Step-by-Step Guidance for Statistics Test 2 Review (Probability Distributions & Normal Curves)

Q1. Create a probability distribution for the number of wireless devices per household using the given table.

Background

Topic: Discrete Probability Distributions

This question tests your ability to construct a probability distribution from frequency data. You will convert the counts of households into probabilities for each possible number of wireless devices.

Key Terms and Formulas:

• Probability Distribution: A table that lists each possible value of a random variable and its probability.

• Probability Formula: $P(x) = \frac{\text{Number of households with } x \text{ devices}}{\text{Total number of households}}$

Step-by-Step Guidance

1. Add up the total number of households by summing all the frequencies given for each number of devices.

2. For each value of wireless devices (0, 1, 2, 3, 4, 5), calculate the probability by dividing the number of households with that many devices by the total number of households.

3. Fill in a table with each value of wireless devices and its corresponding probability.

4. Check that the sum of all probabilities is 1 (or very close, due to rounding).

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Q2. Use the discrete probability distribution for number of cats owned to answer the following:

Background

Topic: Discrete Probability Distributions, Mean, and Standard Deviation

This question asks you to interpret a probability distribution, calculate probabilities for certain events, and find the mean and standard deviation of a discrete random variable.

Key Terms and Formulas:

• Probability of an event: Add up the probabilities for the relevant values of $x$.

• Mean (Expected Value): $\mu = E(x) = \sum [x \cdot P(x)]$

• Standard Deviation: $\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$

Step-by-Step Guidance

1. For part (a), identify the probability associated with $x = 2$ from the table.

2. For part (b), add the probabilities for $x = 1, 2, 3$ to find the probability a person owns between 1 and 3 cats, inclusive.

3. For part (c), add the probabilities for $x = 3$ and $x = 4$ to find the probability a person owns at least 3 cats.

4. For part (d), calculate the mean using $\mu = \sum [x \cdot P(x)]$ by multiplying each $x$ value by its probability and summing the results.

5. For part (e), use the mean from part (d) to calculate the standard deviation: $\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$.

6. For part (f), recall that the expected value is the same as the mean you calculated in part (d).

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Q3. The five-year success rate of kidney transplant surgery from living donors is 86%. The surgery is performed on sixteen patients.

Background

Topic: Binomial Probability Distribution

This question tests your understanding of the binomial distribution, including how to find the mean and standard deviation, and how to justify that a scenario fits the binomial model.

Key Terms and Formulas:

• Binomial Distribution: Used when there are a fixed number of independent trials, each with two possible outcomes (success/failure), and the probability of success is constant.

• Mean: $\mu = n p$

• Standard Deviation: $\sigma = \sqrt{n p (1-p)}$

Step-by-Step Guidance

1. Identify the number of trials ($n = 16$) and the probability of success ($p = 0.86$).

2. For part (a), use the formula $\mu = n p$ to set up the calculation for the mean number of successful surgeries.

3. For part (b), use the formula $\sigma = \sqrt{n p (1-p)}$ to set up the calculation for the standard deviation.

4. For part (c), list the five properties of a binomial experiment and briefly explain how this scenario meets each one (fixed number of trials, only two outcomes, constant probability, independent trials, and each trial is identical).

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Q4. For a standard normal curve, find the area to the right of $z = -1.68$.

Background

Topic: Standard Normal Distribution (Z-scores)

This question tests your ability to use the standard normal table (z-table) to find areas (probabilities) under the normal curve.

Key Terms and Formulas:

• Z-score: The number of standard deviations a value is from the mean.

• Standard Normal Table: Gives the area to the left of a given z-score.

Step-by-Step Guidance

1. Look up the area to the left of $z = -1.68$ in the standard normal table.

2. Since you want the area to the right, subtract the area you found from 1: $P(Z > -1.68) = 1 - P(Z < -1.68)$.

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Q5. For the standard normal curve, find the area to the left of $z = 2.23$.

Background

Topic: Standard Normal Distribution (Z-scores)

This question tests your ability to use the z-table to find the cumulative probability to the left of a given z-score.

Key Terms and Formulas:

• Z-score

• Standard Normal Table

Step-by-Step Guidance

1. Find $z = 2.23$ in the standard normal table.

2. The value you find is the area to the left of $z = 2.23$.

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Q6. For the standard normal curve, find the area between $z = -0.47$ and $z = 2.39$.

Background

Topic: Standard Normal Distribution (Z-scores)

This question tests your ability to find the probability between two z-scores using the standard normal table.

Key Terms and Formulas:

• Area between two z-scores: $P(a < Z < b) = P(Z < b) - P(Z < a)$

Step-by-Step Guidance

1. Find the area to the left of $z = 2.39$ using the z-table.

2. Find the area to the left of $z = -0.47$ using the z-table.

3. Subtract the area for $z = -0.47$ from the area for $z = 2.39$ to get the area between the two z-scores.

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Q7. In a standardized IQ test, scores are normally distributed with a mean of 100 and a standard deviation of 15.

Background

Topic: Normal Distribution, Z-scores, and Percentiles

This question tests your ability to use the normal distribution to find probabilities and cutoff scores for given percentiles.

Key Terms and Formulas:

• Z-score formula: $z = \frac{x - \mu}{\sigma}$

• Normal Distribution Table

Step-by-Step Guidance

1. For each part, convert the IQ score(s) to z-scores using $z = \frac{x - 100}{15}$.

2. Use the z-table to find the area (probability) corresponding to each z-score.

3. For probabilities "higher than" or "lower than" a value, use $1 - P(Z < z)$ or $P(Z < z)$ as appropriate.

4. For cutoff scores (percentiles), find the z-score that corresponds to the desired percentile, then solve for $x$ using $x = \mu + z \sigma$.

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Q8. Find the area to the left of $z = 0.33$ for a standard normal curve.

Background

Topic: Standard Normal Distribution

This question tests your ability to use the z-table to find the cumulative probability to the left of a given z-score.

Key Terms and Formulas:

• Z-score

• Standard Normal Table

Step-by-Step Guidance

1. Look up $z = 0.33$ in the standard normal table.

2. The value you find is the area to the left of $z = 0.33$.

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Q9. Find the area to the right of $z = -0.57$ for a standard normal curve.

Background

Topic: Standard Normal Distribution

This question tests your ability to find the area to the right of a given z-score using the z-table.

Key Terms and Formulas:

• Area to the right: $P(Z > z) = 1 - P(Z < z)$

Step-by-Step Guidance

1. Find the area to the left of $z = -0.57$ using the z-table.

2. Subtract this value from 1 to get the area to the right.

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Q10. Find the area between $z = 0.51$ and $z = 2.68$ for a standard normal curve.

Background

Topic: Standard Normal Distribution

This question tests your ability to find the probability between two z-scores using the standard normal table.

Key Terms and Formulas:

• Area between two z-scores: $P(a < Z < b) = P(Z < b) - P(Z < a)$

Step-by-Step Guidance

1. Find the area to the left of $z = 2.68$ using the z-table.

2. Find the area to the left of $z = 0.51$ using the z-table.

3. Subtract the area for $z = 0.51$ from the area for $z = 2.68$ to get the area between the two z-scores.

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Q11. Yearly amounts of black carbon emissions from cars in India are normally distributed with a mean of 14.7 gigagrams and a standard deviation of 11.5 gigagrams. Find the probability that the amount of black carbon emissions for a randomly selected year is...

Background

Topic: Normal Distribution, Z-scores

This question tests your ability to use the normal distribution to find probabilities for values less than, greater than, or between certain values.

Key Terms and Formulas:

• Z-score formula: $z = \frac{x - \mu}{\sigma}$

Step-by-Step Guidance

1. For each part, convert the relevant gigagram value(s) to z-scores using $z = \frac{x - 14.7}{11.5}$.

2. Use the z-table to find the area (probability) corresponding to each z-score.

3. For "less than" or "more than" questions, use the area to the left or right as appropriate.

4. For "between" questions, subtract the area for the lower value from the area for the higher value.

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Q12. Find the z-score that corresponds to the 46th percentile for a standard normal curve.

Background

Topic: Standard Normal Distribution, Percentiles

This question tests your ability to use the z-table in reverse: finding the z-score that corresponds to a given cumulative probability (percentile).

Key Terms and Formulas:

• Percentile: The area to the left of the z-score.

Step-by-Step Guidance

1. Convert the percentile to a decimal (e.g., 46% = 0.46).

2. Look up 0.46 in the body of the z-table to find the closest value, then find the corresponding z-score.

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Q13. Find the z-score that has 30.5% area to the right for a standard normal curve.

Background

Topic: Standard Normal Distribution, Percentiles

This question tests your ability to find the z-score for a given area to the right (which means you need to find the area to the left first).

Key Terms and Formulas:

• Area to the right: $P(Z > z)$

• Area to the left: $P(Z < z) = 1 - P(Z > z)$

Step-by-Step Guidance

1. Subtract 0.305 from 1 to get the area to the left: $1 - 0.305 = 0.695$.

2. Look up 0.695 in the body of the z-table to find the closest value, then find the corresponding z-score.

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Q14. On a dry surface, the braking distance of a sedan is normally distributed with a mean of 132 ft and a standard deviation of 4.53 ft.

Background

Topic: Normal Distribution, Percentiles, Cutoff Scores

This question tests your ability to find cutoff values (scores) for given percentiles using the normal distribution.

Key Terms and Formulas:

• Z-score formula: $z = \frac{x - \mu}{\sigma}$

• Solving for x: $x = \mu + z \sigma$

Step-by-Step Guidance

1. For each part, determine the percentile (e.g., bottom 20% = 0.20, top 15% = 0.85 for the cutoff point).

2. Use the z-table to find the z-score corresponding to the desired percentile.

3. Plug the z-score into $x = 132 + z \times 4.53$ to find the cutoff braking distance.

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