document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163125>Question 1163125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help needed please....\r<BR>\n" );
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document.write( " 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.) \r<BR>\n" );
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document.write( "p-value<BR>\n" );
document.write( " a. x = 46; n = 118 _______ <BR>\n" );
document.write( " b. x = 112; n = 311 _______ <BR>\n" );
document.write( " c. p¯ = 0.38; n = 53 _______ <BR>\n" );
document.write( " d. p¯ = 0.38; n = 430 _______  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854392 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=854392\">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( "def calculate_p_value(n, p0, x=None, p_hat=None):<BR>\n" );
document.write( "    if p_hat is None:<BR>\n" );
document.write( "        p_hat = x / n<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # Standard Error<BR>\n" );
document.write( "    se = math.sqrt(p0 * (1 - p0) / n)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # Test statistic z<BR>\n" );
document.write( "    z = (p_hat - p0) / se<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # Round z to 2 decimal places as per instructions<BR>\n" );
document.write( "    z_rounded = round(z, 2)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # Lower-tailed p-value<BR>\n" );
document.write( "    p_val = norm.cdf(z_rounded)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    return {<BR>\n" );
document.write( "        \"p_hat\": round(p_hat, 6),<BR>\n" );
document.write( "        \"se\": round(se, 6),<BR>\n" );
document.write( "        \"z\": round(z, 6),<BR>\n" );
document.write( "        \"z_rounded\": z_rounded,<BR>\n" );
document.write( "        \"p_value\": round(p_val, 4)<BR>\n" );
document.write( "    }\r<BR>\n" );
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document.write( "p0 = 0.47\r<BR>\n" );
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document.write( "results = {<BR>\n" );
document.write( "    \"a\": calculate_p_value(n=118, p0=p0, x=46),<BR>\n" );
document.write( "    \"b\": calculate_p_value(n=311, p0=p0, x=112),<BR>\n" );
document.write( "    \"c\": calculate_p_value(n=53, p0=p0, p_hat=0.38),<BR>\n" );
document.write( "    \"d\": calculate_p_value(n=430, p0=p0, p_hat=0.38)<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "import json<BR>\n" );
document.write( "print(json.dumps(results, indent=2))\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "{<BR>\n" );
document.write( "  \"a\": {<BR>\n" );
document.write( "    \"p_hat\": 0.389831,<BR>\n" );
document.write( "    \"se\": 0.045946,<BR>\n" );
document.write( "    \"z\": -1.744871,<BR>\n" );
document.write( "    \"z_rounded\": -1.74,<BR>\n" );
document.write( "    \"p_value\": 0.0409<BR>\n" );
document.write( "  },<BR>\n" );
document.write( "  \"b\": {<BR>\n" );
document.write( "    \"p_hat\": 0.360129,<BR>\n" );
document.write( "    \"se\": 0.028301,<BR>\n" );
document.write( "    \"z\": -3.8822,<BR>\n" );
document.write( "    \"z_rounded\": -3.88,<BR>\n" );
document.write( "    \"p_value\": 0.0001<BR>\n" );
document.write( "  },<BR>\n" );
document.write( "  \"c\": {<BR>\n" );
document.write( "    \"p_hat\": 0.38,<BR>\n" );
document.write( "    \"se\": 0.068557,<BR>\n" );
document.write( "    \"z\": -1.312785,<BR>\n" );
document.write( "    \"z_rounded\": -1.31,<BR>\n" );
document.write( "    \"p_value\": 0.0951<BR>\n" );
document.write( "  },<BR>\n" );
document.write( "  \"d\": {<BR>\n" );
document.write( "    \"p_hat\": 0.38,<BR>\n" );
document.write( "    \"se\": 0.024069,<BR>\n" );
document.write( "    \"z\": -3.739296,<BR>\n" );
document.write( "    \"z_rounded\": -3.74,<BR>\n" );
document.write( "    \"p_value\": 0.0001<BR>\n" );
document.write( "  }<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To compute the p-values for these lower-tailed hypothesis tests, we use the following steps:<BR>\n" );
document.write( "1.  **Calculate the sample proportion ($\hat{p}$):** $\hat{p} = \frac{x}{n}$ (if not already given).<BR>\n" );
document.write( "2.  **Calculate the standard error ($SE$):** $SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$ where $p_0 = 0.47$.<BR>\n" );
document.write( "3.  **Calculate the test statistic ($z$):** $z = \frac{\hat{p} - p_0}{SE}$.<BR>\n" );
document.write( "4.  **Find the p-value:** Since $H_A: p < 0.47$, the p-value is $P(Z \leq z_{rounded})$.\r<BR>\n" );
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document.write( "Here are the step-by-step calculations for each part:\r<BR>\n" );
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document.write( "### a. $x = 46$; $n = 118$<BR>\n" );
document.write( "* **Sample Proportion ($\hat{p}$):** $46 / 118 \approx 0.3898$<BR>\n" );
document.write( "* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{118}} \approx 0.0459$<BR>\n" );
document.write( "* **Test Statistic ($z$):** $\frac{0.3898 - 0.47}{0.0459} \approx -1.74$<BR>\n" );
document.write( "* **p-value:** $P(Z \leq -1.74) = \mathbf{0.0409}$\r<BR>\n" );
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document.write( "### b. $x = 112$; $n = 311$<BR>\n" );
document.write( "* **Sample Proportion ($\hat{p}$):** $112 / 311 \approx 0.3601$<BR>\n" );
document.write( "* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{311}} \approx 0.0283$<BR>\n" );
document.write( "* **Test Statistic ($z$):** $\frac{0.3601 - 0.47}{0.0283} \approx -3.88$<BR>\n" );
document.write( "* **p-value:** $P(Z \leq -3.88) = \mathbf{0.0001}$ (approx.)\r<BR>\n" );
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document.write( "### c. $\bar{p} = 0.38$; $n = 53$<BR>\n" );
document.write( "* **Sample Proportion ($\hat{p}$):** $0.38$<BR>\n" );
document.write( "* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{53}} \approx 0.0686$<BR>\n" );
document.write( "* **Test Statistic ($z$):** $\frac{0.38 - 0.47}{0.0686} \approx -1.31$<BR>\n" );
document.write( "* **p-value:** $P(Z \leq -1.31) = \mathbf{0.0951}$\r<BR>\n" );
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document.write( "### d. $\bar{p} = 0.38$; $n = 430$<BR>\n" );
document.write( "* **Sample Proportion ($\hat{p}$):** $0.38$<BR>\n" );
document.write( "* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{430}} \approx 0.0241$<BR>\n" );
document.write( "* **Test Statistic ($z$):** $\frac{0.38 - 0.47}{0.0241} \approx -3.74$<BR>\n" );
document.write( "* **p-value:** $P(Z \leq -3.74) = \mathbf{0.0001}$ (approx.)\r<BR>\n" );
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document.write( "---<BR>\n" );
document.write( "**Final Answers:**<BR>\n" );
document.write( "* a. **0.0409**<BR>\n" );
document.write( "* b. **0.0001**<BR>\n" );
document.write( "* c. **0.0951**<BR>\n" );
document.write( "* d. **0.0001** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );