PSCI 2702 Chapter 4

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Questions and Answers

What does a Z score of 0.00 represent in the standard normal curve table?

Area between the mean and Z is 0.0040

Area between Z and mean is 0.3413

Area beyond Z is 0.5000 (correct)

Area beyond Z is 0.0000

Which Z score has an area between the Z score and the mean of 0.3461?

1.01

1.03

0.01

1.02 (correct)

What is the area between a Z score of 1.50 and the mean?

0.5000

0.4332 (correct)

0.0668

0.3413

How can the areas presented in the standard normal curve table be expressed?

In percentages after multiplying by 100% Signup and view all the answers

What area does a Z score of 1.03 correspond to for the area beyond Z?

0.1515 Signup and view all the answers

If you find an area of 0.4880 in the table, what is the corresponding Z score?

0.03 Signup and view all the answers

What would be the area between the mean and a Z score of 1.51?

0.4345 Signup and view all the answers

What is the primary purpose of the standard normal curve table?

To determine areas under the normal curve Signup and view all the answers

What does a Z score of positive 1.00 imply about the original score?

The score lies one standard deviation above the mean. Signup and view all the answers

Which formula is used to convert original scores to Z scores?

$Z = \frac{X_i - X}{s}$ Signup and view all the answers

If the mean of a sample is 30 and the standard deviation is 14.14, what is the Z score for a score of 10?

$-1.414$ Signup and view all the answers

How is the normal curve standardized when using Z scores?

The mean is converted to zero and the standard deviation to one. Signup and view all the answers

What are the Z score equivalents of the scores 20 and 40 in a sample where the mean is 30 and standard deviation is 14.14?

-0.707 and 0.707, respectively. Signup and view all the answers

If a score of 30 corresponds to a Z score of 0.000, what does this indicate?

The score is exactly at average. Signup and view all the answers

Given the Z scores of the sample provided, which original score corresponds to a Z score of 1.414?

50 Signup and view all the answers

What is the purpose of converting scores to Z scores?

To standardize the distribution. Signup and view all the answers

What is the area between the Z scores of +0.65 and +1.05?

0.1111 Signup and view all the answers

When finding the area between two scores on the same side of the mean, the area to be subtracted is...

the smaller area Signup and view all the answers

If a score is below the mean, what should the procedure be for finding the area?

Add the areas between each score and the mean Signup and view all the answers

In a standard normal distribution, if the mean score is known, how do you compute the Z score for a given test score?

Z = (score - mean) / standard deviation Signup and view all the answers

In the context of test scores normally distributed with a mean of 67 and standard deviation of 5, a score of 78 would yield a Z score of...

2.2 Signup and view all the answers

Which statement best describes the procedure when two Z scores are on opposite sides of the mean?

You add the areas between each score and the mean. Signup and view all the answers

If you find the area between a Z score of +0.65 and the mean to be 0.2422, what percentage does this represent?

24.22% Signup and view all the answers

To find the area under the normal distribution curve, which of the following approaches is used?

Finding areas for both sides of the mean. Signup and view all the answers

What does a continuous probability distribution describe?

Probabilities for a range of values under the curve Signup and view all the answers

What is the purpose of converting an original score to a Z score?

To simplify intricate calculations and use published tables Signup and view all the answers

In the context of the provided example, what does 'success' refer to?

Selecting a subject with an IQ score within the range of 95 to 100 Signup and view all the answers

How are probabilities presented for continuous variables under the normal distribution?

In proportion form already available in published tables Signup and view all the answers

What is the function of Appendix A in the context of probabilities?

To offer published probabilities for areas under the normal curve Signup and view all the answers

What is the Z score calculated in the example when estimating IQ scores?

-0.25 Signup and view all the answers

Why might doing detailed probability calculations often be unnecessary?

Because reliable probability tables are readily available Signup and view all the answers

What is the importance of the mean in the context of IQ scores?

It provides a reference to compare individual scores against Signup and view all the answers

What is the probability that a randomly selected driver was going between 55 and 65 km/h, given that the average speed is 59 km/h and the standard deviation is 4?

0.68 Signup and view all the answers

If a student has a Z score of 3, what is their raw score if the mean is 18 days and the standard deviation is 3?

30 Signup and view all the answers

In a normal distribution with a mean of 31 and a standard deviation of 5, what percentage of the sample had scores above 35?

15.87% Signup and view all the answers

What is the raw score for a test writer at the 35th percentile, given a mean score of 1,000 and a standard deviation of 50?

965 Signup and view all the answers

What is the probability that the number of burglaries next year will be less than 250 if the average is 311 with a standard deviation of 50?

0.0228 Signup and view all the answers

If a second test writer is at the 50th percentile on a test, what is her raw score if the mean is 100 and standard deviation is 10?

100 Signup and view all the answers

What is the probability of finding a score greater than 80 in a normal distribution with a mean of 65 and a standard deviation of 15?

0.1587 Signup and view all the answers

In a context where a score of 40 or more is considered ‘highly discriminatory’, what is the probability that a randomly selected human resources manager has a score in that range if the distribution is normal?

0.70 Signup and view all the answers

Study Notes

Z-Scores and Normal Distribution

Z-scores standardize data by converting original units (like IQ points, centimeters, dollars) into a distribution with a mean of 0 and a standard deviation of 1.
Formula for Z-score: Z = (Xi - X) / s
- Xi = Individual score
- X = Mean
- s = Standard deviation
Positive Z-score indicates a score above the mean; negative Z-score indicates a score below the mean.
Standard Normal Curve Table (Z-score table): Provides areas under the normal curve for different Z-scores.
Example: A Z-score of +1.00 indicates the original score is one standard deviation above the mean.

Using the Standard Normal Curve Table

Column (a): Z-score
Column (b): Area between the Z-score and the mean
Column (c): Area beyond the Z-score
To find the area between a Z-score and the mean: Look up the Z-score in column (a) and find the corresponding value in column (b).
To find the area between two Z-scores on the same side of the mean: Find the areas between each score and the mean (in column (b)), then subtract the smaller area from the larger area.
To find the area between two Z-scores on opposite sides of the mean: Find the areas between each score and the mean (in column (b)), then add the two areas together.

Understanding Probabilities

Discrete probability distribution: Describes the probability of specific events of a discrete variable.
Continuous probability distribution: Describes the probability of a range of values under a curve.
Normal Distribution: A continuous probability distribution used for many naturally occurring phenomena.

Estimating Probabilities Using the Normal Curve

To find the probability of an event occurring in a normal distribution, convert the original score to a Z-score using the formula.
Example: To find the probability of a subject having an IQ score between 95 and the mean of 100 (assuming IQ scores are normally distributed), find the Z-score for 95 and look up the corresponding area in Appendix A.

Using Z-Scores to Solve Problems

Example: If your score on a driving test is 78, the mean score is 67, and the standard deviation is 5, you can calculate the Z-score for your score to see how it compares to the distribution of all test scores.
Example: You can use Z-scores to determine the probability of a driver exceeding a certain speed limit on a road with a known average speed and standard deviation.

Percentile Scores

Percentile: Indicates the percentage of scores that fall below a specific score.
Example: A score at the 35th percentile means 35% of the scores fall below it.

Applications in Real-World Scenarios

Career preparation aptitude tests: Use percentile scores to report test performance.
Burglary rates: Use the normal distribution to estimate the probability of the number of burglaries in a specific year.
Human Resources: Can use the normal distribution to analyze personality test scores to assess potential employees.

Studying That Suits You

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Description

Test your understanding of Z-scores and the standard normal distribution with this quiz. Explore the formulas, interpretations, and applications of Z-scores, and learn how to use the standard normal curve table effectively. Perfect for students looking to solidify their knowledge in statistics.