If Sodium Intake In A Certain Population Of Men Is Normal

If sodium intake in a certain population of men are normally distrib

If sodium intake in a certain population of men are normally distributed with a mean of 2490 mg and a standard deviation of 745 mg, calculate the following probabilities and percentiles:

  1. Find the probability that an individual picked at random from this population will have a sodium intake between 2721 mg and 4204 mg.
  2. What proportion of the sodium intakes are greater than 1988 mg?
  3. Find the probability that an individual picked at random from this population will have a sodium intake less than 3637 mg.
  4. What proportion of the sodium intakes are between 1074 mg and 3796 mg?
  5. Find the 95th percentile of sodium intake.
  6. Find the 10th percentile of sodium intake.
  7. Find the 14th percentile of sodium intake.

Paper For Above instruction

Understanding the distribution of sodium intake within a specific population is critical for public health strategies and nutritional policy development. When the data distribution is approximately normal, statistical techniques such as z-scores and standard normal distribution tables enable precise calculation of probabilities and percentiles. In this paper, we analyze sodium intake data for a population of men with a mean of 2490 mg and a standard deviation of 745 mg, addressing multiple statistical questions to interpret the data comprehensively.

Calculating Probabilities Using the Standard Normal Distribution

The normal distribution is characterized by its mean (μ) and standard deviation (σ). To compute probabilities for specific ranges, we standardize the values using the z-score formula:

z = (X - μ) / σ

where X is the value of interest. Once the z-score is calculated, probabilities can be obtained from standard normal distribution tables or software tools.

1. Probability Between 2721 mg and 4204 mg

First, calculate the z-scores for the bounds:

  • z for 2721 mg: (2721 - 2490) / 745 ≈ 0.319
  • z for 4204 mg: (4204 - 2490) / 745 ≈ 2.278

Using standard normal tables or software, the probabilities corresponding to these z-scores are approximately:

  • P(Z
  • P(Z

The probability that sodium intake falls between 2721 mg and 4204 mg is the difference between these two cumulative probabilities:

P(2721

2. Proportion Greater Than 1988 mg

Calculate the z-score:

z = (1988 - 2490) / 745 ≈ -0.667

Corresponding probability from the Z-table: P(Z

Therefore, the proportion greater than 1988 mg is:

P(X > 1988) = 1 - 0.2525 ≈ 0.7475

3. Probability Less Than 3637 mg

Calculate z:

z = (3637 - 2490) / 745 ≈ 1.502

P(Z

Thus, the probability is approximately 93.32% that intake is less than 3637 mg.

4. Proportion Between 1074 mg and 3796 mg

Z-scores:

  • 1074 mg: (1074 - 2490) / 745 ≈ -1.819
  • 3796 mg: (3796 - 2490) / 745 ≈ 1.810

From the Z-table:

  • P(Z
  • P(Z

The proportion between these values is:

0.9649 - 0.0347 ≈ 0.9302

Percentile Calculations

Percentiles correspond to specific z-scores in the standard normal distribution.

5. 95th Percentile of Sodium Intake

The z-score for the 95th percentile is approximately 1.645.

X = μ + zσ = 2490 + 1.645 * 745 ≈ 2490 + 1225 ≈ 3715 mg

6. 10th Percentile of Sodium Intake

The z-score for the 10th percentile is approximately -1.281.

X = 2490 + (-1.281) * 745 ≈ 2490 - 956 ≈ 1534 mg

7. 14th Percentile of Sodium Intake

Z-score for 14th percentile is approximately -1.076.

X = 2490 + (-1.076) * 745 ≈ 2490 - 803 ≈ 1687 mg

Conclusion

This analysis demonstrates the power of the normal distribution in interpreting health-related data such as sodium intake. By converting raw data to z-scores, we confidently calculate probabilities and percentiles, enabling public health officials to identify at-risk groups and develop targeted interventions. Such statistical insights support evidence-based nutritional policies and emphasize the importance of data-driven health strategies.

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