document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170653>Question 1170653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The conversion rate of our website is 2.5%, meaning that from 1000 website visitors 25 place an order (purchase).<BR>\n" );
document.write( "If the purchase amount per conversion is 15$ and the today' website visitors will be 220,<BR>\n" );
document.write( "a) What is the probability that we will make exactly 150$ today?<BR>\n" );
document.write( "b) What is the probability that we will make between 90$ and 210$ today?<BR>\n" );
document.write( "c) You now want to improve your webpage by changing the UX design, but you are not sure if this will<BR>\n" );
document.write( "increase the conversion rate. So, you run a split A/B test with the old version (A) and the new UX-enhanced<BR>\n" );
document.write( "version (B) of your web page and measure the following results on Google Analytics:<BR>\n" );
document.write( " * Equal sample size of 678 for A and B respectively<BR>\n" );
document.write( " * 17 conversions on the A-page<BR>\n" );
document.write( " * 28 conversions on the B page.<BR>\n" );
document.write( "Can you claim with 95% confidence, that the new (B) page has a higher conversion rate, than A?<BR>\n" );
document.write( "d) What is the min. sample size (assuming equal sample sizes for A and B) needed in order to conclude from a<BR>\n" );
document.write( "difference in sample conversion rates of just 0.1% (convA=2.5% and convB=2.6%) that B is superior with<BR>\n" );
document.write( "95% confidence?<BR>\n" );
document.write( "e) The marketing manager finally decides to change the design of the web page to design B. Previously the<BR>\n" );
document.write( "average time spent on the page was 3 minutes and 20 seconds and the standard deviation was 50 seconds.<BR>\n" );
document.write( "Now after the new design a sample of 100 visits show an average time spent on page of 3 minutes 30<BR>\n" );
document.write( "seconds. She claims that this is an improvement, meaning that people spend now more time on the page.<BR>\n" );
document.write( "Can she be 99% confident that she’s right? Prove your answer<BR>\n" );
document.write( "f) What if the confidence level in e) was relaxed to 95%?<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 #851120 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=851120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each part of this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**a) Probability of making exactly $150 today**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate expected conversions:**<BR>\n" );
document.write( "    * Expected conversions = (Visitors * Conversion rate) = 220 * 0.025 = 5.5 conversions.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Calculate the number of conversions to make $150:**<BR>\n" );
document.write( "    * Conversions needed = $150 / $15 per conversion = 10 conversions.\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Use the binomial distribution:**<BR>\n" );
document.write( "    * P(X = k) = (nCk) * p^k * q^(n-k)<BR>\n" );
document.write( "    * P(X = 10) = (220C10) * (0.025)^10 * (0.975)^210<BR>\n" );
document.write( "    * This is a very small probability, and it's difficult to calculate directly due to the large numbers. We'll use a normal approximation.\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Normal approximation:**<BR>\n" );
document.write( "    * Mean (μ) = np = 5.5<BR>\n" );
document.write( "    * Standard deviation (σ) = √(npq) = √(220 * 0.025 * 0.975) ≈ 2.316<BR>\n" );
document.write( "    * z = (x - μ) / σ<BR>\n" );
document.write( "    * z1 = (9.5 - 5.5) / 2.316 ≈ 1.727<BR>\n" );
document.write( "    * z2 = (10.5 - 5.5) / 2.316 ≈ 2.159<BR>\n" );
document.write( "    * P(9.5 < X < 10.5) = P(1.727 < Z < 2.159)<BR>\n" );
document.write( "    * P(Z < 2.159) ≈ 0.9846<BR>\n" );
document.write( "    * P(Z < 1.727) ≈ 0.9579<BR>\n" );
document.write( "    * P(9.5 < X < 10.5) = 0.9846 - 0.9579 ≈ 0.0267\r<BR>\n" );
document.write( "\n" );
document.write( "5.  **Answer:**<BR>\n" );
document.write( "    * The probability of making exactly $150 is approximately 0.0267.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Probability of making between $90 and $210 today**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate conversions for $90 and $210:**<BR>\n" );
document.write( "    * Conversions for $90 = $90 / $15 = 6 conversions<BR>\n" );
document.write( "    * Conversions for $210 = $210 / $15 = 14 conversions.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Normal approximation:**<BR>\n" );
document.write( "    * z1 = (5.5 - 5.5) / 2.316 ≈ 0<BR>\n" );
document.write( "    * z2 = (14.5 - 5.5) / 2.316 ≈ 3.886<BR>\n" );
document.write( "    * z3 = (6.5 - 5.5) / 2.316 ≈ 0.432<BR>\n" );
document.write( "    * P(6.5 < X < 14.5) = P(0.432 < Z < 3.886)<BR>\n" );
document.write( "    * P(Z < 3.886) ≈ 1<BR>\n" );
document.write( "    * P(Z < 0.432) ≈ 0.667<BR>\n" );
document.write( "    * P(0.432 < Z < 3.886) = 1 - 0.667 = 0.333\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Answer:**<BR>\n" );
document.write( "    * The probability of making between $90 and $210 is approximately 0.333.\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Claiming B has a higher conversion rate with 95% confidence**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate conversion rates:**<BR>\n" );
document.write( "    * ConvA = 17 / 678 ≈ 0.0251 = 2.51%<BR>\n" );
document.write( "    * ConvB = 28 / 678 ≈ 0.0413 = 4.13%\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Calculate standard errors:**<BR>\n" );
document.write( "    * SE_A = √[ConvA * (1 - ConvA) / 678] ≈ 0.00607<BR>\n" );
document.write( "    * SE_B = √[ConvB * (1 - ConvB) / 678] ≈ 0.00778\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Calculate the standard error of the difference:**<BR>\n" );
document.write( "    * SE_diff = √(SE_A² + SE_B²) ≈ √(0.00607² + 0.00778²) ≈ 0.00987\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Calculate the z-score:**<BR>\n" );
document.write( "    * z = (ConvB - ConvA) / SE_diff = (0.0413 - 0.0251) / 0.00987 ≈ 1.641\r<BR>\n" );
document.write( "\n" );
document.write( "5.  **Compare to critical z-value:**<BR>\n" );
document.write( "    * For 95% confidence (one-tailed), z_critical = 1.645\r<BR>\n" );
document.write( "\n" );
document.write( "6.  **Conclusion:**<BR>\n" );
document.write( "    * Since 1.641 < 1.645, you cannot claim with 95% confidence that B has a higher conversion rate.\r<BR>\n" );
document.write( "\n" );
document.write( "**d) Minimum sample size for 0.1% difference with 95% confidence**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Given:**<BR>\n" );
document.write( "    * ConvA = 0.025<BR>\n" );
document.write( "    * ConvB = 0.026<BR>\n" );
document.write( "    * Difference = 0.001<BR>\n" );
document.write( "    * z_critical = 1.96 (two-tailed)\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Formula:**<BR>\n" );
document.write( "    * n = [z² * (p1 * (1-p1) + p2 * (1-p2))] / (p1 - p2)²\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Calculate:**<BR>\n" );
document.write( "    * n = [1.96² * (0.025 * 0.975 + 0.026 * 0.974)] / 0.001²<BR>\n" );
document.write( "    * n ≈ [3.8416 * (0.024375 + 0.025324)] / 0.000001<BR>\n" );
document.write( "    * n ≈ [3.8416 * 0.049699] / 0.000001<BR>\n" );
document.write( "    * n ≈ 190828\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Answer:**<BR>\n" );
document.write( "    * The minimum sample size needed is approximately 190,828 per group.\r<BR>\n" );
document.write( "\n" );
document.write( "**e) 99% confidence about time spent on page**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Given:**<BR>\n" );
document.write( "    * μ0 = 200 seconds (3 min 20 sec)<BR>\n" );
document.write( "    * μ = 210 seconds (3 min 30 sec)<BR>\n" );
document.write( "    * s = 50 seconds<BR>\n" );
document.write( "    * n = 100<BR>\n" );
document.write( "    * α = 0.01 (one-tailed)\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Calculate the t-statistic:**<BR>\n" );
document.write( "    * t = (μ - μ0) / (s / √n) = (210 - 200) / (50 / √100) = 10 / 5 = 2\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Find critical t-value:**<BR>\n" );
document.write( "    * df = 99<BR>\n" );
document.write( "    * t_critical ≈ 2.365\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Conclusion:**<BR>\n" );
document.write( "    * Since 2 < 2.365, she cannot be 99% confident.\r<BR>\n" );
document.write( "\n" );
document.write( "**f) 95% confidence about time spent on page**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Critical t-value:**<BR>\n" );
document.write( "    * df = 99<BR>\n" );
document.write( "    * t_critical ≈ 1.66\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Conclusion:**<BR>\n" );
document.write( "    * Since 2 > 1.66, she can be 95% confident.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );