document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168059>Question 1168059</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Individual test scores in a large statistics class are normally distributed with ( m= 84.2 )\r<BR>\n" );
document.write( "\n" );
document.write( "       and (s = 12.8 ) Let X represent the test score of a randomly selected student from this\r<BR>\n" );
document.write( "\n" );
document.write( "       class.  Find the probability P( 80 < X < 86 ).  Your solution must include a completely\r<BR>\n" );
document.write( "\n" );
document.write( "       labeled sketch of the distribution with the area of interest shaded.    </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851677 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851677\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the probability $P(80 < X < 86)$, including a completely labeled sketch of the normal distribution:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Understand the Problem:**\r<BR>\n" );
document.write( "\n" );
document.write( "We have a normally distributed random variable $X$ representing test scores with a mean ($\mu$) of 84.2 and a standard deviation ($\sigma$) of 12.8. We want to find the probability that a randomly selected student's score falls between 80 and 86.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Convert the Scores to Z-Scores:**\r<BR>\n" );
document.write( "\n" );
document.write( "To find the probability using the standard normal distribution table or calculator, we need to convert the given scores (80 and 86) into z-scores using the formula:\r<BR>\n" );
document.write( "\n" );
document.write( "$z = \frac{X - \mu}{\sigma}$\r<BR>\n" );
document.write( "\n" );
document.write( "* For $X = 80$:<BR>\n" );
document.write( "    $z_1 = \frac{80 - 84.2}{12.8} = \frac{-4.2}{12.8} \approx -0.3281$\r<BR>\n" );
document.write( "\n" );
document.write( "* For $X = 86$:<BR>\n" );
document.write( "    $z_2 = \frac{86 - 84.2}{12.8} = \frac{1.8}{12.8} \approx 0.1406$\r<BR>\n" );
document.write( "\n" );
document.write( "So, the probability $P(80 < X < 86)$ is equivalent to $P(-0.3281 < Z < 0.1406)$, where $Z$ is the standard normal random variable.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Find the Probabilities Using the Standard Normal Distribution:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to find the area under the standard normal curve between $z_1 = -0.3281$ and $z_2 = 0.1406$. We can do this by finding the cumulative probabilities $P(Z < 0.1406)$ and $P(Z < -0.3281)$ and then subtracting the smaller from the larger.\r<BR>\n" );
document.write( "\n" );
document.write( "* $P(Z < 0.1406) \approx 0.5557$ (using a standard normal distribution table or calculator)<BR>\n" );
document.write( "* $P(Z < -0.3281) \approx 0.3713$ (using a standard normal distribution table or calculator)\r<BR>\n" );
document.write( "\n" );
document.write( "Now, subtract the probabilities:\r<BR>\n" );
document.write( "\n" );
document.write( "$P(-0.3281 < Z < 0.1406) = P(Z < 0.1406) - P(Z < -0.3281) \approx 0.5557 - 0.3713 = 0.1844$\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Sketch of the Distribution:**\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "                      Normal Distribution<BR>\n" );
document.write( "                           / \<BR>\n" );
document.write( "                          /   \<BR>\n" );
document.write( "                         /     \<BR>\n" );
document.write( "                        /       \<BR>\n" );
document.write( "                       /         \<BR>\n" );
document.write( "                      /           \<BR>\n" );
document.write( "                     /             \<BR>\n" );
document.write( "        ----------------------------------------------------<BR>\n" );
document.write( "       -3s    -2s    -1s     μ     +1s    +2s    +3s   Z-scores\r<BR>\n" );
document.write( "\n" );
document.write( "        -3(12.8) -2(12.8) -1(12.8) 84.2 +1(12.8) +2(12.8) +3(12.8) Raw Scores<BR>\n" );
document.write( "        (-38.4)  (-25.6)  (-12.8)      (12.8)   (25.6)   (38.4)  Deviations from Mean\r<BR>\n" );
document.write( "\n" );
document.write( "        55.8     68.6     71.4    84.2   97.0    109.8   122.6  Approximate Raw Scores\r<BR>\n" );
document.write( "\n" );
document.write( "                                 |       |<BR>\n" );
document.write( "                                 80      86<BR>\n" );
document.write( "                                 ^       ^<BR>\n" );
document.write( "                                 |       |<BR>\n" );
document.write( "                             z=-0.33  z=0.14\r<BR>\n" );
document.write( "\n" );
document.write( "        <----------------------- Shaded Area -----------------------><BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "**Labeling the Sketch:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Horizontal Axis:** Labeled as \"Test Scores (X)\" and also showing the corresponding \"Z-scores\" below.<BR>\n" );
document.write( "* **Vertical Axis:** Represents the probability density (not explicitly labeled with values, but the curve shows the shape of the distribution).<BR>\n" );
document.write( "* **Curve:** A bell-shaped curve representing the normal distribution.<BR>\n" );
document.write( "* **Mean (μ):** A vertical line drawn at the mean score of 84.2, labeled with \"μ = 84.2\".<BR>\n" );
document.write( "* **Standard Deviations (s):** Approximate locations of 1, 2, and 3 standard deviations away from the mean are indicated on the raw score axis.<BR>\n" );
document.write( "* **Scores of Interest:** Vertical lines are drawn at the scores 80 and 86 on the horizontal axis.<BR>\n" );
document.write( "* **Corresponding Z-scores:** The z-scores calculated for 80 (-0.33 approximately) and 86 (0.14 approximately) are indicated below the respective raw scores.<BR>\n" );
document.write( "* **Shaded Area:** The area under the curve between the vertical lines at 80 and 86 (or their corresponding z-scores) is shaded. This shaded area represents the probability $P(80 < X < 86)$.\r<BR>\n" );
document.write( "\n" );
document.write( "**Answer:**\r<BR>\n" );
document.write( "\n" );
document.write( "The probability $P(80 < X < 86) \approx \boxed{0.1844}$. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );