document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163233>Question 1163233</A>: <I><SPAN STYLE=\"word-break: break-all;\"> H0: μ = 73<BR>\n" );
document.write( "HA: μ ≠ 73\r<BR>\n" );
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document.write( "Find the p-value for this test based on the following sample information. (You may find it useful to reference the appropriate table: z table or t table)\r<BR>\n" );
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document.write( "1. x¯ = 70; s = 6.9; n = 35\r<BR>\n" );
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document.write( "  A.  p-value < 0.01<BR>\n" );
document.write( "  B.  p-value > or equal to 0.10<BR>\n" );
document.write( "  C.  0.01 < or equal to p-value < 0.02    <BR>\n" );
document.write( "  D.  0.05 < or equal to p-value < 0.10<BR>\n" );
document.write( "  E.  0.02 < or equal to p-value < 0.05\r<BR>\n" );
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document.write( "2. x¯ = 76; s = 6.9; n = 35\r<BR>\n" );
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document.write( "  A.  0.01 < or equal to p-value < 0.02<BR>\n" );
document.write( "  B.  p-value < 0.01<BR>\n" );
document.write( "  C.  p-value > or equal to 0.10<BR>\n" );
document.write( "  D.  0.05 < or equal to p-value < 0.10<BR>\n" );
document.write( "  E.  0.02 < or equal to p-value < 0.05 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854391 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=854391\">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 import stats\r<BR>\n" );
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document.write( "mu0 = 73<BR>\n" );
document.write( "s = 6.9<BR>\n" );
document.write( "n = 35<BR>\n" );
document.write( "df = n - 1\r<BR>\n" );
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document.write( "# Scenario 1<BR>\n" );
document.write( "xbar1 = 70<BR>\n" );
document.write( "t1 = (xbar1 - mu0) / (s / math.sqrt(n))<BR>\n" );
document.write( "p_value1 = 2 * stats.t.sf(abs(t1), df)\r<BR>\n" );
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document.write( "# Scenario 2<BR>\n" );
document.write( "xbar2 = 76<BR>\n" );
document.write( "t2 = (xbar2 - mu0) / (s / math.sqrt(n))<BR>\n" );
document.write( "p_value2 = 2 * stats.t.sf(abs(t2), df)\r<BR>\n" );
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document.write( "print(f\"{t1=}\")<BR>\n" );
document.write( "print(f\"{p_value1=}\")<BR>\n" );
document.write( "print(f\"{t2=}\")<BR>\n" );
document.write( "print(f\"{p_value2=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "t1=-2.572208601347659<BR>\n" );
document.write( "p_value1=0.014646137580801655<BR>\n" );
document.write( "t2=2.572208601347659<BR>\n" );
document.write( "p_value2=0.014646137580801655\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the $p$-value for these hypothesis tests, we use the $t$-test for the population mean ($\mu$) because the population standard deviation is unknown and we are using the sample standard deviation ($s$).\r<BR>\n" );
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document.write( "### Common Variables<BR>\n" );
document.write( "* **Null Hypothesis ($H_0$):** $\mu = 73$<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_A$):** $\mu \neq 73$ (This is a **two-tailed test**)<BR>\n" );
document.write( "* **Sample size ($n$):** $35$<BR>\n" );
document.write( "* **Degrees of freedom ($df$):** $n - 1 = 34$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 1. Calculation for $\bar{x} = 70$, $s = 6.9$, $n = 35$\r<BR>\n" );
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document.write( "**Step 1: Calculate the Test Statistic ($t$)**<BR>\n" );
document.write( "$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{70 - 73}{6.9 / \sqrt{35}} = \frac{-3}{1.1663} \approx -2.57$$\r<BR>\n" );
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document.write( "**Step 2: Determine the $p$-value**<BR>\n" );
document.write( "Since it is a two-tailed test, the $p$-value is $2 \times P(T > | -2.57 |)$ with $34$ degrees of freedom.<BR>\n" );
document.write( "Looking at a $t$-table for $df = 34$:<BR>\n" );
document.write( "* The area for $t = 2.441$ is $0.01$ (one tail).<BR>\n" );
document.write( "* The area for $t = 2.728$ is $0.005$ (one tail).<BR>\n" );
document.write( "* Our value ($2.57$) falls between these two, meaning $0.005 < \text{one-tail } p < 0.01$.<BR>\n" );
document.write( "* Multiplying by $2$ for a two-tailed test: $0.01 < p\text{-value} < 0.02$.\r<BR>\n" );
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document.write( "**Correct Option: C. $0.01 \leq p\text{-value} < 0.02$**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 2. Calculation for $\bar{x} = 76$, $s = 6.9$, $n = 35$\r<BR>\n" );
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document.write( "**Step 1: Calculate the Test Statistic ($t$)**<BR>\n" );
document.write( "$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{76 - 73}{6.9 / \sqrt{35}} = \frac{3}{1.1663} \approx 2.57$$\r<BR>\n" );
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document.write( "**Step 2: Determine the $p$-value**<BR>\n" );
document.write( "Because the distribution is symmetric and the distance from the mean is the same as in the first problem ($|3|$ units), the test statistic magnitude and the $p$-value remain identical.<BR>\n" );
document.write( "* $t = 2.57$<BR>\n" );
document.write( "* $p\text{-value} = 2 \times P(T > 2.57)$<BR>\n" );
document.write( "* Using the same $t$-table values for $df = 34$, we find $0.01 < p\text{-value} < 0.02$.\r<BR>\n" );
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document.write( "**Correct Option: A. $0.01 \leq p\text{-value} < 0.02$** </SPAN>\n" );
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