document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170482>Question 1170482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A consumer advocacy group suspects that a local supermarket’s 500 grams of sugar actually weigh less than 500 grams. The group took a random sample of 20 such packages, weighed each one, and found the mean weight for the sample to be 496 grams with a standard deviation of 8 grams. Using 10% significance level, would you conclude that the mean weight is less than 500 grams? Also, find the 90% confidence interval of the true mean.\r<BR>\n" );
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document.write( "Hypothesis:<BR>\n" );
document.write( "H_0: <BR>\n" );
document.write( "H_1: <BR>\n" );
document.write( "The level of significance is a = _____, df = ____, and the critical region is t = ____.<BR>\n" );
document.write( "Compute for the value of one sample t-test.<BR>\n" );
document.write( "Decision rule: <BR>\n" );
document.write( "Conclusion:<BR>\n" );
document.write( "Compute for the confidence interval.\r<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 #851174 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=851174\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's perform the hypothesis test and calculate the confidence interval.\r<BR>\n" );
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document.write( "**1. Define Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The mean weight of sugar packages is 500 grams (μ = 500).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The mean weight of sugar packages is less than 500 grams (μ < 500).\r<BR>\n" );
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document.write( "**2. Set Significance Level (α)**\r<BR>\n" );
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document.write( "* α = 0.10\r<BR>\n" );
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document.write( "**3. Determine Degrees of Freedom (df)**\r<BR>\n" );
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document.write( "* Sample size (n) = 20<BR>\n" );
document.write( "* df = n - 1 = 20 - 1 = 19\r<BR>\n" );
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document.write( "**4. Find the Critical t-value**\r<BR>\n" );
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document.write( "* This is a left-tailed test since H1: μ < 500.<BR>\n" );
document.write( "* Using a t-distribution table or calculator with α = 0.10 and df = 19, the critical t-value is approximately -1.729.\r<BR>\n" );
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document.write( "**5. Compute the t-statistic**\r<BR>\n" );
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document.write( "* Sample mean (x̄) = 496 grams<BR>\n" );
document.write( "* Sample standard deviation (s) = 8 grams<BR>\n" );
document.write( "* Population mean (μ) = 500 grams<BR>\n" );
document.write( "* t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "* t = (496 - 500) / (8 / √20)<BR>\n" );
document.write( "* t = -4 / (8 / 4.472)<BR>\n" );
document.write( "* t = -4 / 1.789<BR>\n" );
document.write( "* t ≈ -2.236\r<BR>\n" );
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document.write( "**6. Decision Rule**\r<BR>\n" );
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document.write( "* Reject H0 if the calculated t-statistic is less than the critical t-value (-1.729).\r<BR>\n" );
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document.write( "**7. Conclusion**\r<BR>\n" );
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document.write( "* Since the calculated t-statistic (-2.236) is less than the critical t-value (-1.729), we reject the null hypothesis.<BR>\n" );
document.write( "* Therefore, we conclude that there is sufficient evidence at the 10% significance level to suggest that the mean weight of the sugar packages is less than 500 grams.\r<BR>\n" );
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document.write( "**8. Compute the 90% Confidence Interval**\r<BR>\n" );
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document.write( "* Confidence level = 90%<BR>\n" );
document.write( "* α = 1 - 0.90 = 0.10<BR>\n" );
document.write( "* α / 2 = 0.05<BR>\n" );
document.write( "* Using a t-distribution table or calculator with α/2 = 0.05 and df = 19, the t-value is approximately 1.729.<BR>\n" );
document.write( "* Confidence interval = x̄ ± t * (s / √n)<BR>\n" );
document.write( "* Confidence interval = 496 ± 1.729 * (8 / √20)<BR>\n" );
document.write( "* Confidence interval = 496 ± 1.729 * 1.789<BR>\n" );
document.write( "* Confidence interval = 496 ± 3.093<BR>\n" );
document.write( "* Confidence interval = (492.907, 499.093)\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **Hypothesis:**<BR>\n" );
document.write( "    * H0: μ = 500<BR>\n" );
document.write( "    * H1: μ < 500<BR>\n" );
document.write( "* **Level of significance:** α = 0.10<BR>\n" );
document.write( "* **Degrees of freedom:** df = 19<BR>\n" );
document.write( "* **Critical region:** t < -1.729<BR>\n" );
document.write( "* **t-statistic:** t ≈ -2.236<BR>\n" );
document.write( "* **Decision rule:** Reject H0 if t < -1.729.<BR>\n" );
document.write( "* **Conclusion:** Reject H0. The mean weight is less than 500 grams.<BR>\n" );
document.write( "* **90% Confidence Interval:** (492.907, 499.093)<BR>\n" );
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