document.write( "Question 1168332: Normally distributed observations such as a person's weight, height, or shoe size occur quite frequently in nature. Business people who are aware of this use it to their advantage. A purchasing agent for a large retailer buying 15,000 pairs of women's shoes used the normal curve to decide on the order quantities for the various sizes. If women's average shoe size is 7.5 with a standard deviation of 1.5, how many pairs should be ordered between sizes 6.5 and 9? \n" ); document.write( "
Algebra.Com's Answer #851578 by CPhill(1959)\"\" \"About 
You can put this solution on YOUR website!
Let's solve this problem step-by-step using the normal distribution.\r
\n" ); document.write( "\n" ); document.write( "**1. Understand the Problem**\r
\n" ); document.write( "\n" ); document.write( "* Women's shoe sizes are normally distributed.
\n" ); document.write( "* Mean (μ) = 7.5
\n" ); document.write( "* Standard deviation (σ) = 1.5
\n" ); document.write( "* Total number of pairs ordered = 15,000
\n" ); document.write( "* We need to find the number of pairs to order between sizes 6.5 and 9.\r
\n" ); document.write( "\n" ); document.write( "**2. Convert Shoe Sizes to Z-scores**\r
\n" ); document.write( "\n" ); document.write( "We need to find the z-scores corresponding to shoe sizes 6.5 and 9.\r
\n" ); document.write( "\n" ); document.write( "* **Z-score for 6.5:**
\n" ); document.write( " * z = (X - μ) / σ
\n" ); document.write( " * z = (6.5 - 7.5) / 1.5
\n" ); document.write( " * z = -1 / 1.5
\n" ); document.write( " * z = -2/3 ≈ -0.67\r
\n" ); document.write( "\n" ); document.write( "* **Z-score for 9:**
\n" ); document.write( " * z = (X - μ) / σ
\n" ); document.write( " * z = (9 - 7.5) / 1.5
\n" ); document.write( " * z = 1.5 / 1.5
\n" ); document.write( " * z = 1\r
\n" ); document.write( "\n" ); document.write( "**3. Find the Probabilities**\r
\n" ); document.write( "\n" ); document.write( "* **Probability for z = -0.67:**
\n" ); document.write( " * Using a z-table or calculator, the cumulative probability for z = -0.67 is approximately 0.2514.\r
\n" ); document.write( "\n" ); document.write( "* **Probability for z = 1:**
\n" ); document.write( " * Using a z-table or calculator, the cumulative probability for z = 1 is approximately 0.8413.\r
\n" ); document.write( "\n" ); document.write( "**4. Find the Probability Between 6.5 and 9**\r
\n" ); document.write( "\n" ); document.write( "* The probability of a shoe size being between 6.5 and 9 is the difference between the two cumulative probabilities.
\n" ); document.write( "* P(6.5 < X < 9) = P(z < 1) - P(z < -0.67)
\n" ); document.write( "* P(6.5 < X < 9) = 0.8413 - 0.2514
\n" ); document.write( "* P(6.5 < X < 9) = 0.5899\r
\n" ); document.write( "\n" ); document.write( "**5. Calculate the Number of Pairs**\r
\n" ); document.write( "\n" ); document.write( "* Multiply the probability by the total number of pairs ordered.
\n" ); document.write( "* Number of pairs = 0.5899 * 15,000
\n" ); document.write( "* Number of pairs = 8848.5\r
\n" ); document.write( "\n" ); document.write( "**6. Round to the Nearest Whole Number**\r
\n" ); document.write( "\n" ); document.write( "* Since we can't order fractions of pairs, round to the nearest whole number.
\n" ); document.write( "* Number of pairs ≈ 8849\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the purchasing agent should order approximately 8849 pairs of shoes between sizes 6.5 and 9.**
\n" ); document.write( " \n" ); document.write( "
\n" );