Question 1170653
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.