document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163084>Question 1163084</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Three supermarket chains in Quezon City each claim to have the lowest overall prices.  As part of an investigative study on supermarket advertising, a particular news company selected nine available grocery items on a particular day.  Then, the price of each selected item was checked   at each of the three chains as shown in the table.  Use   = 0.05 to compare the prices.\r<BR>\n" );
document.write( "\n" );
document.write( "Item	Supermarket 1	Supermarket 2	Supermarket 3<BR>\n" );
document.write( "1	  55.00	  50.10	  52.57<BR>\n" );
document.write( "2	  56.00	  54.05	  59.45<BR>\n" );
document.write( "3	  84.50	  93.85	102.20<BR>\n" );
document.write( "4	109.80	102.65	114.00<BR>\n" );
document.write( "5	117.90	103.15	113.00<BR>\n" );
document.write( "6	198.50	212.25	203.90<BR>\n" );
document.write( "7	248.10	243.20	248.10<BR>\n" );
document.write( "8	229.95	202.95	229.45<BR>\n" );
document.write( "9	271.20	268.25	287.90<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854393 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=854393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=1<BR>\n" );
document.write( "import pandas as pd<BR>\n" );
document.write( "import scipy.stats as stats\r<BR>\n" );
document.write( "\n" );
document.write( "data = {<BR>\n" );
document.write( "    'S1': [55.00, 56.00, 84.50, 109.80, 117.90, 198.50, 248.10, 229.95, 271.20],<BR>\n" );
document.write( "    'S2': [50.10, 54.05, 93.85, 102.65, 103.15, 212.25, 243.20, 202.95, 268.25],<BR>\n" );
document.write( "    'S3': [52.57, 59.45, 102.20, 114.00, 113.00, 203.90, 248.10, 229.45, 287.90]<BR>\n" );
document.write( "}\r<BR>\n" );
document.write( "\n" );
document.write( "df = pd.DataFrame(data)\r<BR>\n" );
document.write( "\n" );
document.write( "# Randomized Complete Block Design (RCBD)<BR>\n" );
document.write( "# We treat \"Items\" as blocks because the price variance between a bag of rice and a candy bar<BR>\n" );
document.write( "# is huge, but the variance across supermarkets for the SAME item is what matters.\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate means<BR>\n" );
document.write( "means = df.mean()\r<BR>\n" );
document.write( "\n" );
document.write( "# Perform Two-Way ANOVA without replication (since each item is a block)<BR>\n" );
document.write( "f_stat, p_val = stats.f_oneway(df['S1'], df['S2'], df['S3'])\r<BR>\n" );
document.write( "\n" );
document.write( "# However, standard f_oneway is for Independent groups. <BR>\n" );
document.write( "# For RCBD we need to extract the block effect.<BR>\n" );
document.write( "# SS_total = SS_between_supermarkets + SS_between_items + SS_error\r<BR>\n" );
document.write( "\n" );
document.write( "# Total Mean<BR>\n" );
document.write( "grand_mean = df.values.mean()\r<BR>\n" );
document.write( "\n" );
document.write( "# SS Total<BR>\n" );
document.write( "ss_total = ((df.values - grand_mean)**2).sum()\r<BR>\n" );
document.write( "\n" );
document.write( "# SS Blocks (Items)<BR>\n" );
document.write( "ss_blocks = 3 * ((df.mean(axis=1) - grand_mean)**2).sum()\r<BR>\n" );
document.write( "\n" );
document.write( "# SS Treatments (Supermarkets)<BR>\n" );
document.write( "ss_treatments = 9 * ((df.mean(axis=0) - grand_mean)**2).sum()\r<BR>\n" );
document.write( "\n" );
document.write( "# SS Error<BR>\n" );
document.write( "ss_error = ss_total - ss_blocks - ss_treatments\r<BR>\n" );
document.write( "\n" );
document.write( "# Degrees of Freedom<BR>\n" );
document.write( "df_blocks = 8<BR>\n" );
document.write( "df_treatments = 2<BR>\n" );
document.write( "df_error = df_blocks * df_treatments\r<BR>\n" );
document.write( "\n" );
document.write( "# Mean Squares<BR>\n" );
document.write( "ms_treatments = ss_treatments / df_treatments<BR>\n" );
document.write( "ms_error = ss_error / df_error\r<BR>\n" );
document.write( "\n" );
document.write( "# F-statistic for Supermarkets<BR>\n" );
document.write( "f_calc = ms_treatments / ms_error<BR>\n" );
document.write( "p_value_final = 1 - stats.f.cdf(f_calc, df_treatments, df_error)\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"{means=}\")<BR>\n" );
document.write( "print(f\"{ss_total=}\")<BR>\n" );
document.write( "print(f\"{ss_blocks=}\")<BR>\n" );
document.write( "print(f\"{ss_treatments=}\")<BR>\n" );
document.write( "print(f\"{ss_error=}\")<BR>\n" );
document.write( "print(f\"{f_calc=}\")<BR>\n" );
document.write( "print(f\"{p_value_final=}\")\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=1<BR>\n" );
document.write( "means=S1    152.327778<BR>\n" );
document.write( "S2    147.827778<BR>\n" );
document.write( "S3    156.730000<BR>\n" );
document.write( "dtype: float64<BR>\n" );
document.write( "ss_total=174336.29727407405<BR>\n" );
document.write( "ss_blocks=173160.07200740743<BR>\n" );
document.write( "ss_treatments=356.63736296296634<BR>\n" );
document.write( "ss_error=819.5879037036509<BR>\n" );
document.write( "f_calc=3.4811383755309335<BR>\n" );
document.write( "p_value_final=0.05556998578932093\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "To compare the prices across the three supermarket chains, we use a **Randomized Complete Block Design (RCBD)**. In this study, the **Items** act as \"Blocks\" because the price difference between items (e.g., item 1 vs item 9) is much larger than the price difference between supermarkets.\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. Hypotheses<BR>\n" );
document.write( "* **$H_0$ (Null Hypothesis):** There is no significant difference in the mean prices across the three supermarkets ($\mu_1 = \mu_2 = \mu_3$).<BR>\n" );
document.write( "* **$H_A$ (Alternative Hypothesis):** At least one supermarket has a significantly different mean price.\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Summary Statistics<BR>\n" );
document.write( "* **Supermarket 1 Mean:** 152.33<BR>\n" );
document.write( "* **Supermarket 2 Mean:** 147.83<BR>\n" );
document.write( "* **Supermarket 3 Mean:** 156.73\r<BR>\n" );
document.write( "\n" );
document.write( "### 3. ANOVA Table (RCBD)<BR>\n" );
document.write( "Using the data provided, we calculate the Sum of Squares ($SS$), Degrees of Freedom ($df$), and Mean Squares ($MS$).\r<BR>\n" );
document.write( "\n" );
document.write( "| Source of Variation | Sum of Squares ($SS$) | $df$ | Mean Square ($MS$) | $F$-calc |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| **Supermarkets (Treatments)** | 356.64 | 2 | 178.32 | **3.48** |<BR>\n" );
document.write( "| **Items (Blocks)** | 173,160.07 | 8 | 21,645.01 | - |<BR>\n" );
document.write( "| **Error** | 819.59 | 16 | 51.22 | - |<BR>\n" );
document.write( "| **Total** | 174,336.30 | 26 | - | - |\r<BR>\n" );
document.write( "\n" );
document.write( "### 4. Decision Rule<BR>\n" );
document.write( "* **Significance Level ($\alpha$):** 0.05<BR>\n" );
document.write( "* **Degrees of Freedom:** $df_1 = 2$, $df_2 = 16$<BR>\n" );
document.write( "* **Critical Value ($F_{crit}$):** Using an F-table for $F(2, 16)$ at $\alpha = 0.05$, the critical value is **3.63**.<BR>\n" );
document.write( "* **p-value:** 0.0556\r<BR>\n" );
document.write( "\n" );
document.write( "### 5. Conclusion<BR>\n" );
document.write( "Since the calculated $F$-value (**3.48**) is **less than** the critical value (**3.63**), and the $p$-value (**0.0556**) is **greater than** $\alpha = 0.05$, we **fail to reject the null hypothesis**.\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Interpretation:** At the 5% level of significance, there is **not enough evidence** to conclude that there is a significant difference in the overall prices among the three supermarket chains in Quezon City. While Supermarket 2 has a lower numerical average in this sample, the variation is not statistically significant.\r<BR>\n" );
document.write( "\n" );
document.write( "Does this result surprise you, or does it align with what you'd expect from \"lowest price\" advertising wars? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );