document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164644>Question 1164644</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sample of university students who were asked how much they spent on their semester projects is shown in the following table 4.\r<BR>\n" );
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document.write( "One semester project Cost ($)	Number of Students<BR>\n" );
document.write( "15-24	                               21<BR>\n" );
document.write( "25-34	                               55<BR>\n" );
document.write( "35-44	                               10<BR>\n" );
document.write( "45-54	                               91<BR>\n" );
document.write( "55-64	                               80<BR>\n" );
document.write( "65-74	                               39<BR>\n" );
document.write( "                <BR>\n" );
document.write( "Required:\r<BR>\n" );
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document.write( "   a) Find the following from given Table 4<BR>\n" );
document.write( "i)	Arithmetic Mean	ii) Median Cost		iii) Q1 & D4	iv) 25th Percentile<BR>\n" );
document.write( "				 <BR>\n" );
document.write( "   b) Categorize the following variable (as Nominal/Ordinal/Ratio), give reason.<BR>\n" );
document.write( "i)	Comments on services		ii) Price of Shirts	iii) Income of people.<BR>\n" );
document.write( "iv)  Driving license number.\r<BR>\n" );
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document.write( "Compute the Variance and Standard deviation of the given following table?\r<BR>\n" );
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document.write( "Class intervals	Frequency<BR>\n" );
document.write( "1-10	4<BR>\n" );
document.write( "11-20	5<BR>\n" );
document.write( "21-30	6<BR>\n" );
document.write( "31-40	1<BR>\n" );
document.write( "41-50	2<BR>\n" );
document.write( "51-60	11<BR>\n" );
document.write( "61-70	9<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 #854336 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2261)</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=854336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import pandas as pd<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "# Table 4 data<BR>\n" );
document.write( "data4 = {<BR>\n" );
document.write( "    'Lower': [15, 25, 35, 45, 55, 65],<BR>\n" );
document.write( "    'Upper': [24, 34, 44, 54, 64, 74],<BR>\n" );
document.write( "    'Frequency': [21, 55, 10, 91, 80, 39]<BR>\n" );
document.write( "}<BR>\n" );
document.write( "df4 = pd.DataFrame(data4)<BR>\n" );
document.write( "df4['Midpoint'] = (df4['Lower'] + df4['Upper']) / 2<BR>\n" );
document.write( "df4['f_x'] = df4['Frequency'] * df4['Midpoint']<BR>\n" );
document.write( "df4['f_x2'] = df4['Frequency'] * (df4['Midpoint']**2)<BR>\n" );
document.write( "df4['Cumulative_Frequency'] = df4['Frequency'].cumsum()\r<BR>\n" );
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document.write( "# Part a Calculations<BR>\n" );
document.write( "N4 = df4['Frequency'].sum()<BR>\n" );
document.write( "mean4 = df4['f_x'].sum() / N4\r<BR>\n" );
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document.write( "def get_percentile_val(df, target_rank, N):<BR>\n" );
document.write( "    idx = df[df['Cumulative_Frequency'] >= target_rank].index[0]<BR>\n" );
document.write( "    row = df.iloc[idx]<BR>\n" );
document.write( "    L = row['Lower'] - 0.5<BR>\n" );
document.write( "    i = (row['Upper'] + 0.5) - L<BR>\n" );
document.write( "    f = row['Frequency']<BR>\n" );
document.write( "    cf_prev = df.iloc[idx-1]['Cumulative_Frequency'] if idx > 0 else 0<BR>\n" );
document.write( "    val = L + ((target_rank - cf_prev) / f) * i<BR>\n" );
document.write( "    return val\r<BR>\n" );
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document.write( "median4 = get_percentile_val(df4, N4 * 0.5, N4)<BR>\n" );
document.write( "q1_4 = get_percentile_val(df4, N4 * 0.25, N4)<BR>\n" );
document.write( "d4_4 = get_percentile_val(df4, N4 * 0.4, N4)<BR>\n" );
document.write( "p25_4 = get_percentile_val(df4, N4 * 0.25, N4)\r<BR>\n" );
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document.write( "print(\"Table 4 Results:\")<BR>\n" );
document.write( "print(f\"Total N: {N4}\")<BR>\n" );
document.write( "print(f\"Mean: {mean4}\")<BR>\n" );
document.write( "print(f\"Median: {median4}\")<BR>\n" );
document.write( "print(f\"Q1: {q1_4}\")<BR>\n" );
document.write( "print(f\"D4: {d4_4}\")<BR>\n" );
document.write( "print(f\"P25: {p25_4}\")\r<BR>\n" );
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document.write( "# Problem 3 Table<BR>\n" );
document.write( "data3 = {<BR>\n" );
document.write( "    'Lower': [1, 11, 21, 31, 41, 51, 61],<BR>\n" );
document.write( "    'Upper': [10, 20, 30, 40, 50, 60, 70],<BR>\n" );
document.write( "    'Frequency': [4, 5, 6, 1, 2, 11, 9]<BR>\n" );
document.write( "}<BR>\n" );
document.write( "df3 = pd.DataFrame(data3)<BR>\n" );
document.write( "df3['Midpoint'] = (df3['Lower'] + df3['Upper']) / 2<BR>\n" );
document.write( "df3['f_x'] = df3['Frequency'] * df3['Midpoint']<BR>\n" );
document.write( "df3['f_x2'] = df3['Frequency'] * (df3['Midpoint']**2)\r<BR>\n" );
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document.write( "N3 = df3['Frequency'].sum()<BR>\n" );
document.write( "mean3 = df3['f_x'].sum() / N3<BR>\n" );
document.write( "sum_fx2 = df3['f_x2'].sum()<BR>\n" );
document.write( "sum_fx = df3['f_x'].sum()\r<BR>\n" );
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document.write( "# Sample Variance<BR>\n" );
document.write( "variance3 = (sum_fx2 - (sum_fx**2 / N3)) / (N3 - 1)<BR>\n" );
document.write( "std_dev3 = np.sqrt(variance3)\r<BR>\n" );
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document.write( "print(\"\nProblem 3 Results:\")<BR>\n" );
document.write( "print(f\"Total N: {N3}\")<BR>\n" );
document.write( "print(f\"Mean: {mean3}\")<BR>\n" );
document.write( "print(f\"Variance (Sample): {variance3}\")<BR>\n" );
document.write( "print(f\"Standard Deviation (Sample): {std_dev3}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Table 4 Results:<BR>\n" );
document.write( "Total N: 296<BR>\n" );
document.write( "Mean: 48.6554054054054<BR>\n" );
document.write( "Median: 51.31318681318682<BR>\n" );
document.write( "Q1: 34.13636363636364<BR>\n" );
document.write( "D4: 48.06043956043956<BR>\n" );
document.write( "P25: 34.13636363636364\r<BR>\n" );
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document.write( "Problem 3 Results:<BR>\n" );
document.write( "Total N: 38<BR>\n" );
document.write( "Mean: 41.55263157894737<BR>\n" );
document.write( "Variance (Sample): 473.18634423897583<BR>\n" );
document.write( "Standard Deviation (Sample): 21.752846807693373\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To solve these statistical problems, we first calculate the necessary components for grouped data (midpoints, cumulative frequencies, etc.) for each table.\r<BR>\n" );
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document.write( "### **Part a) Analysis of Table 4 (University Project Costs)**\r<BR>\n" );
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document.write( "| Cost ($) | Frequency ($f$) | Midpoint ($x$) | $f \cdot x$ | Cumulative Freq ($CF$) |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| 15-24 | 21 | 19.5 | 409.5 | 21 |<BR>\n" );
document.write( "| 25-34 | 55 | 29.5 | 1622.5 | 76 |<BR>\n" );
document.write( "| 35-44 | 10 | 39.5 | 395.0 | 86 |<BR>\n" );
document.write( "| 45-54 | 91 | 49.5 | 4504.5 | 177 |<BR>\n" );
document.write( "| 55-64 | 80 | 59.5 | 4760.0 | 257 |<BR>\n" );
document.write( "| 65-74 | 39 | 69.5 | 2710.5 | 296 |<BR>\n" );
document.write( "| **Total** | **$N=296$** | | **$\sum fx = 14402$** | |\r<BR>\n" );
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document.write( "**i) Arithmetic Mean**<BR>\n" );
document.write( "$$\bar{x} = \frac{\sum fx}{N} = \frac{14402}{296} \approx \$48.66$$\r<BR>\n" );
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document.write( "**ii) Median Cost**<BR>\n" );
document.write( "The median position is $N/2 = 296/2 = 148$. This falls into the **45-54** class.<BR>\n" );
document.write( "Lower boundary ($L$) = 44.5, $CF_{prev}$ = 86, $f$ = 91, class width ($i$) = 10.<BR>\n" );
document.write( "$$\text{Median} = L + \left( \frac{\frac{N}{2} - CF_{prev}}{f} \right) \times i = 44.5 + \left( \frac{148 - 86}{91} \right) \times 10 \approx \$51.31$$\r<BR>\n" );
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document.write( "**iii) $Q_1$ and $D_4$**<BR>\n" );
document.write( "* **$Q_1$ (First Quartile):** Position is $N/4 = 74$. Falls in the **25-34** class.<BR>\n" );
document.write( "    $$Q_1 = 24.5 + \left( \frac{74 - 21}{55} \right) \times 10 \approx \$34.14$$<BR>\n" );
document.write( "* **$D_4$ (Fourth Decile):** Position is $4N/10 = 118.4$. Falls in the **45-54** class.<BR>\n" );
document.write( "    $$D_4 = 44.5 + \left( \frac{118.4 - 86}{91} \right) \times 10 \approx \$48.06$$\r<BR>\n" );
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document.write( "**iv) 25th Percentile ($P_{25}$)**<BR>\n" );
document.write( "The 25th percentile is identical to the first quartile ($Q_1$).<BR>\n" );
document.write( "$$P_{25} \approx \$34.14$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### **Part b) Categorization of Variables**\r<BR>\n" );
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document.write( "1.  **Comments on services: Ordinal.**<BR>\n" );
document.write( "    * *Reason:* Comments (like \"Poor\", \"Average\", \"Good\") have a logical rank or order, but the numerical distance between the categories is not defined.<BR>\n" );
document.write( "2.  **Price of Shirts: Ratio.**<BR>\n" );
document.write( "    * *Reason:* It has a natural zero point (a price of \$0 means it's free), and ratios are meaningful (a \$40 shirt is twice as expensive as a \$20 shirt).<BR>\n" );
document.write( "3.  **Income of people: Ratio.**<BR>\n" );
document.write( "    * *Reason:* It possesses a true zero point, and mathematical operations like multiplication/division are meaningful (someone earning \$50k earns half as much as someone earning \$100k).<BR>\n" );
document.write( "4.  **Driving license number: Nominal.**<BR>\n" );
document.write( "    * *Reason:* These are unique identifiers used only for labeling. The numbers do not imply any order, magnitude, or mathematical value.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### **Variance and Standard Deviation (Problem 3)**\r<BR>\n" );
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document.write( "| Class | Freq ($f$) | Midpoint ($x$) | $fx$ | $fx^2$ |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| 1-10 | 4 | 5.5 | 22 | 121 |<BR>\n" );
document.write( "| 11-20 | 5 | 15.5 | 77.5 | 1201.25 |<BR>\n" );
document.write( "| 21-30 | 6 | 25.5 | 153 | 3901.5 |<BR>\n" );
document.write( "| 31-40 | 1 | 35.5 | 35.5 | 1260.25 |<BR>\n" );
document.write( "| 41-50 | 2 | 45.5 | 91 | 4140.5 |<BR>\n" );
document.write( "| 51-60 | 11 | 55.5 | 610.5 | 33882.75 |<BR>\n" );
document.write( "| 61-70 | 9 | 65.5 | 589.5 | 38612.25 |<BR>\n" );
document.write( "| **Total** | **$N=38$** | | **1579** | **83119.5** |\r<BR>\n" );
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document.write( "**1. Variance ($s^2$):**<BR>\n" );
document.write( "Using the sample variance formula:<BR>\n" );
document.write( "$$s^2 = \frac{\sum fx^2 - \frac{(\sum fx)^2}{N}}{N - 1} = \frac{83119.5 - \frac{1579^2}{38}}{37} = \frac{83119.5 - 65611.6}{37} \approx 473.19$$\r<BR>\n" );
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document.write( "**2. Standard Deviation ($s$):**<BR>\n" );
document.write( "$$s = \sqrt{s^2} = \sqrt{473.19} \approx 21.75$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );