers

What is the area to the left of z = -1.32?

0.0934 Signup and view all the answers

What is the area to the left of z = -0.40?

0.3446 Signup and view all the answers

What is the area to the left of z = 0.54?

0.7054 Signup and view all the answers

What is the area to the left of z = 0.78?

0.7823 Signup and view all the answers

What is the area to the right of z = 1.63?

0.0516 Signup and view all the answers

What is the area to the right of z = -1.14?

0.8729 Signup and view all the answers

What is the area to the right of z = -2.12?

0.983 Signup and view all the answers

What is the area between z = 0 and z = 2.46?

0.4931 Signup and view all the answers

What is the area between z = 0 and z = -1.97?

0.4756 Signup and view all the answers

What is the area between z = -2.29 and z = 1.33?

0.8972 Signup and view all the answers

What is the area between z = -1.46 and z = 1.93?

0.9011 Signup and view all the answers

What is the area between z = 0.31 and z = 1.84?

0.3454 Signup and view all the answers

What is the area between z = 1.32 and z = 2.15?

0.0776 Signup and view all the answers

What is the probability that P(z ≤ 0) for a standard normal variable?

0.5 Signup and view all the answers

What is the probability that P(z ≤ -0.12) for a standard normal variable?

0.4522 Signup and view all the answers

What is the probability that P(z ≤ -2.04) for a standard normal variable?

0.0207 Signup and view all the answers

What is the probability that P(z ≤ 1.24) for a standard normal variable?

0.8925 Signup and view all the answers

What is the probability that P(z ≥ 1.44) for a standard normal variable?

0.0749 Signup and view all the answers

What is the probability that P(z ≥ 2.09) for a standard normal variable?

0.0183 Signup and view all the answers

What is the probability that P(z ≥ -1.27) for a standard normal variable?

0.8979 Signup and view all the answers

What is the probability that P(-1.10 ≤ z ≤ 2.64) for a standard normal variable?

0.8602 Signup and view all the answers

What is the probability that P(-2.02 ≤ z ≤ 1.08) for a standard normal variable?

0.8382 Signup and view all the answers

What is the probability that P(-2.13 ≤ z ≤ -0.36) for a standard normal variable?

0.3428 Signup and view all the answers

What is the z-score corresponding to x = 19 for a normal distribution with mean 𝜇 = 24 and standard deviation 𝜎 = 5?

-1 Signup and view all the answers

What is the z-score corresponding to x = 37 for a normal distribution with mean 𝜇 = 24 and standard deviation 𝜎 = 5?

2.6 Signup and view all the answers

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?

168.74 Signup and view all the answers

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?

175.54 Signup and view all the answers

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?

117.06 Signup and view all the answers

What is the final exam score for Joel if the mean score is 𝜇 = 148?

148 Signup and view all the answers

What is the final exam score for Jan if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17?

135.08 Signup and view all the answers

What is the final exam score for Linda if the mean score is 𝜇 = 148 and standard deviation 𝜎 = 17?

178.6 Signup and view all the answers

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.

z < -0.29 Signup and view all the answers

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.

z > -2.94 Signup and view all the answers

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.

0.88 < z < 1.76 Signup and view all the answers

If a fawn weighs 14 kilograms, would you say it is an unusually small animal?

Yes. Signup and view all the answers

If a fawn is unusually large, what kind of z value would you expect?

It would have a large positive z, such as 3. Signup and view all the answers

What is the probability of selecting a value greater than 26 from a normal distribution with mean 26 and standard deviation 7?

0.5 Signup and view all the answers

If 90% of the area under the standard normal curve lies to the right of z, then z is positive.

False Signup and view all the answers

If 25% of the area under the standard normal curve lies to the left of z, then z is positive.

False Signup and view all the answers

For a normal distribution with mean 𝜇 = 8 and standard deviation 𝜎 = 2, what is P(7 ≤ x ≤ 10)?

0.5328 Signup and view all the answers

For a normal distribution with mean 𝜇 = 16.0 and standard deviation 𝜎 = 4.5, what is P(10 ≤ x ≤ 26)?

0.895 Signup and view all the answers

For a normal distribution with mean 𝜇 = 40 and standard deviation 𝜎 = 15, what is P(50 ≤ x ≤ 70)?

0.2286 Signup and view all the answers

For a normal distribution with mean 𝜇 = 5.1 and standard deviation 𝜎 = 1.7, what is P(7 ≤ x ≤ 9)?

0.1204 Signup and view all the answers

For a normal distribution with mean 𝜇 = 28 and standard deviation 𝜎 = 4.2, what is P(x ≥ 30)?

0.3156 Signup and view all the answers

For a normal distribution with mean 𝜇 = 104 and standard deviation 𝜎 = 11, what is P(x ≥ 120)?

0.0735 Signup and view all the answers

For a normal distribution with mean 𝜇 = 100 and standard deviation 𝜎 = 19, what is P(x ≥ 90)?

0.7019 Signup and view all the answers

For a normal distribution with mean 𝜇 = 2.1 and standard deviation 𝜎 = 0.33, what is P(x ≥ 2)?

0.6179 Signup and view all the answers

Find z such that 70% of the standard normal curve lies to the left of z.

z = 0.52 Signup and view all the answers

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.

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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!