Questions: Using the accompanying table of data, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.5. (All units are 1000 cells/ L.) Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? At least 75% of women have platelet counts within 2 standard deviations of the mean. (Round to the nearest integer as needed.) What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? The minimum possible platelet count within 2 standard deviations of the mean is . The maximum possible platelet count within 2 standard deviations of the mean is (Type integers or decimals rounded to one decimal place as needed.)

Using the accompanying table of data, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.5. (All units are 1000 cells/ L.) Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? At least 75% of women have platelet counts within 2 standard deviations of the mean. (Round to the nearest integer as needed.) What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? The minimum possible platelet count within 2 standard deviations of the mean is . The maximum possible platelet count within 2 standard deviations of the mean is (Type integers or decimals rounded to one decimal place as needed.)
Transcript text: Using the accompanying table of data, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.5 . (All units are 1000 cells/ $\mathrm{L}$.) Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? At least $75 \%$ of women have platelet counts within 2 standard deviations of the mean. (Round to the nearest integer as needed.) What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? The minimum possible platelet count within 2 standard deviations of the mean is $\square$ . The maximum possible platelet count within 2 standard deviations of the mean is $\square$ (Type integers or decimals rounded to one decimal place as needed.)
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Solution

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Solution Steps

To solve this problem, we will use Chebyshev's theorem, which states that for any distribution, at least \(1 - \frac{1}{k^2}\) of the data lies within \(k\) standard deviations of the mean. For \(k = 2\), this means at least 75% of the data is within 2 standard deviations. We will calculate the minimum and maximum platelet counts by adding and subtracting 2 times the standard deviation from the mean.

Step 1: Apply Chebyshev's Theorem

Chebyshev's theorem states that for any distribution, at least \(1 - \frac{1}{k^2}\) of the data lies within \(k\) standard deviations of the mean. For \(k = 2\), this means at least \(1 - \frac{1}{4} = \frac{3}{4} = 75\%\) of the data is within 2 standard deviations.

Step 2: Calculate Minimum and Maximum Platelet Counts

Given:

  • Mean \(\mu = 255.1\)
  • Standard deviation \(\sigma = 65.5\)
  • \(k = 2\)

The minimum platelet count within 2 standard deviations is: \[ \mu - k \cdot \sigma = 255.1 - 2 \cdot 65.5 = 124.1 \]

The maximum platelet count within 2 standard deviations is: \[ \mu + k \cdot \sigma = 255.1 + 2 \cdot 65.5 = 386.1 \]

Final Answer

  • At least \(75\%\) of women have platelet counts within 2 standard deviations of the mean.
  • Minimum platelet count: \(\boxed{124.1}\)
  • Maximum platelet count: \(\boxed{386.1}\)
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