document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168277>Question 1168277</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In New York State, the mean salary for high school teachers in 2017 was $98,210 with a standard deviation of $9,560. Only Alaska’s mean salary was higher! Assume New York’s state salaries follow a normal distribution.\r<BR>\n" );
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document.write( "a. What percent of New York’s state high school teachers earn between $84,000 and $89,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.)\r<BR>\n" );
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document.write( "b. What percent of New York’s state high school teachers earn between $89,000 and $104,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.)\r<BR>\n" );
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document.write( "c. What percent of New York’s state high school teachers earn less than $74,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851659 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=851659\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $X$ be the random variable representing the salary of a high school teacher in New York State in 2017. We are given that $X$ follows a normal distribution with a mean ($\mu$) of $98,210 and a standard deviation ($\sigma$) of $9,560.\r<BR>\n" );
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document.write( "We need to calculate the percentages for different salary ranges by first converting the salary values to z-scores using the formula:<BR>\n" );
document.write( "$z = \frac{x - \mu}{\sigma}$\r<BR>\n" );
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document.write( "**a. What percent of New York’s state high school teachers earn between $84,000 and $89,000?**\r<BR>\n" );
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document.write( "First, calculate the z-scores for $x_1 = 84,000$ and $x_2 = 89,000$:<BR>\n" );
document.write( "$z_1 = \frac{84,000 - 98,210}{9,560} = \frac{-14,210}{9,560} \approx -1.49$<BR>\n" );
document.write( "$z_2 = \frac{89,000 - 98,210}{9,560} = \frac{-9,210}{9,560} \approx -0.96$\r<BR>\n" );
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document.write( "Now, we need to find the area under the standard normal curve between $z_1 = -1.49$ and $z_2 = -0.96$. We can find the cumulative probabilities for each z-score from a standard normal distribution table or using a calculator.\r<BR>\n" );
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document.write( "$P(Z < -1.49) \approx 0.0681$<BR>\n" );
document.write( "$P(Z < -0.96) \approx 0.1685$\r<BR>\n" );
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document.write( "The percentage of teachers earning between $84,000 and $89,000 is:<BR>\n" );
document.write( "$P(-1.49 < Z < -0.96) = P(Z < -0.96) - P(Z < -1.49) = 0.1685 - 0.0681 = 0.1004$\r<BR>\n" );
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document.write( "Rounding the final answer to 2 decimal places, the percentage is $10.04\%$.\r<BR>\n" );
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document.write( "**b. What percent of New York’s state high school teachers earn between $89,000 and $104,000?**\r<BR>\n" );
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document.write( "First, calculate the z-scores for $x_1 = 89,000$ (which we already calculated as $z_2 \approx -0.96$) and $x_3 = 104,000$:<BR>\n" );
document.write( "$z_3 = \frac{104,000 - 98,210}{9,560} = \frac{5,790}{9,560} \approx 0.61$\r<BR>\n" );
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document.write( "Now, we need to find the area under the standard normal curve between $z_2 = -0.96$ and $z_3 = 0.61$. We can find the cumulative probabilities for each z-score from a standard normal distribution table or using a calculator.\r<BR>\n" );
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document.write( "$P(Z < -0.96) \approx 0.1685$<BR>\n" );
document.write( "$P(Z < 0.61) \approx 0.7291$\r<BR>\n" );
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document.write( "The percentage of teachers earning between $89,000 and $104,000 is:<BR>\n" );
document.write( "$P(-0.96 < Z < 0.61) = P(Z < 0.61) - P(Z < -0.96) = 0.7291 - 0.1685 = 0.5606$\r<BR>\n" );
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document.write( "Rounding the final answer to 2 decimal places, the percentage is $56.06\%$.\r<BR>\n" );
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document.write( "**c. What percent of New York’s state high school teachers earn less than $74,000?**\r<BR>\n" );
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document.write( "First, calculate the z-score for $x_4 = 74,000$:<BR>\n" );
document.write( "$z_4 = \frac{74,000 - 98,210}{9,560} = \frac{-24,210}{9,560} \approx -2.53$\r<BR>\n" );
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document.write( "Now, we need to find the area under the standard normal curve to the left of $z_4 = -2.53$. We can find the cumulative probability for this z-score from a standard normal distribution table or using a calculator.\r<BR>\n" );
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document.write( "$P(Z < -2.53) \approx 0.0057$\r<BR>\n" );
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document.write( "Rounding the final answer to 2 decimal places, the percentage is $0.57\%$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{a) 10.04\%, b) 56.06\%, c) 0.57\%}$ </SPAN>\n" );
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