A bank analyzes customer adoption of its new mobile banking app. Historically, 45% of customers use online banking services. Use a normal distribution to approximate the probability that between 62 and 70 customers out of a sample of 100 will adopt the online banking service.

• A. 0.00165
• B. 0.000463
• C. 0.99
• D. 0.01

Answer: (B)

Verified step by step guidance

1. Identify the problem as a binomial distribution problem where n = 100 (sample size) and p = 0.45 (probability of success).
2. Use the normal approximation to the binomial distribution. Calculate the mean (μ) and standard deviation (σ) of the binomial distribution using the formulas: μ = n * p and σ = sqrt(n * p * (1 - p)).
3. Convert the binomial problem to a normal distribution problem by applying the continuity correction. For the range 62 to 70, adjust the values to 61.5 and 70.5.
4. Calculate the z-scores for 61.5 and 70.5 using the formula: z = (X - μ) / σ, where X is the value for which you are calculating the z-score.
5. Use the standard normal distribution table to find the probabilities corresponding to the calculated z-scores and determine the probability that the number of customers is between 62 and 70 by finding the difference between these probabilities.