document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171064>Question 1171064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The average daily expenditure on the food of a middle-income family is P220, with a standard deviation of P15. A sample of 10 families showed the following expenditure: P220, P190, P205, P180, P215, P200, P187, P195, P182, and P201. Test the hypothesis that the average daily expenditure is still 200. Use 0.01 (one-tailed) level. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850996 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=850996\">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( "**1. State the Hypotheses**\r<BR>\n" );
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document.write( "* We are testing if the average daily expenditure is still 220, so we will use 220 as our population mean.<BR>\n" );
document.write( "* Since the problem does not specify a direction, we will test if the mean is different from 220. This indicates a two-tailed test.<BR>\n" );
document.write( "* **Null Hypothesis (H₀):** The average daily expenditure is P220.<BR>\n" );
document.write( "    * H₀: μ = 220<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The average daily expenditure is not P220.<BR>\n" );
document.write( "    * H₁: μ ≠ 220 (two-tailed test)\r<BR>\n" );
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document.write( "**2. Calculate the Sample Mean and Standard Deviation**\r<BR>\n" );
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document.write( "* Sample data: 220, 190, 205, 180, 215, 200, 187, 195, 182, 201<BR>\n" );
document.write( "* Sample size (n) = 10<BR>\n" );
document.write( "* Calculate the sample mean (x̄):<BR>\n" );
document.write( "    * x̄ = (220 + 190 + 205 + 180 + 215 + 200 + 187 + 195 + 182 + 201) / 10 = 1975 / 10 = 197.5<BR>\n" );
document.write( "* Calculate the sample standard deviation (s):<BR>\n" );
document.write( "    * First, calculate the squared differences from the mean:<BR>\n" );
document.write( "        * (220 - 197.5)^2 = 506.25<BR>\n" );
document.write( "        * (190 - 197.5)^2 = 56.25<BR>\n" );
document.write( "        * (205 - 197.5)^2 = 56.25<BR>\n" );
document.write( "        * (180 - 197.5)^2 = 306.25<BR>\n" );
document.write( "        * (215 - 197.5)^2 = 306.25<BR>\n" );
document.write( "        * (200 - 197.5)^2 = 6.25<BR>\n" );
document.write( "        * (187 - 197.5)^2 = 110.25<BR>\n" );
document.write( "        * (195 - 197.5)^2 = 6.25<BR>\n" );
document.write( "        * (182 - 197.5)^2 = 240.25<BR>\n" );
document.write( "        * (201 - 197.5)^2 = 12.25<BR>\n" );
document.write( "    * Sum of squared differences = 1606.5<BR>\n" );
document.write( "    * Sample variance (s^2) = 1606.5 / (10 - 1) = 1606.5 / 9 = 178.5<BR>\n" );
document.write( "    * Sample standard deviation (s) = √178.5 ≈ 13.36\r<BR>\n" );
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document.write( "**3. Determine the Test Statistic**\r<BR>\n" );
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document.write( "* Since the population standard deviation is unknown and the sample size is small (n < 30), we will use a t-test.<BR>\n" );
document.write( "* t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "* t = (197.5 - 220) / (13.36 / √10)<BR>\n" );
document.write( "* t = -22.5 / (13.36 / 3.162)<BR>\n" );
document.write( "* t = -22.5 / 4.225<BR>\n" );
document.write( "* t ≈ -5.326\r<BR>\n" );
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document.write( "**4. Determine the Critical Value**\r<BR>\n" );
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document.write( "* Significance level (α) = 0.01<BR>\n" );
document.write( "* Degrees of freedom (df) = n - 1 = 10 - 1 = 9<BR>\n" );
document.write( "* Type of test: Two-tailed<BR>\n" );
document.write( "* Using a t-table or calculator, we find the critical t-values for α = 0.01 and df = 9.<BR>\n" );
document.write( "* Critical t-values ≈ ±3.250\r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* Calculated t-statistic: -5.326<BR>\n" );
document.write( "* Critical t-values: ±3.250<BR>\n" );
document.write( "* Since |-5.326| > 3.250, we 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 0.01 significance level to conclude that the average daily expenditure on food is different from P220.<BR>\n" );
document.write( " </SPAN>\n" );
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