document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165189>Question 1165189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A psychiatrist is testing a new antianxiety drug, which seems to have the potentially harmful side effect of lowering the heart rate. For a sample of 50 medical students whose pulse was measured after 6 weeks of taking the drug, the mean heart rate was 70 beats per minute (bpm). If the mean heart rate for the population is 72 bpm with a standard deviation of 12, can the psychiatrist conclude that the new drug lowers heart rate significantly? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853112 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=853112\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a hypothesis testing problem. We will use a one-sample $Z$-test for the mean, as the population standard deviation ($\sigma$) is known and the sample size ($n=50$) is large.\r<BR>\n" );
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document.write( "## 1. State the Hypotheses\r<BR>\n" );
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document.write( "The psychiatrist is concerned that the drug **lowers** the heart rate.\r<BR>\n" );
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document.write( "* **Null Hypothesis ($H_0$):** The drug has no effect; the mean heart rate ($\mu$) is still $72$ bpm.<BR>\n" );
document.write( "    $$H_0: \mu = 72$$<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_1$):** The drug lowers the mean heart rate ($\mu$). (This is a left-tailed test.)<BR>\n" );
document.write( "    $$H_1: \mu < 72$$\r<BR>\n" );
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document.write( "## 2. Identify the Given Data\r<BR>\n" );
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document.write( "| Variable | Symbol | Value |<BR>\n" );
document.write( "| :---: | :---: | :---: |<BR>\n" );
document.write( "| Population Mean (Hypothesized) | $\mu_0$ | $72 \text{ bpm}$ |<BR>\n" );
document.write( "| Population Standard Deviation | $\sigma$ | $12 \text{ bpm}$ |<BR>\n" );
document.write( "| Sample Size | $n$ | $50$ |<BR>\n" );
document.write( "| Sample Mean | $\bar{x}$ | $70 \text{ bpm}$ |<BR>\n" );
document.write( "| Significance Level | $\alpha$ | $0.05$ (Standard assumption since none is given) |\r<BR>\n" );
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document.write( "## 3. Calculate the Test Statistic ($Z_{\text{test}}$)\r<BR>\n" );
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document.write( "We use the $Z$-test formula:<BR>\n" );
document.write( "$$Z_{\text{test}} = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$\r<BR>\n" );
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document.write( "### a. Calculate the Standard Error ($\text{SE}$)<BR>\n" );
document.write( "$$\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{50}} \approx \frac{12}{7.071}$$<BR>\n" );
document.write( "$$\text{SE} \approx 1.697 \text{ bpm}$$\r<BR>\n" );
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document.write( "### b. Calculate the $Z$-score<BR>\n" );
document.write( "$$Z_{\text{test}} = \frac{70 - 72}{1.697} = \frac{-2}{1.697}$$<BR>\n" );
document.write( "$$Z_{\text{test}} \approx -1.1786$$\r<BR>\n" );
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document.write( "## 4. Determine the P-value\r<BR>\n" );
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document.write( "Since this is a left-tailed test, the $P$-value is the probability of observing a $Z$-score of $-1.1786$ or lower.<BR>\n" );
document.write( "$$P\text{-value} = P(Z \le -1.1786)$$\r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator:<BR>\n" );
document.write( "$$P\text{-value} \approx 0.1192$$\r<BR>\n" );
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document.write( "## 5. Make a Decision and Conclusion\r<BR>\n" );
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document.write( "We compare the $P$-value to the assumed significance level, $\alpha = 0.05$.\r<BR>\n" );
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document.write( "$$P\text{-value} (0.1192) > \alpha (0.05)$$\r<BR>\n" );
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document.write( "**Decision:** Since the $P$-value is greater than $\alpha$, we **fail to reject the null hypothesis ($H_0$)**.\r<BR>\n" );
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document.write( "**Conclusion:**<BR>\n" );
document.write( "The psychiatrist **cannot conclude** that the new drug significantly lowers the heart rate. The sample mean of $70 \text{ bpm}$ is not far enough below the population mean of $72 \text{ bpm}$ to be considered statistically significant at the $0.05$ level. The observed difference could easily be due to random sampling variation. </SPAN>\n" );
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