Question

Daily demand for a business is normally distributed around a mean of 48 with a standard deviation of 4.

1. what is the probability of receiving 40 or less customers in one day?
2. what is the probability of receiving exactly 50 customers in one day?
3. what is the probability of receiving 47 to 56 customers in one day?
4. what value does the probability density function return for when daily demand equals 42?
5. what level of demand is only exceeded 10% of the time?
6. what is the probability of a stock having a negative yearly return?
7. what is the probability of a stock having a yearly return between 5% and 8%?
8. what is the relative frequency of stocks, from the random sample on the stock returns worksheet, that had a negative yearly return?
9. what is the relative frequency of stocks, from the random sample on the stock returns worksheet, that had between 5% and 8% for a yearly return?
10 what yearly return is exceeded 80% of the time?

Answer 1

The given question involves calculating probabilities and using the probability density function for a normally distributed demand for a business. The question also covers the probability of stock returns and relative frequencies of returns. These calculations require a deep understanding of probability and statistics.

Step-by-step explanation:

The given question is related to the concept of probability and distributions. In this scenario, the daily demand for a business follows a normal distribution with a mean of 48 and a standard deviation of 4. Let's address each part of the question:

1. The probability of receiving 40 or less customers in one day can be found by calculating the area under the normal curve below 40. To find this probability, we need to standardize the value using the z-score formula. The z-score can be calculated as (40 - mean) / standard deviation. Once we have the z-score, we can use a standard normal distribution table or a calculator to find the corresponding probability.
2. If the distribution is truly normal, the probability of receiving exactly 50 customers in one day would be zero, as the probability of any specific value in a continuous distribution is zero.
3. The probability of receiving between 47 and 56 customers in one day can be found by calculating the area under the normal curve between these two values. We need to standardize each value using the z-score formula, find the corresponding areas, and subtract them to find the probability.
4. The probability density function (PDF) represents the relative likelihood of obtaining a specific value from a continuous distribution. To find the PDF value for daily demand equals 42, we need to calculate the corresponding z-score using the z-score formula, and then use the formula for the probability density function of the normal distribution.
5. The level of demand that is exceeded 10% of the time can be found by calculating the z-score corresponding to the 10th percentile of the normal distribution and then converting it back to the original scale using the mean and standard deviation.
6. The probability of a stock having a negative yearly return can be determined by analyzing the historical data of stock returns and calculating the proportion of negative returns.
7. The probability of a stock having a yearly return between 5% and 8% can again be found by analyzing the historical data of stock returns and calculating the proportion of returns falling within this range.
8. The relative frequency of stocks that had a negative yearly return can be calculated by dividing the number of stocks with a negative yearly return by the total number of stocks in the random sample on the stock returns worksheet.
9. The relative frequency of stocks that had between 5% and 8% for a yearly return can be calculated by dividing the number of stocks with a yearly return in that range by the total number of stocks in the random sample on the stock returns worksheet.
10. To find the yearly return that is exceeded 80% of the time, we can calculate the z-score corresponding to the 80th percentile of the normal distribution and then convert it back to the original scale using the mean and standard deviation.

These calculations require a deep understanding of probability, statistics, and the normal distribution. Make sure to double-check the formulas and calculations or consult with a teacher or tutor if needed.