document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171423>Question 1171423</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researched wants to study the attitudes of college students towards credit cards. She plants to randomly select a sample of n students and give each a survey.  She would like to have at least 50 students in the sample who regularly use credit cards.  If 25% of all college students regularly use credit cards, estimate the minimum size of n so that the sample will contain 50 students who regularly use credit cards with a probability of at least 0.95.\r<BR>\n" );
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document.write( "I know this is a binomial approximation but I am not sure how to set this up. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850936 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=850936\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're right, this is a binomial distribution problem, and we'll use a normal approximation to solve it. Let's break down how to set it up:\r<BR>\n" );
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document.write( "**1. Define the Variables**\r<BR>\n" );
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document.write( "* **p:** The proportion of college students who regularly use credit cards (p = 0.25).<BR>\n" );
document.write( "* **n:** The sample size we need to find.<BR>\n" );
document.write( "* **X:** The number of students in the sample who regularly use credit cards.<BR>\n" );
document.write( "* We want to find the minimum 'n' such that P(X ≥ 50) ≥ 0.95.\r<BR>\n" );
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document.write( "**2. Apply the Normal Approximation**\r<BR>\n" );
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document.write( "Since 'n' is unknown, we'll use the normal approximation to the binomial distribution. For this to be valid, we need np and n(1-p) to be sufficiently large (generally, at least 10). We'll check this condition after finding 'n'.\r<BR>\n" );
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document.write( "* **Mean (μ):** μ = np = 0.25n<BR>\n" );
document.write( "* **Standard Deviation (σ):** σ = √(np(1-p)) = √(0.25n * 0.75) = √(0.1875n)\r<BR>\n" );
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document.write( "**3. Set up the Probability Statement**\r<BR>\n" );
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document.write( "We want P(X ≥ 50) ≥ 0.95. We need to convert this to a standard normal distribution (Z-score).\r<BR>\n" );
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document.write( "* **Continuity Correction:** Since we're approximating a discrete distribution with a continuous one, we use a continuity correction. Instead of X ≥ 50, we use X ≥ 49.5.<BR>\n" );
document.write( "* **Z-score:** Z = (X - μ) / σ = (49.5 - 0.25n) / √(0.1875n)\r<BR>\n" );
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document.write( "So, we want:\r<BR>\n" );
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document.write( "* P(Z ≥ (49.5 - 0.25n) / √(0.1875n)) ≥ 0.95\r<BR>\n" );
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document.write( "**4. Find the Z-value**\r<BR>\n" );
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document.write( "Since we want the probability to be at least 0.95, we need to find the Z-value that corresponds to the area to the *left* being 0.05 (1 - 0.95), because we are looking for the area to the right to be 0.95.\r<BR>\n" );
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document.write( "* Using a Z-table or calculator, the Z-value corresponding to a left-tail area of 0.05 is approximately -1.645.\r<BR>\n" );
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document.write( "**5. Set up the Inequality**\r<BR>\n" );
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document.write( "Now we have:\r<BR>\n" );
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document.write( "* (49.5 - 0.25n) / √(0.1875n) ≤ -1.645\r<BR>\n" );
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document.write( "**6. Solve for n**\r<BR>\n" );
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document.write( "* 49.5 - 0.25n ≤ -1.645√(0.1875n)<BR>\n" );
document.write( "* 49.5 - 0.25n ≤ -0.7115√(n)\r<BR>\n" );
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document.write( "Let's rearrange the inequality:\r<BR>\n" );
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document.write( "* 0.25n - 0.7115√n - 49.5 >= 0\r<BR>\n" );
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document.write( "This is a bit tricky to solve directly. Let's make a substitution to simplify:\r<BR>\n" );
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document.write( "* Let y = √n, so n = y².<BR>\n" );
document.write( "* 0.25y² - 0.7115y - 49.5 >= 0\r<BR>\n" );
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document.write( "Now, we can use the quadratic formula to solve for y:\r<BR>\n" );
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document.write( "* y = [0.7115 ± √(0.7115² - 4 * 0.25 * -49.5)] / (2 * 0.25)<BR>\n" );
document.write( "* y = [0.7115 ± √(0.50623 + 49.5)] / 0.5<BR>\n" );
document.write( "* y = [0.7115 ± √49.99623] / 0.5<BR>\n" );
document.write( "* y = [0.7115 ± 7.0708] / 0.5\r<BR>\n" );
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document.write( "Since √n must be positive, we take the positive solution:\r<BR>\n" );
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document.write( "* y = (0.7115 + 7.0708) / 0.5<BR>\n" );
document.write( "* y = 7.7823 / 0.5<BR>\n" );
document.write( "* y = 15.5646\r<BR>\n" );
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document.write( "Now, substitute back √n = y:\r<BR>\n" );
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document.write( "* √n = 15.5646<BR>\n" );
document.write( "* n = (15.5646)²<BR>\n" );
document.write( "* n ≈ 242.26\r<BR>\n" );
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document.write( "Since n must be an integer, we round up to ensure the probability is at least 0.95.\r<BR>\n" );
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document.write( "* n = 243\r<BR>\n" );
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document.write( "**7. Check the Normal Approximation Conditions**\r<BR>\n" );
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document.write( "* np = 243 * 0.25 = 60.75 (≥ 10)<BR>\n" );
document.write( "* n(1-p) = 243 * 0.75 = 182.25 (≥ 10)\r<BR>\n" );
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document.write( "The conditions are met.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "The minimum sample size 'n' should be 243.<BR>\n" );
document.write( " </SPAN>\n" );
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