document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179992>Question 1179992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The personnel director of a large corporation wishes to study absenteeism among<BR>\n" );
document.write( "clerical workers at the corporation• s central office during the year. A random sample of 25<BR>\n" );
document.write( "clerical workers reveals the following: [25 Marks]<BR>\n" );
document.write( "€ Absenteeism: X = 9.7 days, S = 4.0 days.<BR>\n" );
document.write( "€ 12 clerical workers were absent more than 10 days.<BR>\n" );
document.write( "a. Construct a 95% confidence interval estimate for the mean number of absences for<BR>\n" );
document.write( "clerical workers during the year.\r<BR>\n" );
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document.write( "b. Construct a 95% confidence interval estimate for the population proportion of clerical<BR>\n" );
document.write( "workers absent more than 10 days during the year. Suppose that the personnel director<BR>\n" );
document.write( "also wishes to take a survey in a branch office. Answer these questions<BR>\n" );
document.write( "c. What sample size is needed to have 95% confidence in estimating the population<BR>\n" );
document.write( "mean absenteeism to within ± 1.5 days if the population standard deviation is<BR>\n" );
document.write( "estimated to be 4.5 days?<BR>\n" );
document.write( "d. How many clerical workers need to be selected to have 90% confidence in estimating<BR>\n" );
document.write( "the population proportion to within ± 0.075 if no previous estimate is available?<BR>\n" );
document.write( "e. Based on (c) and (d), what sample size is needed if a single survey is being<BR>\n" );
document.write( "conducted?<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 #850129 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=850129\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem step-by-step:\r<BR>\n" );
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document.write( "**a. 95% Confidence Interval for Mean Absences:**\r<BR>\n" );
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document.write( "1.  **Identify Given Information:**<BR>\n" );
document.write( "    * Sample mean (x̄) = 9.7 days<BR>\n" );
document.write( "    * Sample standard deviation (s) = 4.0 days<BR>\n" );
document.write( "    * Sample size (n) = 25<BR>\n" );
document.write( "    * Confidence level = 95%\r<BR>\n" );
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document.write( "2.  **Determine Degrees of Freedom:**<BR>\n" );
document.write( "    * df = n - 1 = 25 - 1 = 24\r<BR>\n" );
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document.write( "3.  **Find the Critical t-Value:**<BR>\n" );
document.write( "    * For a 95% confidence level and df = 24, the critical t-value (t*) is approximately 2.064 (from a t-table or calculator).\r<BR>\n" );
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document.write( "4.  **Calculate the Margin of Error (E):**<BR>\n" );
document.write( "    * E = t* * (s / √n)<BR>\n" );
document.write( "    * E = 2.064 * (4.0 / √25)<BR>\n" );
document.write( "    * E = 2.064 * (4.0 / 5)<BR>\n" );
document.write( "    * E = 2.064 * 0.8 = 1.6512\r<BR>\n" );
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document.write( "5.  **Construct the Confidence Interval:**<BR>\n" );
document.write( "    * Lower Bound = x̄ - E = 9.7 - 1.6512 = 8.0488<BR>\n" );
document.write( "    * Upper Bound = x̄ + E = 9.7 + 1.6512 = 11.3512\r<BR>\n" );
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document.write( "    The 95% confidence interval is (8.0488, 11.3512) days.\r<BR>\n" );
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document.write( "**b. 95% Confidence Interval for Proportion of Absences > 10 Days:**\r<BR>\n" );
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document.write( "1.  **Calculate Sample Proportion (p̂):**<BR>\n" );
document.write( "    * p̂ = 12 / 25 = 0.48\r<BR>\n" );
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document.write( "2.  **Find the Critical z-Value:**<BR>\n" );
document.write( "    * For a 95% confidence level, the critical z-value (z*) is 1.96.\r<BR>\n" );
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document.write( "3.  **Calculate the Margin of Error (E):**<BR>\n" );
document.write( "    * E = z* * √(p̂(1 - p̂) / n)<BR>\n" );
document.write( "    * E = 1.96 * √(0.48 * 0.52 / 25)<BR>\n" );
document.write( "    * E = 1.96 * √(0.2496 / 25)<BR>\n" );
document.write( "    * E = 1.96 * √0.009984<BR>\n" );
document.write( "    * E = 1.96 * 0.09992 ≈ 0.1958\r<BR>\n" );
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document.write( "4.  **Construct the Confidence Interval:**<BR>\n" );
document.write( "    * Lower Bound = p̂ - E = 0.48 - 0.1958 = 0.2842<BR>\n" );
document.write( "    * Upper Bound = p̂ + E = 0.48 + 0.1958 = 0.6758\r<BR>\n" );
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document.write( "    The 95% confidence interval is (0.2842, 0.6758).\r<BR>\n" );
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document.write( "**c. Sample Size for Mean Absenteeism (± 1.5 days):**\r<BR>\n" );
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document.write( "1.  **Identify Given Information:**<BR>\n" );
document.write( "    * Margin of error (E) = 1.5 days<BR>\n" );
document.write( "    * Population standard deviation (σ) = 4.5 days<BR>\n" );
document.write( "    * Confidence level = 95% (z* = 1.96)\r<BR>\n" );
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document.write( "2.  **Use the Sample Size Formula:**<BR>\n" );
document.write( "    * n = (z* * σ / E)²<BR>\n" );
document.write( "    * n = (1.96 * 4.5 / 1.5)²<BR>\n" );
document.write( "    * n = (8.82 / 1.5)²<BR>\n" );
document.write( "    * n = 5.88²<BR>\n" );
document.write( "    * n = 34.5744\r<BR>\n" );
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document.write( "3.  **Round Up:**<BR>\n" );
document.write( "    * Since sample size must be a whole number, round up to 35.\r<BR>\n" );
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document.write( "**d. Sample Size for Proportion (± 0.075):**\r<BR>\n" );
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document.write( "1.  **Identify Given Information:**<BR>\n" );
document.write( "    * Margin of error (E) = 0.075<BR>\n" );
document.write( "    * Confidence level = 90% (z* = 1.645)<BR>\n" );
document.write( "    * No previous estimate, so use p̂ = 0.5\r<BR>\n" );
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document.write( "2.  **Use the Sample Size Formula:**<BR>\n" );
document.write( "    * n = (z*² * p̂ * (1 - p̂)) / E²<BR>\n" );
document.write( "    * n = (1.645² * 0.5 * 0.5) / 0.075²<BR>\n" );
document.write( "    * n = (2.706025 * 0.25) / 0.005625<BR>\n" );
document.write( "    * n = 0.67650625 / 0.005625<BR>\n" );
document.write( "    * n = 120.2678\r<BR>\n" );
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document.write( "3.  **Round Up:**<BR>\n" );
document.write( "    * Round up to 121.\r<BR>\n" );
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document.write( "**e. Sample Size for a Single Survey:**\r<BR>\n" );
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document.write( "To conduct a single survey that satisfies both requirements, you need to use the *larger* of the two calculated sample sizes.\r<BR>\n" );
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document.write( "Therefore, the required sample size is 121.<BR>\n" );
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