document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163238>Question 1163238</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following hypotheses: \r<BR>\n" );
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document.write( " H0: p ≥ 0.47 <BR>\n" );
document.write( " HA: p < 0.47 \r<BR>\n" );
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document.write( "Compute the p-value based on the following sample information. (You may find it useful to reference the appropriate table: z table or t table) (Round \"z\" value to 2 decimal places. Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.) <BR>\n" );
document.write( "                   <BR>\n" );
document.write( "                   p value<BR>\n" );
document.write( "a. x = 46; n = 118 _______  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854389 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854389\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import math<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
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document.write( "p0 = 0.47<BR>\n" );
document.write( "x = 46<BR>\n" );
document.write( "n = 118\r<BR>\n" );
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document.write( "# 1. Calculate sample proportion<BR>\n" );
document.write( "p_hat = x / n\r<BR>\n" );
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document.write( "# 2. Calculate standard error<BR>\n" );
document.write( "se = math.sqrt(p0 * (1 - p0) / n)\r<BR>\n" );
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document.write( "# 3. Calculate z-score<BR>\n" );
document.write( "z = (p_hat - p0) / se\r<BR>\n" );
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document.write( "# Round z to 2 decimal places<BR>\n" );
document.write( "z_rounded = round(z, 2)\r<BR>\n" );
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document.write( "# 4. Calculate p-value for a lower-tailed test<BR>\n" );
document.write( "# Using rounded z as per instructions \"Round 'z' value to 2 decimal places.\"<BR>\n" );
document.write( "p_value = norm.cdf(z_rounded)\r<BR>\n" );
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document.write( "print(f\"{p_hat=}\")<BR>\n" );
document.write( "print(f\"{se=}\")<BR>\n" );
document.write( "print(f\"{z=}\")<BR>\n" );
document.write( "print(f\"{z_rounded=}\")<BR>\n" );
document.write( "print(f\"{p_value=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "p_hat=0.3898305084745763<BR>\n" );
document.write( "se=0.04594580447823873<BR>\n" );
document.write( "z=-1.7448707762510565<BR>\n" );
document.write( "z_rounded=-1.74<BR>\n" );
document.write( "p_value=0.040929508978807365\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To compute the p-value for the given hypothesis test, we follow these steps:\r<BR>\n" );
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document.write( "### 1. State the Hypotheses<BR>\n" );
document.write( "* **Null Hypothesis ($H_0$):** $p \geq 0.47$<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_A$):** $p < 0.47$ (This is a lower-tailed test)\r<BR>\n" );
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document.write( "### 2. Identify Sample Information<BR>\n" );
document.write( "* Sample size ($n$) = $118$<BR>\n" );
document.write( "* Number of successes ($x$) = $46$<BR>\n" );
document.write( "* Hypothesized proportion ($p_0$) = $0.47$\r<BR>\n" );
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document.write( "### 3. Calculate the Sample Proportion ($\hat{p}$)<BR>\n" );
document.write( "$$\hat{p} = \frac{x}{n} = \frac{46}{118} \approx 0.389831$$\r<BR>\n" );
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document.write( "### 4. Calculate the Standard Error ($SE$)<BR>\n" );
document.write( "The standard error is calculated using the hypothesized proportion $p_0$:<BR>\n" );
document.write( "$$SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.47(1 - 0.47)}{118}} = \sqrt{\frac{0.47 \times 0.53}{118}} \approx 0.045946$$\r<BR>\n" );
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document.write( "### 5. Calculate the Test Statistic ($z$)<BR>\n" );
document.write( "$$z = \frac{\hat{p} - p_0}{SE} = \frac{0.389831 - 0.47}{0.045946} \approx -1.7449$$<BR>\n" );
document.write( "Rounding the **$z$ value to 2 decimal places** as requested:<BR>\n" );
document.write( "$$z = -1.74$$\r<BR>\n" );
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document.write( "### 6. Determine the p-value<BR>\n" );
document.write( "For a lower-tailed test, the p-value is the area to the left of our $z$-score in the standard normal distribution ($P(Z \leq -1.74)$). \r<BR>\n" );
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document.write( "Using a standard normal ($z$) table:<BR>\n" );
document.write( "* $P(Z \leq -1.74) = 0.0409$\r<BR>\n" );
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document.write( "**p-value:** **0.0409** </SPAN>\n" );
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