The Blood Platelet Counts Of A Group Of Women Have A Bell Sh

The Blood Platelet Counts Of A Group Of Women Have A Bell Shaped Dist

The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 250.8 and a standard deviation of 65.5. (All units are 1000 cells/μL.) Using the empirical rule, find each approximate percentage below.

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The empirical rule, also known as the 68-95-99.7 rule, provides a quick way to estimate the percentage of data that falls within certain ranges for a normal distribution. Given the mean (μ) of 250.8 and a standard deviation (σ) of 65.5, we can apply this rule to estimate the proportions of women with specific platelet counts.

Part a: Percentage within 1 standard deviation of the mean

According to the empirical rule, approximately 68% of data in a normal distribution falls within one standard deviation of the mean, i.e., between μ - σ and μ + σ. In this case, that range is calculated as follows:

  • Lower bound: 250.8 - 65.5 = 185.3
  • Upper bound: 250.8 + 65.5 = 316.3

Thus, approximately 68% of women have platelet counts between 185.3 and 316.3 (1000 cells/μL).

Part b: Percentage between 54.3 and 447.3

The range from 54.3 to 447.3 spans more than two standard deviations from the mean in both directions. To determine this, we calculate how many standard deviations these bounds are from the mean:

  • Lower bound deviation: (54.3 - 250.8) / 65.5 ≈ -3.22
  • Upper bound deviation: (447.3 - 250.8) / 65.5 ≈ 2.95

This range approximately covers from about -3.2σ to +3σ around the mean. Using the empirical rule:

  • Within 3 standard deviations from the mean, approximately 99.7% of the data falls.

Since the bounds are just over three standard deviations below and above the mean, the approximate percentage of women with platelet counts between 54.3 and 447.3 is very close to 99.7%. Slight deviations slightly below or above 3σ change the percentage minimally, but for practical purposes, the estimated coverage is about 99.7%.

Conclusion

In summary, based on the empirical rule, approximately 68% of women have platelet counts within one standard deviation of the mean, between 185.3 and 316.3. Nearly all women, approximately 99.7%, have counts between 54.3 and 447.3, within about three standard deviations of the mean.

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