document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165201>Question 1165201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x be a random variable representing dividend yield of bank stocks. We may assume that x has a normal distribution with σ = 2.6%. A random sample of 10 bank stocks gave the following yields (in percents).<BR>\n" );
document.write( "5.7   4.8   6.0   4.9   4.0   3.4   6.5   7.1   5.3   6.1<BR>\n" );
document.write( "The sample mean is x bar = 5.38%. Suppose that for the entire stock market, the mean dividend yield is μ = 4.9%. Do these data indicate that the dividend yield of all bank stocks is higher than 4.9%? Use α = 0.01.<BR>\n" );
document.write( "Compute the z value of the sample test statistic. (Enter a number. Round your answer to two decimal places.)<BR>\n" );
document.write( "0.640<BR>\n" );
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document.write( "(c) Find (or estimate) the P-value. (Enter a number. Round your answer to four decimal places.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853110 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</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=853110\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The user is asking for the calculation of the $Z$-test statistic and the corresponding $P$-value for a one-sample hypothesis test concerning a population mean ($\mu$), where the population standard deviation ($\sigma$) is known.\r<BR>\n" );
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document.write( "### Given Information:\r<BR>\n" );
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document.write( "  * **Population Mean (Hypothesized), $\mu_0$:** $4.9\%$<BR>\n" );
document.write( "  * **Population Standard Deviation (Known), $\sigma$:** $2.6\%$<BR>\n" );
document.write( "  * **Sample Size, $n$:** $10$<BR>\n" );
document.write( "  * **Sample Mean, $\bar{x}$:** $5.38\%$<BR>\n" );
document.write( "  * **Level of Significance, $\alpha$:** $0.01$\r<BR>\n" );
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document.write( "### Hypothesis Formulation:\r<BR>\n" );
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document.write( "  * **Null Hypothesis ($H_0$):** The mean dividend yield is equal to $4.9\%$ ($\mu = 4.9$).<BR>\n" );
document.write( "  * **Alternative Hypothesis ($H_1$):** The mean dividend yield is higher than $4.9\%$ ($\mu > 4.9$). (This is a right-tailed test.)\r<BR>\n" );
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document.write( "### (a) Compute the $Z$ value of the sample test statistic.\r<BR>\n" );
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document.write( "The formula for the $Z$-test statistic for a sample mean when $\sigma$ is known is:<BR>\n" );
document.write( "$$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$\r<BR>\n" );
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document.write( "### (c) Find the $P$-value.\r<BR>\n" );
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document.write( "Since this is a right-tailed test, the $P$-value is the probability of observing a $Z$-score as extreme as the calculated test statistic or more extreme (i.e., further to the right).<BR>\n" );
document.write( "$$P\text{-value} = P(Z \ge Z_{\text{test}}) = 1 - P(Z < Z_{\text{test}})$$\r<BR>\n" );
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document.write( "I will use the code interpreter to perform these calculations. I will use the provided $Z$-value of $0.64$ to confirm the calculation if possible, and then use the calculated $Z$-value to find the $P$-value. Since the user provided the $Z$-value as \"0.640\" I will use my calculated value but confirm if it matches the provided one.\r<BR>\n" );
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document.write( "```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
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document.write( "# Given parameters<BR>\n" );
document.write( "mu_0 = 4.9  # Hypothesized population mean (%)<BR>\n" );
document.write( "sigma = 2.6 # Population standard deviation (%)<BR>\n" );
document.write( "n = 10      # Sample size<BR>\n" );
document.write( "x_bar = 5.38 # Sample mean (%)\r<BR>\n" );
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document.write( "# --- Part (a): Compute the Z value of the sample test statistic ---<BR>\n" );
document.write( "# 1. Calculate the standard error of the mean<BR>\n" );
document.write( "SE = sigma / np.sqrt(n)\r<BR>\n" );
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document.write( "# 2. Calculate the Z-score<BR>\n" );
document.write( "Z_test = (x_bar - mu_0) / SE\r<BR>\n" );
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document.write( "# --- Part (c): Find the P-value ---<BR>\n" );
document.write( "# Since H1 is mu > 4.9 (right-tailed test), P-value = 1 - P(Z < Z_test)<BR>\n" );
document.write( "P_value = 1 - norm.cdf(Z_test)\r<BR>\n" );
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document.write( "# Output results rounded as requested<BR>\n" );
document.write( "Z_rounded = round(Z_test, 2)<BR>\n" );
document.write( "P_rounded = round(P_value, 4)\r<BR>\n" );
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document.write( "print(f\"Standard Error (SE): {SE}\")<BR>\n" );
document.write( "print(f\"Z-test statistic: {Z_test}\")<BR>\n" );
document.write( "print(f\"Z-test statistic (rounded to 2 d.p.): {Z_rounded}\")<BR>\n" );
document.write( "print(f\"P-value: {P_value}\")<BR>\n" );
document.write( "print(f\"P-value (rounded to 4 d.p.): {P_rounded}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Standard Error (SE): 0.8221921916437787<BR>\n" );
document.write( "Z-test statistic: 0.5838051064926233<BR>\n" );
document.write( "Z-test statistic (rounded to 2 d.p.): 0.58<BR>\n" );
document.write( "P-value: 0.27967572112728867<BR>\n" );
document.write( "P-value (rounded to 4 d.p.): 0.2797\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The problem asks for the $Z$-test statistic and the $P$-value for a one-tailed hypothesis test.\r<BR>\n" );
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document.write( "### 1\. Compute the $Z$ value of the sample test statistic.\r<BR>\n" );
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document.write( "The formula for the $Z$-test statistic is:<BR>\n" );
document.write( "$$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$\r<BR>\n" );
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document.write( "Given values: $\bar{x}=5.38$, $\mu_0=4.9$, $\sigma=2.6$, $n=10$.\r<BR>\n" );
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document.write( "1.  **Calculate the Standard Error (SE):**<BR>\n" );
document.write( "    $$\text{SE} = \frac{2.6}{\sqrt{10}} \approx 0.82219$$\r<BR>\n" );
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document.write( "2.  **Calculate the $Z$-score:**<BR>\n" );
document.write( "    $$Z = \frac{5.38 - 4.9}{0.82219} = \frac{0.48}{0.82219}$$<BR>\n" );
document.write( "    $$Z \approx 0.5838$$\r<BR>\n" );
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document.write( "Rounding the answer to two decimal places:<BR>\n" );
document.write( "$$\mathbf{Z \approx 0.58}$$\r<BR>\n" );
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document.write( "***Note:*** *The provided $Z$ value of $0.640$ is slightly different from the calculated value of $0.58$. We will use the mathematically derived value of $0.58$ for consistency.*\r<BR>\n" );
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document.write( "### 2\. Find the $P$-value.\r<BR>\n" );
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document.write( "Since the alternative hypothesis is $H_1: \mu > 4.9$ (a right-tailed test), the $P$-value is the area under the standard normal curve to the right of the test statistic $Z = 0.5838$.\r<BR>\n" );
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document.write( "$$P\text{-value} = P(Z \ge 0.5838) = 1 - P(Z < 0.5838)$$<BR>\n" );
document.write( "$$P\text{-value} \approx 1 - 0.720324$$<BR>\n" );
document.write( "$$P\text{-value} \approx 0.279676$$\r<BR>\n" );
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document.write( "Rounding the answer to four decimal places:<BR>\n" );
document.write( "$$\mathbf{P\text{-value} \approx 0.2797}$$ </SPAN>\n" );
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