document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186605>Question 1186605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6. A school administrator claims that less than 50% of the students of the school are dissatisfied by the community cafeteria service. Test this claim by using sample data obtained from a survey of 500 students of the school where 54% indicated their dissatisfaction of the community cafeteria service. Use alpha = 0.05\r<BR>\n" );
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document.write( "answer the problem using the steps in hypothesis testing. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849423 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=849423\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct a hypothesis test for the school administrator's claim:\r<BR>\n" );
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document.write( "**1. State the Hypotheses:**\r<BR>\n" );
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document.write( "*   **Null Hypothesis (H₀):**  p ≥ 0.50 (The proportion of dissatisfied students is greater than or equal to 50%)<BR>\n" );
document.write( "*   **Alternative Hypothesis (H₁ or Ha):** p < 0.50 (The proportion of dissatisfied students is less than 50%)  This is a *left-tailed test*.\r<BR>\n" );
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document.write( "**2. Determine the Level of Significance (alpha):**\r<BR>\n" );
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document.write( "*   α = 0.05 (This is given in the problem.)\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic:**\r<BR>\n" );
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document.write( "*   **Sample Proportion (p̂):** p̂ = 0.54 (54% of the 500 students surveyed)<BR>\n" );
document.write( "*   **Hypothesized Proportion (p₀):** p₀ = 0.50 (The proportion under the null hypothesis)<BR>\n" );
document.write( "*   **Standard Error (SE):** SE = sqrt[(p₀ * (1 - p₀)) / n] = sqrt[(0.50 * (1 - 0.50)) / 500] = sqrt(0.0005) ≈ 0.0224<BR>\n" );
document.write( "*   **Z-score:** z = (p̂ - p₀) / SE = (0.54 - 0.50) / 0.0224 ≈ 1.79\r<BR>\n" );
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document.write( "**4. Determine the Critical Value or the P-value:**\r<BR>\n" );
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document.write( "*   **Critical Value (Left-tailed test):** For α = 0.05, the critical z-value is -1.645. (You can find this using a z-table or a calculator.)<BR>\n" );
document.write( "*   **P-value:** Since this is a left-tailed test, the p-value is the probability of observing a z-score as extreme as, or more extreme than, 1.79. Using a z-table or calculator, we find the area to the *left* of z = 1.79. Because this is the opposite of the tail we are interested in, we subtract this from 1. So p-value = 1 - 0.9633 = 0.0367.\r<BR>\n" );
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document.write( "**5. Make a Decision:**\r<BR>\n" );
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document.write( "*   **Using Critical Values:** Our calculated z-score (1.79) is *greater* than the critical z-value (-1.645). Therefore, we fail to reject the null hypothesis.<BR>\n" );
document.write( "*   **Using P-value:** The p-value (0.0367) is *less* than the level of significance (0.05). Because this is a left tailed test, we are interested in the values to the left of the critical value and therefore, reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Conclusion:**\r<BR>\n" );
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document.write( "There *is* sufficient evidence at the 0.05 level of significance to support the school administrator's claim that less than 50% of students are dissatisfied with the cafeteria service.<BR>\n" );
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