Question 1170653: The conversion rate of our website is 2.5%, meaning that from 1000 website visitors 25 place an order (purchase).
If the purchase amount per conversion is 15$ and the today' website visitors will be 220,
a) What is the probability that we will make exactly 150$ today?
b) What is the probability that we will make between 90$ and 210$ today?
c) You now want to improve your webpage by changing the UX design, but you are not sure if this will
increase the conversion rate. So, you run a split A/B test with the old version (A) and the new UX-enhanced
version (B) of your web page and measure the following results on Google Analytics:
* Equal sample size of 678 for A and B respectively
* 17 conversions on the A-page
* 28 conversions on the B page.
Can you claim with 95% confidence, that the new (B) page has a higher conversion rate, than A?
d) What is the min. sample size (assuming equal sample sizes for A and B) needed in order to conclude from a
difference in sample conversion rates of just 0.1% (convA=2.5% and convB=2.6%) that B is superior with
95% confidence?
e) The marketing manager finally decides to change the design of the web page to design B. Previously the
average time spent on the page was 3 minutes and 20 seconds and the standard deviation was 50 seconds.
Now after the new design a sample of 100 visits show an average time spent on page of 3 minutes 30
seconds. She claims that this is an improvement, meaning that people spend now more time on the page.
Can she be 99% confident that she’s right? Prove your answer
f) What if the confidence level in e) was relaxed to 95%?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down each part of this problem step-by-step.
**a) Probability of making exactly $150 today**
1. **Calculate expected conversions:**
* Expected conversions = (Visitors * Conversion rate) = 220 * 0.025 = 5.5 conversions.
2. **Calculate the number of conversions to make $150:**
* Conversions needed = $150 / $15 per conversion = 10 conversions.
3. **Use the binomial distribution:**
* P(X = k) = (nCk) * p^k * q^(n-k)
* P(X = 10) = (220C10) * (0.025)^10 * (0.975)^210
* This is a very small probability, and it's difficult to calculate directly due to the large numbers. We'll use a normal approximation.
4. **Normal approximation:**
* Mean (μ) = np = 5.5
* Standard deviation (σ) = √(npq) = √(220 * 0.025 * 0.975) ≈ 2.316
* z = (x - μ) / σ
* z1 = (9.5 - 5.5) / 2.316 ≈ 1.727
* z2 = (10.5 - 5.5) / 2.316 ≈ 2.159
* P(9.5 < X < 10.5) = P(1.727 < Z < 2.159)
* P(Z < 2.159) ≈ 0.9846
* P(Z < 1.727) ≈ 0.9579
* P(9.5 < X < 10.5) = 0.9846 - 0.9579 ≈ 0.0267
5. **Answer:**
* The probability of making exactly $150 is approximately 0.0267.
**b) Probability of making between $90 and $210 today**
1. **Calculate conversions for $90 and $210:**
* Conversions for $90 = $90 / $15 = 6 conversions
* Conversions for $210 = $210 / $15 = 14 conversions.
2. **Normal approximation:**
* z1 = (5.5 - 5.5) / 2.316 ≈ 0
* z2 = (14.5 - 5.5) / 2.316 ≈ 3.886
* z3 = (6.5 - 5.5) / 2.316 ≈ 0.432
* P(6.5 < X < 14.5) = P(0.432 < Z < 3.886)
* P(Z < 3.886) ≈ 1
* P(Z < 0.432) ≈ 0.667
* P(0.432 < Z < 3.886) = 1 - 0.667 = 0.333
3. **Answer:**
* The probability of making between $90 and $210 is approximately 0.333.
**c) Claiming B has a higher conversion rate with 95% confidence**
1. **Calculate conversion rates:**
* ConvA = 17 / 678 ≈ 0.0251 = 2.51%
* ConvB = 28 / 678 ≈ 0.0413 = 4.13%
2. **Calculate standard errors:**
* SE_A = √[ConvA * (1 - ConvA) / 678] ≈ 0.00607
* SE_B = √[ConvB * (1 - ConvB) / 678] ≈ 0.00778
3. **Calculate the standard error of the difference:**
* SE_diff = √(SE_A² + SE_B²) ≈ √(0.00607² + 0.00778²) ≈ 0.00987
4. **Calculate the z-score:**
* z = (ConvB - ConvA) / SE_diff = (0.0413 - 0.0251) / 0.00987 ≈ 1.641
5. **Compare to critical z-value:**
* For 95% confidence (one-tailed), z_critical = 1.645
6. **Conclusion:**
* Since 1.641 < 1.645, you cannot claim with 95% confidence that B has a higher conversion rate.
**d) Minimum sample size for 0.1% difference with 95% confidence**
1. **Given:**
* ConvA = 0.025
* ConvB = 0.026
* Difference = 0.001
* z_critical = 1.96 (two-tailed)
2. **Formula:**
* n = [z² * (p1 * (1-p1) + p2 * (1-p2))] / (p1 - p2)²
3. **Calculate:**
* n = [1.96² * (0.025 * 0.975 + 0.026 * 0.974)] / 0.001²
* n ≈ [3.8416 * (0.024375 + 0.025324)] / 0.000001
* n ≈ [3.8416 * 0.049699] / 0.000001
* n ≈ 190828
4. **Answer:**
* The minimum sample size needed is approximately 190,828 per group.
**e) 99% confidence about time spent on page**
1. **Given:**
* μ0 = 200 seconds (3 min 20 sec)
* μ = 210 seconds (3 min 30 sec)
* s = 50 seconds
* n = 100
* α = 0.01 (one-tailed)
2. **Calculate the t-statistic:**
* t = (μ - μ0) / (s / √n) = (210 - 200) / (50 / √100) = 10 / 5 = 2
3. **Find critical t-value:**
* df = 99
* t_critical ≈ 2.365
4. **Conclusion:**
* Since 2 < 2.365, she cannot be 99% confident.
**f) 95% confidence about time spent on page**
1. **Critical t-value:**
* df = 99
* t_critical ≈ 1.66
2. **Conclusion:**
* Since 2 > 1.66, she can be 95% confident.
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