University Of Phoenix Material: Time To Practice Week Two

University Of Phoenix Materialtime To Practice Week Twocompletepart

Part A questions involve understanding standard scores (z scores), probability calculations based on normal distributions, and interpreting z scores in context. It requires calculating z scores for given raw scores, understanding why standard scores are useful for comparison across different distributions, and applying the properties of the standard normal distribution to find probabilities associated with various score ranges. Additionally, it involves determining raw scores needed for specific percentile rankings, comparing student performances based on z scores, and interpreting what specific z scores mean in a normal distribution context.

Paper For Above instruction

Introduction

Understanding the concept of z scores and their application in statistics is essential for interpreting data across different distributions. Z scores are standardized scores that enable comparison of scores from different datasets or tests by transforming raw scores into a common scale. This paper explores the properties and usefulness of z scores, calculations of probabilities within normal distributions, and interpretation of these scores in practical scenarios such as student assessments and fitness certifications.

What is a z score and why are standard scores useful?

A z score is a type of standard score that indicates how many standard deviations a particular raw score is from the mean of its distribution. Mathematically, it is calculated by subtracting the mean from the raw score and then dividing by the standard deviation. Z scores are crucial because they provide a normalized scale, allowing comparisons between scores from different distributions with possibly different means and standard deviations.

Standard scores can be used to compare scores from different distributions because they represent each score's position relative to its own mean, standardized by its standard deviation. For example, a z score of 1.5 indicates the score is 1.5 standard deviations above the mean, regardless of the original distribution's units or scale. This normalization makes it easier to interpret relative performance and to compare scores across test types or subject areas.

Calculating Z scores for given raw scores

• Given mean = 70, standard deviation = 8:
• Raw score 68: \( z = \frac{68 - 70}{8} = -0.25 \)
• Raw score 82: \( z = \frac{82 - 70}{8} = 1.5 \)
• Raw score 69: \( z = \frac{69 - 70}{8} = -0.125 \)
• Raw score 85: \( z = \frac{85 - 70}{8} = 1.875 \)
• Raw score 72: \( z = \frac{72 - 70}{8} = 0.25 \)

Probability calculations in normal distributions

Using the mean of 70 and standard deviation of 6.38, we convert raw scores to z scores and then find the probabilities. To find the probability of a score falling within a range, we identify the corresponding z scores and consult standard normal distribution tables or use statistical software.

For example, to find the probability between 70 and 80:

• Calculate z scores:
• 70: \( z = \frac{70 - 70}{6.38} = 0 \)
• 80: \( z = \frac{80 - 70}{6.38} \approx 1.57 \)

Consulting the standard normal table, the area between z=0 and z=1.57 is approximately 0.471 (or 47.1%). Therefore, the probability of falling between raw scores 70 and 80 is about 47.1%.

Similarly, probabilities for other ranges are calculated by converting raw scores to z scores and then using the standard normal distribution:

• Score above 80 corresponds to z > 1.57, area to the right of z=1.57 is about 0.058 (5.8%).
• Between 81 and 83: z scores are approximately 2.53 and 2.76; using tables, the area between these z scores is roughly 0.027.
• Below 63: z = \( \frac{63 - 70}{6.38} \approx -1.09 \). The area to the left of z=-1.09 is approximately 0.138.

Determining raw scores for specific percentiles and comparisons

To find the raw score needed to be in the top 10%, find the z score corresponding to the 90th percentile, which is approximately 1.28. Applying the formula:

Raw score = (z * standard deviation) + mean

In this case:

Raw score = (1.28 * 5.5) + 78 ≈ 7.04 + 78 = 85.04.

Thus, Jake needs approximately a score of 85.04 to be in the top 10%.

To compare students' performance, the student with the higher z score is relatively better within their distribution. For example, Talya's z scores can be computed and compared to Noah's to determine who performed better relative to their classmates.

Understanding what z scores of 1 and the percentage of scores between specific deviations mean

A z score of 1 signifies a score that is exactly one standard deviation above the mean. In a normal distribution, approximately 34.13% of scores fall between the mean (z=0) and z=1. Combining both sides, about 68.26% of scores lie within ±1 standard deviation from the mean.

Extending further, approximately 99.7% of scores fall within ±3 standard deviations of the mean, according to the empirical rule. These properties help us understand the distribution of data points in real-world scenarios like test scores, physical fitness levels, and academic performance.

Conclusion

The concept of z scores and the properties of the standard normal distribution are fundamental for making meaningful comparisons and calculating probabilities in statistics. By standardizing raw scores, statisticians and educators can better interpret data, predict outcomes, and assess individual performance relative to a population. Practical applications of these principles include determining cut-off scores for certifications, comparing academic scores across different subjects, and understanding the distribution of results in large data sets.

References

• Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Freeman, S., et al. (2014). Elementary Statistics. Pearson.
• Moore, D., et al. (2018). The Practice of Statistics. W. H. Freeman.
• Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Duxbury Press.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
• Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
• Zou, G. (2004). A Modified Poisson Regression Approach to Prospective Studies with Binary Data. American Journal of Epidemiology.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.