document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171198>Question 1171198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that historically the average age at which Canadians have been happier than at any other time in their lives is 37 years. A random sample of 50 Canadians taken in 2020 indicated that the mean happiest age was 40 years with a standard deviation equal to 16 years. At the 10 percent level of significance, does this sample indicate that there has been a change in the average age when Canadians are happiest? \r<BR>\n" );
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document.write( "a) State the null and alternative hypothesis. \r<BR>\n" );
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document.write( "b)State the decision rule\r<BR>\n" );
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document.write( "c)Compute the observed value of the test statistic.\r<BR>\n" );
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document.write( "d) What is your decision regarding the null hypothesis?   Don’t forget your concluding statement.\r<BR>\n" );
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document.write( "e) Determine the p-value for this test.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850978 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=850978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this hypothesis test step-by-step.\r<BR>\n" );
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document.write( "**a) State the Null and Alternative Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** The average age at which Canadians are happiest is still 37 years.<BR>\n" );
document.write( "    * H₀: μ = 37<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The average age at which Canadians are happiest has changed (is not 37 years).<BR>\n" );
document.write( "    * H₁: μ ≠ 37 (two-tailed test)\r<BR>\n" );
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document.write( "**b) State the Decision Rule**\r<BR>\n" );
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document.write( "* Significance level (α) = 10% = 0.10<BR>\n" );
document.write( "* Sample size (n) = 50<BR>\n" );
document.write( "* Since the population standard deviation is unknown and the sample size is large (n > 30), we will use a z-test.<BR>\n" );
document.write( "* For a two-tailed test with α = 0.10, the critical z-values are ±z<sub>α/2</sub> = ±z<sub>0.05</sub>.<BR>\n" );
document.write( "* Using a standard normal distribution table or a calculator, we find that z<sub>0.05</sub> ≈ 1.645.<BR>\n" );
document.write( "* **Decision Rule:** Reject H₀ if the absolute value of the calculated z-statistic is greater than 1.645 (|z| > 1.645).\r<BR>\n" );
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document.write( "**c) Compute the Observed Value of the Test Statistic**\r<BR>\n" );
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document.write( "* Sample mean (x̄) = 40 years<BR>\n" );
document.write( "* Population mean (μ) = 37 years<BR>\n" );
document.write( "* Sample standard deviation (s) = 16 years<BR>\n" );
document.write( "* Sample size (n) = 50\r<BR>\n" );
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document.write( "We use the z-statistic formula:\r<BR>\n" );
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document.write( "z = (x̄ - μ) / (s / √n)\r<BR>\n" );
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document.write( "z = (40 - 37) / (16 / √50)\r<BR>\n" );
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document.write( "z = 3 / (16 / 7.071)\r<BR>\n" );
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document.write( "z = 3 / 2.2627\r<BR>\n" );
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document.write( "z ≈ 1.326\r<BR>\n" );
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document.write( "**d) What is your Decision Regarding the Null Hypothesis?**\r<BR>\n" );
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document.write( "* The calculated z-statistic is 1.326.<BR>\n" );
document.write( "* The critical z-values are ±1.645.<BR>\n" );
document.write( "* Since |1.326| < 1.645, we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**Concluding Statement:** There is not sufficient evidence at the 10% significance level to conclude that the average age when Canadians are happiest has changed from 37 years.\r<BR>\n" );
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document.write( "**e) Determine the p-value for this Test**\r<BR>\n" );
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document.write( "* The calculated z-statistic is 1.326.<BR>\n" );
document.write( "* Since this is a two-tailed test, we need to find the probability of observing a z-statistic as extreme as 1.326 or -1.326.<BR>\n" );
document.write( "* Using a standard normal distribution table or a calculator, we find the p-value:<BR>\n" );
document.write( "    * P(Z > 1.326) ≈ 0.0924<BR>\n" );
document.write( "    * P(Z < -1.326) ≈ 0.0924<BR>\n" );
document.write( "    * p-value = 2 * 0.0924 ≈ 0.1848\r<BR>\n" );
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document.write( "* Since the p-value (0.1848) is greater than the significance level (0.10), we fail to reject the null hypothesis.<BR>\n" );
document.write( " </SPAN>\n" );
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