document.write( "Question 1185702: The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 370 grams and a standard deviation of 15 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.)
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\n" ); document.write( "Highest 30 percent
\n" ); document.write( "Middle 70 percent
\n" ); document.write( "Highest 90 percent
\n" ); document.write( "Lowest 20 percent \n" ); document.write( "

Algebra.Com's Answer #849624 by CPhill(1987) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here's how to find the weights corresponding to each event, using z-scores and the properties of a normal distribution:\r
\n" ); document.write( "\n" ); document.write( "**1. Highest 30 Percent:**\r
\n" ); document.write( "\n" ); document.write( "* We want to find the weight *x* such that P(X > x) = 0.30. This means that 70% of the weights are *below* this value.
\n" ); document.write( "* Find the z-score corresponding to a cumulative probability of 0.70. Using a z-table or calculator, we find that z ≈ 0.52.
\n" ); document.write( "* Use the z-score formula to find the weight:\r
\n" ); document.write( "\n" ); document.write( " x = μ + zσ
\n" ); document.write( " x = 370 + (0.52 * 15)
\n" ); document.write( " x = 370 + 7.8
\n" ); document.write( " x ≈ 377.80 grams\r
\n" ); document.write( "\n" ); document.write( "**2. Middle 70 Percent:**\r
\n" ); document.write( "\n" ); document.write( "* If the middle 70% is included, that leaves 30% in the tails, or 15% in *each* tail.
\n" ); document.write( "* To find the lower bound, find the z-score corresponding to a cumulative probability of 0.15 (15%). z ≈ -1.04.
\n" ); document.write( "* To find the upper bound, find the z-score corresponding to a cumulative probability of 0.85 (15% in each tail, so 100% - 15% = 85%). z ≈ 1.04.\r
\n" ); document.write( "\n" ); document.write( "Now, calculate the weights:\r
\n" ); document.write( "\n" ); document.write( "* Lower bound: x = 370 + (-1.04 * 15) = 370 - 15.6 ≈ 354.40 grams
\n" ); document.write( "* Upper bound: x = 370 + (1.04 * 15) = 370 + 15.6 ≈ 385.60 grams\r
\n" ); document.write( "\n" ); document.write( "**3. Highest 90 Percent:**\r
\n" ); document.write( "\n" ); document.write( "* If 90% of the weights are below this value, then we're looking for the weight such that P(X > x) = 0.10, so 90% of the weights are *below* this value.
\n" ); document.write( "* Find the z-score corresponding to a cumulative probability of 0.90. z ≈ 1.28
\n" ); document.write( "* Calculate the weight:\r
\n" ); document.write( "\n" ); document.write( " x = 370 + (1.28 * 15)
\n" ); document.write( " x = 370 + 19.2
\n" ); document.write( " x ≈ 389.20 grams\r
\n" ); document.write( "\n" ); document.write( "**4. Lowest 20 Percent:**\r
\n" ); document.write( "\n" ); document.write( "* We want to find the weight *x* such that P(X < x) = 0.20.
\n" ); document.write( "* Find the z-score corresponding to a cumulative probability of 0.20. z ≈ -0.84
\n" ); document.write( "* Calculate the weight:\r
\n" ); document.write( "\n" ); document.write( " x = 370 + (-0.84 * 15)
\n" ); document.write( " x = 370 - 12.6
\n" ); document.write( " x ≈ 357.40 grams\r
\n" ); document.write( "\n" ); document.write( "**Final Answers (rounded to 2 decimal places):**\r
\n" ); document.write( "\n" ); document.write( "* Highest 30 percent: 377.80 grams
\n" ); document.write( "* Middle 70 percent: Between 354.40 and 385.60 grams
\n" ); document.write( "* Highest 90 percent: 389.20 grams
\n" ); document.write( "* Lowest 20 percent: 357.40 grams
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