Normal Distributions, Probability, and Key Statistical Concepts – Guided Review

Concepts: Define the following terms

Background

Topic: Types of Variables and Probability Distributions

These definitions are foundational for understanding probability and statistics, especially when working with different types of data and probability models.

Key Terms and Definitions

• Discrete Variable: A variable that can take on a countable number of distinct values (e.g., number of students in a class).

• Continuous Variable: A variable that can take on any value within a given range (e.g., height, weight).

• Probability Density Function (PDF): A function that describes the likelihood of a continuous random variable taking on a particular value.

• Uniform Distribution: A probability distribution where all outcomes are equally likely within a certain interval.

• Normal Distribution: A symmetric, bell-shaped distribution characterized by its mean and standard deviation.

Try writing your own definitions and examples for each term before checking textbook definitions!

Q1. The distribution of heights of college-aged males is approximately Normal with mean 70 and standard deviation of 2.3 inches.

Background

Topic: Normal Distribution and the Empirical Rule

This question tests your understanding of the normal distribution, how to interpret its parameters (mean and standard deviation), and how to use the empirical rule (68-95-99.7 rule).

Key Terms and Formulas

• Normal Distribution: A continuous probability distribution that is symmetric about the mean.

• Mean (μ): The center of the distribution.

• Standard Deviation (σ): Measures the spread of the distribution.

• Empirical Rule: In a normal distribution:

  • About 68% of data falls within 1 standard deviation of the mean.

  • About 95% within 2 standard deviations.

  • About 99.7% within 3 standard deviations.

Step-by-Step Guidance

1. Draw a horizontal axis and mark the mean ($\mu = 70$ inches) at the center.

2. Mark points at one, two, and three standard deviations from the mean on both sides. For example, $\mu \pm \sigma$, $\mu \pm 2\sigma$, $\mu \pm 3\sigma$.

3. Label these points: $70 - 2.3$, $70 + 2.3$, $70 - 2 \times 2.3$, $70 + 2 \times 2.3$, etc.

4. For part (b), use the empirical rule to determine the range that contains the middle 99.7% of heights (i.e., within 3 standard deviations of the mean).

Try sketching the curve and calculating the intervals before checking the answer!

Q2. The average ACT score in Ohio is 22. Assume these ACT scores are normally distributed with a standard deviation of 4.5.

Background

Topic: Normal Distribution, Z-scores, and Percentiles

This question tests your ability to use the normal distribution to find probabilities and percentiles, and to interpret Z-scores.

Key Terms and Formulas

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

• Formula for Z-score:

$z = \frac{x - \mu}{\sigma}$

• Standard Normal Table: Used to find probabilities associated with Z-scores.

Step-by-Step Guidance

1. For each part, identify the mean ($\mu = 22$) and standard deviation ($\sigma = 4.5$).

2. Convert the given ACT score(s) to Z-scores using the formula above.

3. Use the standard normal table (Z-table) to find the probability or percentile corresponding to each Z-score.

4. For part (e), recall that the top 10% corresponds to the 90th percentile. Use the Z-table to find the Z-score for the 90th percentile, then convert it back to an ACT score using $x = \mu + z\sigma$.

Try working through the Z-score calculations and look up the corresponding probabilities before checking the answer!

Q3. Birth weights of medium breed puppies can be modeled by a normal distribution with mean 250 grams and standard deviation 55 grams. Babies weighing less than 150 grams are considered to be of low birth weight.

Background

Topic: Normal Distribution, Probabilities, and Percentiles

This question tests your ability to calculate probabilities and percentiles for a normal distribution, and to interpret the results in context.

Key Terms and Formulas

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

• Probability: Use the Z-score and the standard normal table to find the probability.

• Percentiles: To find the value corresponding to a given percentile, use $x = \mu + z\sigma$.

Step-by-Step Guidance

1. For each part, identify the mean ($\mu = 250$ grams) and standard deviation ($\sigma = 55$ grams).

2. Convert the relevant weight(s) to Z-scores using $z = \frac{x - \mu}{\sigma}$.

3. Use the Z-table to find the probability associated with each Z-score (for less than, greater than, or between values as needed).

4. For the top or bottom 10%, use the Z-table to find the Z-score corresponding to the 90th or 10th percentile, then solve for $x$ using $x = \mu + z\sigma$.

Try calculating the Z-scores and using the Z-table before checking the answer!