Step-by-Step Guidance for Introductory Statistics Practice Problems (Chapters 6–8)

Q1. Use the following probability distribution for parts a–e:

x	P(x)
1	0.12
2	0.09
3	0.73
4	0.02
5	?

• a) Find the missing probability.

• b) What is the probability that x = 3?

• c) What is the probability that x is more than 2?

• d) What is the probability that x is at least 4?

• e) Calculate the mean and standard deviation.

Background

Topic: Probability Distributions

These questions test your understanding of discrete probability distributions, including how to find missing probabilities, calculate probabilities for specific events, and compute the mean and standard deviation of a distribution.

Key Terms and Formulas

• Probability Distribution: A table showing all possible values of a random variable and their probabilities.

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

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

• Sum of Probabilities: $\sum P(x) = 1$

Step-by-Step Guidance

1. For part (a), recall that the sum of all probabilities in a distribution must equal 1. Add up the given probabilities and subtract from 1 to find the missing value.

2. For part (b), simply identify the probability associated with $x = 3$ from the table.

3. For part (c), "more than 2" means $x = 3, 4, 5$. Add the probabilities for these values.

4. For part (d), "at least 4" means $x = 4$ or $x = 5$. Add the probabilities for these values.

5. For part (e), use the formulas for mean and standard deviation. First, calculate the mean by multiplying each $x$ value by its probability and summing. Then, use the mean to calculate the standard deviation.

Try solving on your own before revealing the answer!

Final Answers:

• a) Missing probability for $x = 5$ is $0.04$.

• b) $P(x = 3) = 0.73$.

• c) $P(x > 2) = 0.73 + 0.02 + 0.04 = 0.79$.

• d) $P(x \geq 4) = 0.02 + 0.04 = 0.06$.

• e) Mean $\mu = 2.94$, Standard deviation $\sigma \approx 0.77$.

Each answer uses the probability distribution rules and formulas for mean and standard deviation.

Q2. In the game of roulette, a player can place a $5 bet on slot #17 and have a 1/38 probability of winning. If the metal ball lands in slot #17 the player wins $175. Otherwise, the casino takes the player’s $5. Calculate the player’s expected winnings.

Background

Topic: Expected Value

This question tests your ability to calculate expected value for a discrete random variable based on possible outcomes and their probabilities.

Key Terms and Formulas

• Expected Value: $E(X) = \sum [\text{Winnings} \times \text{Probability}]$

Step-by-Step Guidance

1. Identify the two possible outcomes: win ($175$) and lose ($-5$).

2. Assign probabilities: win ($\frac{1}{38}$), lose ($\frac{37}{38}$).

3. Multiply each outcome by its probability.

4. Add the products to find the expected value.

Try solving on your own before revealing the answer!

Final Answer: Expected winnings = -$0.53

$E(X) = (175 \times \frac{1}{38}) + (-5 \times \frac{37}{38}) = -0.53$

The expected value is negative, meaning the player loses on average per bet.

Q3. In a sample of 30 teenagers, 70% of them had MP3 music players by the age of 16. Calculate the following probabilities:

• a) Exactly 18 had an MP3 player by the age of 16.

• b) Less than 17 had an MP3 player by the age of 16.

• c) At least 14 had an MP3 player by the age of 16.

Background

Topic: Binomial Probability

These questions test your ability to use the binomial probability formula to calculate probabilities for a fixed number of trials and a constant probability of success.

Key Terms and Formulas

• Binomial Probability: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Where $n = 30$, $p = 0.70$, $k$ is the number of successes.

Step-by-Step Guidance

1. For part (a), use the binomial formula with $k = 18$.

2. For part (b), sum the binomial probabilities for $k = 0$ to $k = 16$.

3. For part (c), sum the binomial probabilities for $k = 14$ to $k = 30$.

4. Use a calculator or statistical software for the computations, as the binomial formula can be tedious for large $n$.

Try solving on your own before revealing the answer!

Final Answers:

• a) $P(X = 18) \approx 0.117$

• b) $P(X < 17) \approx 0.162$

• c) $P(X \geq 14) \approx 0.999$

These probabilities are calculated using the binomial formula with $n = 30$ and $p = 0.70$.

Q4. In a sample of 450 bananas, the probability that a banana is overripe is 6%.

• a) Calculate the mean and standard deviation for the number of overripe bananas in this sample.

• b) Suppose that a produce manager found 35 overripe bananas in the sample of 450. Is this unusual based on the mean and standard deviation? Explain.

Background

Topic: Binomial Distribution (Mean and Standard Deviation)

This question tests your ability to calculate the mean and standard deviation for a binomial distribution and interpret whether an observed value is unusual.

Key Terms and Formulas

• Mean: $\mu = n \cdot p$

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

• Unusual: Typically, a value more than 2 standard deviations from the mean is considered unusual.

Step-by-Step Guidance

1. For part (a), identify $n = 450$ and $p = 0.06$.

2. Calculate the mean using $\mu = n \cdot p$.

3. Calculate the standard deviation using $\sigma = \sqrt{n \cdot p \cdot (1-p)}$.

4. For part (b), compare 35 to the mean and standard deviation. Calculate how many standard deviations away 35 is from the mean.

5. Determine if 35 is more than 2 standard deviations from the mean to assess if it's unusual.

Try solving on your own before revealing the answer!

Final Answers:

• a) Mean $\mu = 27$, Standard deviation $\sigma \approx 5$

• b) 35 is 1.6 standard deviations above the mean, so it is not considered unusual.

Values more than 2 standard deviations from the mean are typically considered unusual.

Q5. Is the following probability experiment binomial?

• a) A random sample of 30 cars in a used lot is obtained and their mileages are recorded.

• b) A poll of 100 registered voters is conducted in which the respondents are asked whether they believe Congress should reform Social Security.

Background

Topic: Binomial Experiments

This question tests your understanding of the criteria for a binomial experiment: fixed number of trials, only two possible outcomes, constant probability, and independence.

Key Terms and Formulas

• Binomial Experiment Criteria:

  • Fixed number of trials ($n$)

  • Each trial has only two possible outcomes (success/failure)

  • Probability of success is constant

  • Trials are independent

Step-by-Step Guidance

1. For part (a), consider whether recording mileage fits the binomial criteria (are there only two outcomes?).

2. For part (b), consider whether asking a yes/no question fits the binomial criteria.

3. Review each scenario against the four binomial experiment criteria.

Try solving on your own before revealing the answer!

Final Answers:

• a) Not binomial (mileage is a quantitative variable, not two outcomes).

• b) Binomial (yes/no response fits two outcomes).

Only experiments with two possible outcomes per trial are binomial.

Q6. Calculate the following z-scores:

• a) Find the z-score so that the area under the standard normal curve to its right is 0.1056.

• b) Find the z-score so that the area under the standard normal curve to its left is 0.2655.

• c) Find the z-scores that separate the middle 10% of the standard normal curve.

Background

Topic: Standard Normal Distribution and z-scores

These questions test your ability to use the standard normal table to find z-scores corresponding to given areas (probabilities).

Key Terms and Formulas

• z-score: Number of standard deviations from the mean.

• Standard Normal Table: Used to find areas/probabilities for z-scores.

Step-by-Step Guidance

1. For part (a), area to the right means you need to find the z-score where $P(Z > z) = 0.1056$. Use the table to find the corresponding z.

2. For part (b), area to the left means $P(Z < z) = 0.2655$. Use the table to find the z-score.

3. For part (c), "middle 10%" means you want z-scores that leave 5% in each tail. Find z-scores where $P(Z < z_1) = 0.05$ and $P(Z < z_2) = 0.95$.

Try solving on your own before revealing the answer!

Final Answers:

• a) $z \approx 1.25$

• b) $z \approx -0.62$

• c) $z_1 \approx -1.645$, $z_2 \approx 1.645$

These values are found using the standard normal table.

Q7. Find the area under the standard normal curve:

• a) To the right of $z = -1.36$.

• b) To the left of $z = 2.34$.

Background

Topic: Standard Normal Distribution Areas

This question tests your ability to use the standard normal table to find areas under the curve for given z-scores.

Key Terms and Formulas

• Area to the right: $P(Z > z)$

• Area to the left: $P(Z < z)$

Step-by-Step Guidance

1. For part (a), use the standard normal table to find $P(Z < -1.36)$, then subtract from 1 to get area to the right.

2. For part (b), use the table to find $P(Z < 2.34)$ directly.

Try solving on your own before revealing the answer!

Final Answers:

• a) Area to the right of $z = -1.36$ is $0.9131$

• b) Area to the left of $z = 2.34$ is $0.9904$

These values are found using the standard normal table.

Q8. The salaries for full-time PhD students are normally distributed with an average of $12,837 per year and a standard deviation of $1,500.

• a) Find the probability that a randomly selected student will make more than $15,000.

• b) If a group of 50 graduate students are selected, find the probability that the average salary is more than $15,000.

Background

Topic: Normal Distribution and Central Limit Theorem

These questions test your ability to use the normal distribution and the central limit theorem to find probabilities for individual values and sample means.

Key Terms and Formulas

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

• For sample mean: $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

• Use the standard normal table to find probabilities.

Step-by-Step Guidance

1. For part (a), calculate the z-score for $x = 15,000$ using the formula.

2. Use the standard normal table to find the probability that $z$ is greater than this value.

3. For part (b), calculate the z-score for the sample mean using $n = 50$.

4. Use the standard normal table to find the probability that the sample mean is greater than $15,000$.

Try solving on your own before revealing the answer!

Final Answers:

• a) Probability is $0.0475$

• b) Probability is $0.0000$ (very close to zero)

For a single student, the probability is about 4.75%. For a group of 50, it's extremely unlikely.

Q9. To qualify for a job at a health club, candidates must score in the upper 40% on a physical fitness test. The test has a mean score of 200 and a standard deviation of 20. Find the lowest possible qualifying score. Round your answer to the nearest whole number.

Background

Topic: Normal Distribution Percentiles

This question tests your ability to find a cutoff score corresponding to a given percentile using the normal distribution.

Key Terms and Formulas

• Percentile: The value below which a given percentage of observations fall.

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

• Find the z-score corresponding to the 60th percentile (since upper 40% means above the 60th percentile).

Step-by-Step Guidance

1. Find the z-score that leaves 40% above it (i.e., 60% below).

2. Use the standard normal table to find the z-score for $P(Z < z) = 0.60$.

3. Use the z-score formula to solve for $x$.

4. Round your answer to the nearest whole number.

Try solving on your own before revealing the answer!

Final Answer: 205

The lowest qualifying score is 205. This is found by using the z-score for the 60th percentile and solving for $x$.