Final answer:
To answer this question, we need to use the properties of the normal distribution. (a) The probability of a randomly chosen salesperson obtaining a total value more than $378 is approximately 0.2432. (b) A salesperson needs a total sales value of approximately $381.99 to qualify for the free meal.
Step-by-step explanation:
To answer this question, we need to use the properties of the normal distribution.
(a) To find the probability that a randomly chosen salesperson will obtain a total value more than $378, we need to find the area under the curve to the right of $378. We can use a standard normal distribution table or a calculator to find this probability. The z-score is calculated using the formula: z = (x - mean) / standard deviation. So, z = (378 - 333.73) / 63.84 = 0.6952. Looking up the z-score in the standard normal distribution table, we find that the probability is approximately 0.2432.
(b) To find the total sales value a salesperson needs to qualify for the free meal, we need to find the cutoff value that corresponds to the top 18% of the distribution. Again, we can use the standard normal distribution table or a calculator to find this cutoff value. The z-score corresponding to the top 18% is approximately 0.920 and can be found using the inverse normal function on a calculator. So, the salesperson would need a total sales value equal to: cutoff value * standard deviation + mean = 0.920 * 63.84 + 333.73 = $381.99 (approximately).