Inferential Statistics And Analytics Deliverable 02 Tutoring

Inferential Statistics And Analyticsdeliverable 02 Tutoring On The No

Assume that a randomly selected subject is given a bone density test. Those tests follow a standard normal distribution. Find the probability that the bone density score for this subject is between -1.93 and 2.37. The student’s initial approach was to find the probabilities corresponding to these z-scores using Excel, noting that for -1.93 the cumulative area to the left was 0.0268, and for 2.37 it was 0.9911. However, there was an error in handling the calculation of the probability between these z-scores. The correct method is to find the difference of the cumulative probabilities: P(-1.93

Paper For Above instruction

Understanding and applying the concepts of inferential statistics, particularly the normal distribution, are essential in numerous scientific and industrial applications. The tasks provided serve to illustrate key steps in calculating probabilities associated with normally distributed variables, converting raw data to z-scores, and interpreting cumulative areas under the standard normal curve.

Problem 1: Calculating the Probability Between Two Z-scores

The first problem involves finding the probability that a bone density score, which follows a standard normal distribution, lies between -1.93 and 2.37. The key is to understand that the standard normal distribution table or computer software like Excel can provide the cumulative probability from the far left (negative infinity) up to a given z-score.

Using Excel or a similar tool, we find that the cumulative probability for z = -1.93 is approximately 0.0268, indicating that about 2.68% of the population has a bone density score less than this value. For z = 2.37, the cumulative probability is approximately 0.9911, meaning 99.11% of the population scores less than this value. To find the probability that the score is between these two z-scores, we subtract the smaller cumulative probability from the larger:

Probability = P(Z

This yields a probability of approximately 0.9643, or 96.43%. This step demonstrates the importance of understanding the difference of areas under the standard normal curve to compute probabilities for an interval.

Problem 2: Percentage of Women Meeting Height Requirement

The second problem addresses the probability that a woman’s height falls between 64 inches and 77 inches, given that the heights are normally distributed with a mean of 63.8 inches and a standard deviation of 2.6 inches. The approach involves standardizing each raw score to a z-score using the formula:

z = (x - μ) / σ

Calculations for the two bounds are as follows:

  • Z for x = 64 inches: z = (64 - 63.8) / 2.6 ≈ 0.077, rounded to 0.08
  • Z for x = 77 inches: z = (77 - 63.8) / 2.6 ≈ 5.077, rounded to 5.08

Next, using the standard normal table or software, find the cumulative probabilities:

P(Z 
P(Z

The probability that a woman’s height falls between 64 and 77 inches is then:

0.999999 - 0.5319 ≈ 0.4681

or approximately 46.81%. This implies that about 47% of women meet the height requirement for pilots, based on the normal distribution.

Problem 3: Calculating the Z-score for a Given Pulse Rate

The third problem involves calculating the z-score for a woman with a pulse rate of 66 beats per minute. The mean pulse rate is 69.4 beats per minute with a standard deviation of 11.3. The z-score is computed as:

Z = (X - μ) / σ = (66 - 69.4) / 11.3 ≈ -0.3009

Rounding to two decimal places, the z-score is approximately -0.30. The calculation shows how many standard deviations the woman's pulse rate is below the mean. Using the z-table, this corresponds roughly to a cumulative area of about 0.3821, signifying the proportion of women with pulse rates less than 66 bpm.

Problem 4: Area Corresponding to a Z-score of -1.645

The fourth task involves determining the cumulative area to the left of a z-score of -1.645. Consulting the standard normal table, this area is about 0.0505, indicating that approximately 5.05% of the population has a z-score less than -1.645. The area on the right of the z-score is simply 1 minus this value:

1 - 0.0505 = 0.9495

Thus, about 94.95% of the population has a z-score greater than -1.645, which is relevant in hypothesis testing and confidence interval calculations involving tail probabilities.

Problem 5: Finding the Z-score from an Area

Given that the area under the normal curve to the right is 0.8980, the corresponding cumulative area to the left is 1 - 0.8980 = 0.102. To find the z-score associated with this area, you can use Excel’s NORM.INV function:

=NORM.INV(0.102, 0, 1) ≈ -1.27

Alternatively, referencing the z-table, an area of 0.102 to the left corresponds approximately to a z-score of -1.27. This z-score indicates that the value is about 1.27 standard deviations below the mean.

Problem 6: Percentage of Men Fitting into a Manhole

The last problem involves determining what percentage of men with shoulder widths normally distributed around a mean of 18.2 inches and a standard deviation of 2.09 inches can fit into a 22-inch diameter manhole. Standardizing the value:

Z = (22 - 18.2) / 2.09 ≈ 1.81

Using the z-table or software, the cumulative probability for Z = 1.81 is approximately 0.9649, meaning about 96.49% of men have shoulder widths less than 22 inches. Therefore, approximately 96.49% of men will fit into the manhole of this diameter, assuming the distribution holds true.

Conclusion

These examples demonstrate foundational skills in normal distribution analysis, including converting raw scores to z-scores, interpreting cumulative areas, and understanding the probabilistic implications in practical contexts. Mastery of these steps enables accurate statistical inference, which is vital in health sciences, engineering, and social sciences, among others.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
  • Mendenhall, W., Ott, L., & Sincich, T. (2017). A First Course in Probability (10th ed.). Pearson.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
  • U.S. Census Bureau. (2020). Statistical Abstract of the United States.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • Ross, S. (2014). Introduction to Probability and Statistics (4th ed.). Academic Press.
  • Heumann, M. (2017). Normal Distribution and Z-scores. Khan Academy.
  • Excel Help Documentation. (2023). NORM.DIST and NORM.INV functions.
  • Gelman, A., Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

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