document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163253>Question 1163253</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following hypotheses:<BR>\n" );
document.write( "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( "a. x¯= 70; s = 6.9; n = 35<BR>\n" );
document.write( "b. x¯ = 76; s = 6.9; n = 35<BR>\n" );
document.write( "c. x¯ = 70; s = 6.4; n = 19<BR>\n" );
document.write( "d. x¯ = 70; s = 6.4; n = 24<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854383 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=854383\">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 scipy.stats as stats<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "def calculate_p_value(x_bar, s, n, mu_0=73):<BR>\n" );
document.write( "    t_stat = (x_bar - mu_0) / (s / np.sqrt(n))<BR>\n" );
document.write( "    df = n - 1<BR>\n" );
document.write( "    # Two-tailed test<BR>\n" );
document.write( "    p_value = 2 * (1 - stats.t.cdf(abs(t_stat), df))<BR>\n" );
document.write( "    return t_stat, p_value\r<BR>\n" );
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document.write( "# Case a<BR>\n" );
document.write( "ta, pa = calculate_p_value(70, 6.9, 35)<BR>\n" );
document.write( "# Case b<BR>\n" );
document.write( "tb, pb = calculate_p_value(76, 6.9, 35)<BR>\n" );
document.write( "# Case c<BR>\n" );
document.write( "tc, pc = calculate_p_value(70, 6.4, 19)<BR>\n" );
document.write( "# Case d<BR>\n" );
document.write( "td, pd = calculate_p_value(70, 6.4, 24)\r<BR>\n" );
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document.write( "print(f\"a: t = {ta:.4f}, p-value = {pa:.4f}\")<BR>\n" );
document.write( "print(f\"b: t = {tb:.4f}, p-value = {pb:.4f}\")<BR>\n" );
document.write( "print(f\"c: t = {tc:.4f}, p-value = {pc:.4f}\")<BR>\n" );
document.write( "print(f\"d: t = {td:.4f}, p-value = {pd:.4f}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "a: t = -2.5722, p-value = 0.0146<BR>\n" );
document.write( "b: t = 2.5722, p-value = 0.0146<BR>\n" );
document.write( "c: t = -2.0432, p-value = 0.0560<BR>\n" );
document.write( "d: t = -2.2964, p-value = 0.0311\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the p-values for these two-tailed hypothesis tests, we use the $t$-test because the population standard deviation is unknown and the sample standard deviation ($s$) is provided.\r<BR>\n" );
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document.write( "The test statistic is calculated as:<BR>\n" );
document.write( "$$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$<BR>\n" );
document.write( "The degrees of freedom are $df = n - 1$. Since it is a two-tailed test ($H_A: \mu \neq 73$), the p-value is $2 \times P(T > |t|)$.\r<BR>\n" );
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document.write( "### a. $\bar{x} = 70, s = 6.9, n = 35$<BR>\n" );
document.write( "* **Test Statistic ($t$):** $\frac{70 - 73}{6.9/\sqrt{35}} \approx -2.572$<BR>\n" );
document.write( "* **Degrees of Freedom:** $35 - 1 = 34$<BR>\n" );
document.write( "* **p-value:** $P(T < -2.572) + P(T > 2.572) \approx \mathbf{0.0146}$\r<BR>\n" );
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document.write( "### b. $\bar{x} = 76, s = 6.9, n = 35$<BR>\n" );
document.write( "* **Test Statistic ($t$):** $\frac{76 - 73}{6.9/\sqrt{35}} \approx 2.572$<BR>\n" );
document.write( "* **Degrees of Freedom:** $35 - 1 = 34$<BR>\n" );
document.write( "* **p-value:** $P(T < -2.572) + P(T > 2.572) \approx \mathbf{0.0146}$<BR>\n" );
document.write( "*(Note: Because the distance from the mean is the same as in part 'a', the p-value is identical.)*\r<BR>\n" );
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document.write( "### c. $\bar{x} = 70, s = 6.4, n = 19$<BR>\n" );
document.write( "* **Test Statistic ($t$):** $\frac{70 - 73}{6.4/\sqrt{19}} \approx -2.043$<BR>\n" );
document.write( "* **Degrees of Freedom:** $19 - 1 = 18$<BR>\n" );
document.write( "* **p-value:** $2 \times P(T > 2.043) \approx \mathbf{0.0560}$\r<BR>\n" );
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document.write( "### d. $\bar{x} = 70, s = 6.4, n = 24$<BR>\n" );
document.write( "* **Test Statistic ($t$):** $\frac{70 - 73}{6.4/\sqrt{24}} \approx -2.296$<BR>\n" );
document.write( "* **Degrees of Freedom:** $24 - 1 = 23$<BR>\n" );
document.write( "* **p-value:** $2 \times P(T > 2.296) \approx \mathbf{0.0311}$ </SPAN>\n" );
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