Final answer:
To find the probability that the mean amount spent in a random sample of pet owners is less than a given value, we can use the Central Limit Theorem and the z-score formula. Given the population mean and standard deviation, we can calculate the z-score and find the corresponding probability.
Step-by-step explanation:
To find the probability that the mean amount spent in a random sample of pet owners is less than a given value, we can use the Central Limit Theorem. Since the distribution of the amount spent is approximately normal, we can assume that the sample mean will also be normally distributed. We can then use the z-score formula to find the desired probability. The z-score is calculated as (sample mean - population mean) / (population standard deviation / sqrt(sample size)). Once we have the z-score, we can find the corresponding probability using a standard normal distribution table or a calculator.
Let's assume the mean amount customers spend for a valentine's gift for their pets is $50 and the standard deviation is $10. If we want to find the probability that the mean amount spent of the sample will be less than $45, we can calculate the z-score as (45 - 50) / (10 / sqrt(sample size)). Let's assume the sample size is 100. The z-score would be -5 / (10 / sqrt(100)) = -5 / 1 = -5.
Using a standard normal distribution table, we can find that the probability associated with a z-score of -5 is approximately 0.0000003, rounding to seven decimal places. Therefore, the probability that the mean amount spent of the sample will be less than $45 is approximately 0.0000003.