document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185790>Question 1185790</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A new weight-reducing technique,consisting of a liquid protein diet, currently tests by the food and Drug Administration (FDA) before it's introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let sample mean 1 be true mean weight of individuals before starting the diet and let sample mean 1 be the true mean weight of individuals after 3 weeks on the diet.<BR>\n" );
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document.write( "                     1.    2.    3.    4.    5<BR>\n" );
document.write( "Weight before diet. 156.  201.  194.  203.  210.  <BR>\n" );
document.write( "Weight after diet.   149.  196.  191.  197.  206. \r<BR>\n" );
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document.write( "Given that SD = 1.58, test to determine if the diet is effective at reducing weight. Use alpha =0.1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849612 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=849612\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a *paired* t-test because we have measurements from the same individuals before and after the diet.  This pairing is crucial because it accounts for individual differences in weight.\r<BR>\n" );
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document.write( "Here's how to conduct the paired t-test:\r<BR>\n" );
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document.write( "1. **Calculate the Differences:**  Find the difference (d) between the \"before\" and \"after\" weights for each person.  d = weight_before - weight_after\r<BR>\n" );
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document.write( "   ```<BR>\n" );
document.write( "   Person | Before | After | Difference (d)<BR>\n" );
document.write( "   -------|--------|-------|---------------<BR>\n" );
document.write( "   1      | 156    | 149   | 7<BR>\n" );
document.write( "   2      | 201    | 196   | 5<BR>\n" );
document.write( "   3      | 194    | 191   | 3<BR>\n" );
document.write( "   4      | 203    | 197   | 6<BR>\n" );
document.write( "   5      | 210    | 206   | 4<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "2. **Calculate the Mean Difference (d̄):**\r<BR>\n" );
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document.write( "   ```<BR>\n" );
document.write( "   d̄ = (7 + 5 + 3 + 6 + 4) / 5 = 25 / 5 = 5<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "3. **Calculate the Standard Deviation of the Differences (sd):** You've already provided this: sd = 1.58\r<BR>\n" );
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document.write( "4. **Hypotheses:**\r<BR>\n" );
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document.write( "   *   **Null Hypothesis (H0):** The diet is *not* effective at reducing weight (the mean difference is zero or positive). μd ≥ 0<BR>\n" );
document.write( "   *   **Alternative Hypothesis (H1):** The diet *is* effective at reducing weight (the mean difference is negative). μd < 0 (This is a left-tailed test).\r<BR>\n" );
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document.write( "5. **Significance Level (alpha):** α = 0.1\r<BR>\n" );
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document.write( "6. **Test Statistic:**\r<BR>\n" );
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document.write( "   ```<BR>\n" );
document.write( "   t = (d̄ - μd) / (sd / √n)<BR>\n" );
document.write( "   t = (5 - 0) / (1.58 / √5)<BR>\n" );
document.write( "   t = 5 / 0.707<BR>\n" );
document.write( "   t ≈ 7.07<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "7. **Degrees of Freedom:** df = n - 1 = 5 - 1 = 4\r<BR>\n" );
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document.write( "8. **Critical Value:** For a one-tailed (left-tailed) t-test with α = 0.1 and df = 4, the critical value is approximately -1.476 (you would look this up in a t-table or use a calculator).\r<BR>\n" );
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document.write( "9. **Decision Rule:** Reject H0 if the calculated t-statistic is less than the critical value.\r<BR>\n" );
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document.write( "10. **Conclusion:**\r<BR>\n" );
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document.write( "The calculated t-statistic (7.07) is *much greater* than the critical value (-1.476).  Even though it is a left-tailed test, the value is so far in the *right tail* that we can automatically reject the null hypothesis.\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "At a 10% significance level, there is *very strong* evidence to conclude that the liquid protein diet is effective at reducing weight.  The very large t-value and correspondingly small p-value strongly suggest that the observed weight loss is not due to random chance.<BR>\n" );
document.write( " </SPAN>\n" );
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