Precalculus Statistics Quiz: Normal Distribution and Z-Scores Study Guidance

Q1. Convert each score to a standard score.

Background

Topic: Standard Scores (Z-Scores) in Normal Distribution

This question tests your ability to convert raw scores to z-scores, which measure how many standard deviations a value is from the mean.

Key formula:

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

Where:

• $x$ = raw score

• $\mu$ = mean

• $\sigma$ = standard deviation

Step-by-Step Guidance

1. Identify the mean ($\mu$) and standard deviation ($\sigma$) given in the problem.

2. For each score, subtract the mean from the raw score to find the difference.

3. Divide the difference by the standard deviation to calculate the z-score.

4. Repeat for each score you are asked to convert.

Try solving on your own before revealing the answer!

Q2. Which score was higher with respect to the rest of the class?

Background

Topic: Comparing Z-Scores

This question asks you to compare two z-scores to determine which score is higher relative to the class distribution.

Key concept:

The higher the z-score, the further above the mean the score is, relative to the standard deviation.

Step-by-Step Guidance

1. Calculate the z-score for each student using the formula $z = \frac{x - \mu}{\sigma}$.

2. Compare the two z-scores numerically.

3. The score with the higher z-score is higher with respect to the rest of the class.

Try solving on your own before revealing the answer!

Q3. For each of the following probabilities, sketch a standard normal curve and shade the area for which you are computing the probability, and find the requested probability.

Background

Topic: Standard Normal Distribution and Probability

This question tests your ability to interpret and calculate probabilities using the standard normal curve (z-distribution).

Key terms:

• Standard normal curve: A normal distribution with mean 0 and standard deviation 1.

• Probability: The area under the curve for a given z-value or range.

Key formula:

Use the z-table to find probabilities associated with z-values.

Step-by-Step Guidance

1. For each probability, identify the z-value(s) given.

2. Sketch the standard normal curve and shade the area corresponding to the probability (e.g., left, right, or between z-values).

3. Use the z-table to look up the area under the curve for the given z-value(s).

4. Set up the calculation for the probability, but stop before computing the final value.

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Q4. The snow pack on the summit of Wolf Creek Pass, Colorado on March 1 has been normally distributed with a mean of 80 inches and a standard deviation of 10.4 inches. Find the probability that the snow pack will be:

• Less than 60 inches

• More than 85 inches

Background

Topic: Normal Distribution Probability

This question tests your ability to calculate probabilities for values in a normal distribution using the mean and standard deviation.

Key formula:

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

Step-by-Step Guidance

1. For each scenario, identify the value ($x$) for which you want to find the probability.

2. Calculate the z-score for each value using $z = \frac{x - \mu}{\sigma}$.

3. Use the z-table to find the probability corresponding to each z-score.

4. Set up the probability calculation, but stop before computing the final value.

Try solving on your own before revealing the answer!