document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164885>Question 1164885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The average production cost for major movies is 60 million dollars and the standard deviation is 19 million dollars. Assume the production cost distribution is normal. Suppose that 44 randomly selected major movies are researched. Answer the following questions.  Give your answers in millions of dollars, not dollars.  Round all answers to 4 decimal places where possible.\r<BR>\n" );
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document.write( "What is the distribution of X?X~ N(_______,_________)\r<BR>\n" );
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document.write( "What is the distribution of ¯x?¯x~ N(_________,_________)\r<BR>\n" );
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document.write( "For a single randomly selected movie, find the probability that this movie's production cost is between 57 and 60 million dollars. \r<BR>\n" );
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document.write( "For the group of 44 movies, find the probability that the average production cost is between 57 and 60 million dollars. \r<BR>\n" );
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document.write( "For part d), is the assumption of normal necessary? Yes or No </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853505 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=853505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python<BR>\n" );
document.write( "import scipy.stats as stats<BR>\n" );
document.write( "import math\r<BR>\n" );
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document.write( "mu = 60<BR>\n" );
document.write( "sigma = 19<BR>\n" );
document.write( "n = 44\r<BR>\n" );
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document.write( "# 1. Distribution of X<BR>\n" );
document.write( "# 2. Distribution of x_bar<BR>\n" );
document.write( "sigma_x_bar = sigma / math.sqrt(n)\r<BR>\n" );
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document.write( "# 3. P(57 < X < 60)<BR>\n" );
document.write( "prob_single = stats.norm.cdf(60, loc=mu, scale=sigma) - stats.norm.cdf(57, loc=mu, scale=sigma)\r<BR>\n" );
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document.write( "# 4. P(57 < x_bar < 60)<BR>\n" );
document.write( "prob_avg = stats.norm.cdf(60, loc=mu, scale=sigma_x_bar) - stats.norm.cdf(57, loc=mu, scale=sigma_x_bar)\r<BR>\n" );
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document.write( "print(f\"{mu=}\")<BR>\n" );
document.write( "print(f\"{sigma=}\")<BR>\n" );
document.write( "print(f\"{sigma_x_bar=}\")<BR>\n" );
document.write( "print(f\"{prob_single=}\")<BR>\n" );
document.write( "print(f\"{prob_avg=}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text<BR>\n" );
document.write( "mu=60<BR>\n" );
document.write( "sigma=19<BR>\n" );
document.write( "sigma_x_bar=2.8643577734887544<BR>\n" );
document.write( "prob_single=0.06273012728359711<BR>\n" );
document.write( "prob_avg=0.3525321063551667\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "To solve these problems, we use the properties of the normal distribution and the Central Limit Theorem.\r<BR>\n" );
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document.write( "### Given Data:\r<BR>\n" );
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document.write( "* Population Mean () =  million dollars<BR>\n" );
document.write( "* Population Standard Deviation () =  million dollars<BR>\n" );
document.write( "* Sample Size () = \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### a) What is the distribution of ?\r<BR>\n" );
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document.write( "The distribution of the individual production costs () is given as normal.\r<BR>\n" );
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document.write( "### b) What is the distribution of ?\r<BR>\n" );
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document.write( "According to the Central Limit Theorem, the sampling distribution of the mean () has the same mean as the population, but a smaller standard deviation (known as the standard error).\r<BR>\n" );
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document.write( "* Mean of  = <BR>\n" );
document.write( "* Standard Error () = \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c) For a single randomly selected movie, find the probability that this movie's production cost is between 57 and 60 million dollars.\r<BR>\n" );
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document.write( "We need to find . We convert these values to z-scores:\r<BR>\n" );
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document.write( "* For : <BR>\n" );
document.write( "* For : \r<BR>\n" );
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document.write( "Using the standard normal table or calculator:\r<BR>\n" );
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document.write( "**Probability = 0.0627**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### d) For the group of 44 movies, find the probability that the average production cost is between 57 and 60 million dollars.\r<BR>\n" );
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document.write( "We need to find . Using the standard error calculated in part (b):\r<BR>\n" );
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document.write( "* For : <BR>\n" );
document.write( "* For : \r<BR>\n" );
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document.write( "**Probability = 0.3525**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### e) For part d), is the assumption of normal necessary?\r<BR>\n" );
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document.write( "**No**. Because the sample size () is greater than 30, the Central Limit Theorem guarantees that the distribution of the sample mean will be approximately normal regardless of the shape of the population distribution. </SPAN>\n" );
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