MTH 110 Statistics Assignments: Data Analysis And Probabilit

MTH 110 Statistics Assignments: Data Analysis and Probability Questions

Perform statistical analyses and probability calculations based on provided data sets, including measures of central tendency, quartiles, interquartile range, fences, outlier detection, normal distribution probability calculations, z-scores, percentile conversions, and understanding of unusual events in a statistical context.

Paper For Above instruction

This paper systematically addresses the assigned statistical questions, integrating concepts of data analysis, probability, and distribution analysis to interpret real-world datasets. The analysis begins with descriptive statistics of the costs of electric glass-top ranges, moves through probability calculations for reading speeds among fourth graders, evaluates the weights of chocolate bars produced by a machine, and concludes with analysis of head circumference data among male soldiers. Each section illustrates applied statistical methods, including median, quartiles, IQR, fences, outliers, standard normal distribution probabilities, z-scores, and percentile rankings, complemented by contextual explanations and implications.

Question 1: Cost Analysis of Electric Glass-Top Ranges

The data set consists of 10 ranges rated very good or excellent by Consumer Reports: $765, $955, $1050, $1100, $1200, $1240, $1250, $1400, $1450, $1650.

Part a: Find the median.

Ordering the data: $765, $955, $1050, $1100, $1200, $1240, $1250, $1400, $1450, $1650.

The median is the average of the 5th and 6th observations: ($1200 + $1240)/2 = $1220.

Part b: Find the first and third quartiles.

First quartile (Q1): median of the lower half (first 5 observations): $765, $955, $1050, $1100, $1200. The median is the 3rd value: $1050.

Third quartile (Q3): median of the upper half (last 5 observations): $1240, $1250, $1400, $1450, $1650. The median is the 3rd value: $1400.

Part c: Find the interquartile range (IQR).

IQR = Q3 - Q1 = $1400 - $1050 = $350.

Part d: Find the upper and lower fences.

Lower fence = Q1 - 1.5 × IQR = $1050 - 1.5 × $350 = $1050 - $525 = $525.

Upper fence = Q3 + 1.5 × IQR = $1400 + $525 = $1925.

Part e: Does the set contain outliers? If so, what are they?

Any data point below $525 or above $1925$ is an outlier. The maximum value is $1650$, which is below $1925$, and the minimum is $765$, above $525$, so there are no outliers in this data set.

Question 2: Reading Speeds of Fourth Graders

The reading speeds are normally distributed, with mean μ = 92 words per minute (wpm) and standard deviation σ = 8 wpm.

Part a: Probability a student reads less than 88 wpm.

Calculate z = (88 - 92) / 8 = -0.5. From standard normal tables, P(Z

Part b: Probability a student reads more than 116 wpm.

Z = (116 - 92) / 8 = 3.0. P(Z > 3.0) = 1 - P(Z

Part c: Is it unusual for a student to read more than 116 wpm?

Since the probability is very low (~0.13%), it is considered unusual for a student to have such a high reading speed.

Part d: Reading rate at the 85th percentile.

Find z for the 85th percentile: z ≈ 1.036. Then, X = μ + zσ = 92 + 1.036 × 8 ≈ 92 + 8.288 ≈ 100.288 wpm.

Part e: Probability the mean of 16 students exceeds 97 wpm.

For sample mean: standard error SE = σ / √n = 8 / 4 = 2.0.

Z = (97 - 92) / 2 = 2.5. P(Z > 2.5) = 1 - 0.9938 = 0.0062. So, about 0.62% chance that the sample mean exceeds 97 wpm.

Question 3: Weights of Chocolate Bars

The weights are normally distributed with mean μ = 7.1 ounces, standard deviation σ = 0.2 ounces.

Part a: Probability a bar weighs less than 6.95 ounces.

Z = (6.95 - 7.1) / 0.2 = -0.75. P(Z

Part b: Probability the mean weight of 16 bars is less than 6.95 ounces.

Standard error SE = 0.2 / √16 = 0.2 / 4 = 0.05.

Z = (6.95 - 7.1) / 0.05 = -3.0. P(Z

Question 4: Head Circumference of Male Soldiers

Distribution is approximately normal with μ = 22.8 inches, σ = 1.1 inches.

Part a: Percentile of a soldier with 23.6 inches.

Z = (23.6 - 22.8) / 1.1 ≈ 0.727. Lookup: P(Z

Part b: Percentage of soldiers with head circumference between 20.6 and 24.2 inches.

Z for 20.6 inches: (20.6 - 22.8) / 1.1 ≈ -2.0.

Z for 24.2 inches: (24.2 - 22.8) / 1.1 ≈ 1.273.

P(20.6

Part c: Is a head circumference of 26.9 inches unusual?

Z = (26.9 - 22.8) / 1.1 ≈ 3.727. P(Z > 3.727) ≈ 0.0001, indicating it is highly unusual for a soldier to have such a large head circumference.

Conclusion

These analyses demonstrate the application of descriptive and inferential statistics to real data, emphasizing the importance of understanding data distribution, outliers, and probability in making informed decisions and assessing unusual events. Whether analyzing the costs of kitchen appliances, reading speeds, chocolate bar weights, or biological measurements, statistical methods provide critical insights into data patterns and deviations.

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.
• Freund, J. E., & Walpole, R. E. (2014). Mathematical Statistics with Applications. Pearson.
• Moore, D. S., Mc Cabe, B., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W.H. Freeman.
• Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
• Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Procedures. CRC Press.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2014). Mathematical Statistics with Applications. Cengage Learning.
• Wilkinson, L., et al. (2005). The Grammar of Graphics. Springer.
• Zar, J. H. (2010). Biostatistical Analysis. Pearson.
• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.