Step-by-Step Guidance: Normal Distributions & Probabilities in Statistics

Q1. What is the probability that a randomly selected adult has a bone density test score less than 1.27?

Background

Topic: Standard Normal Distribution (Z-scores)

This question tests your ability to find the probability (area under the curve) for a given z-score using the standard normal distribution.

Key Terms and Formulas

• Z-score: A standardized value representing the number of standard deviations a data point is from the mean.

• Standard Normal Distribution: A normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$.

• Probability: The area under the normal curve to the left of a given z-score.

Step-by-Step Guidance

1. Recognize that the question is asking for $P(Z < 1.27)$, where $Z$ is a standard normal variable.

2. Use a standard normal table (Z-table) or a calculator (like StatCrunch) to find the area to the left of $z = 1.27$.

3. Enter $z = 1.27$ into the calculator or locate it in the Z-table to find the corresponding probability.

Try solving on your own before revealing the answer!

Q2. What is the probability that a randomly selected adult has a bone density test score above -1.00?

Background

Topic: Standard Normal Distribution (Right-Tail Probability)

This question asks you to find the probability that a z-score is greater than a given value.

Key Terms and Formulas

• Right-tail probability: $P(Z > z_0) = 1 - P(Z < z_0)$

Step-by-Step Guidance

1. Identify that you need $P(Z > -1.00)$.

2. Find $P(Z < -1.00)$ using a Z-table or calculator.

3. Subtract this value from 1 to get the probability above $z = -1.00$.

Try solving on your own before revealing the answer!

Q3. What is the probability that a randomly selected adult has a bone density test score between -2.50 and -1.00?

Background

Topic: Standard Normal Distribution (Probability Between Two Values)

This question tests your ability to find the probability that a z-score falls between two values.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Find $P(Z < -1.00)$ and $P(Z < -2.50)$ using a Z-table or calculator.

2. Subtract $P(Z < -2.50)$ from $P(Z < -1.00)$ to get the probability between the two values.

Try solving on your own before revealing the answer!

Q4. What bone density score corresponds to the 95th percentile (P95)?

Background

Topic: Finding a Z-score for a Given Percentile

This question asks you to find the z-score that separates the bottom 95% from the top 5% of the standard normal distribution.

Key Terms and Formulas

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

• Inverse normal calculation: Find $z$ such that $P(Z < z) = 0.95$.

Step-by-Step Guidance

1. Set the cumulative probability to 0.95.

2. Use the inverse normal function on a calculator or StatCrunch to find the corresponding z-score.

Try solving on your own before revealing the answer!

Q5. What bone density scores separate the bottom 2.5% and the top 2.5%?

Background

Topic: Finding Z-scores for Symmetric Tails

This question asks you to find the z-scores that correspond to the lower and upper 2.5% of the standard normal distribution.

Key Terms and Formulas

• Lower tail: Find $z$ such that $P(Z < z) = 0.025$.

• Upper tail: Find $z$ such that $P(Z > z) = 0.025$, or equivalently $P(Z < z) = 0.975$.

Step-by-Step Guidance

1. For the bottom 2.5%, use the inverse normal function with a cumulative probability of 0.025.

2. For the top 2.5%, use the inverse normal function with a cumulative probability of 0.975.

Try solving on your own before revealing the answer!

Q6. What percentage of women are at least 70 inches tall, given that women's heights are normally distributed with mean 63.8 inches and standard deviation 2.6 inches?

Background

Topic: Normal Distribution (Non-Standard, Using Z-scores)

This question tests your ability to convert a raw score to a z-score and then find the probability above that value.

Key Terms and Formulas

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

• Right-tail probability: $P(X > x_0) = 1 - P(Z < z_0)$

Step-by-Step Guidance

1. Calculate the z-score for $x = 70$ using $\mu = 63.8$ and $\sigma = 2.6$.

2. Find $P(Z < z)$ using a Z-table or calculator.

3. Subtract this value from 1 to get the percentage of women taller than 70 inches.

4. Convert the probability to a percentage by multiplying by 100.

Try solving on your own before revealing the answer!

Q7. What ceiling height will accommodate 95% of men, given that men's heights are normally distributed with mean 69.5 inches and standard deviation 2.4 inches?

Background

Topic: Finding a Value for a Given Percentile in a Normal Distribution

This question asks you to find the raw score (height) corresponding to the 95th percentile.

Key Terms and Formulas

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

• Inverse z-score formula: $x = \mu + z \sigma$

Step-by-Step Guidance

1. Find the z-score corresponding to the 95th percentile (use a Z-table or calculator to find $z$ such that $P(Z < z) = 0.95$).

2. Plug the z-score, mean ($\mu = 69.5$), and standard deviation ($\sigma = 2.4$) into the formula $x = \mu + z \sigma$ to find the required ceiling height.

Try solving on your own before revealing the answer!