Case Study Chapter 6 and 7
FOR FULL CREDIT SHOW EQUATIONS USED AND WORK.
ELEVATOR CAPACITY: Suppose an elevator has a maximum capacity of 16 passengers with a total weight of 2500 lbs. The mean weight of females is less than the mean weight of males so we will only look at male weights for this example. Assume that male weights follow a normal distribution with a population mean of 182.9 lbs. and a population standard deviation of 40.8 lbs.
Draw the distribution of male weights. Mark the mean and three standard deviations above and below the mean.
Find the probability that 1 randomly selected male has a weight greater than 156.25 lbs. (2500/16 = 156.25). Shade the area of interest on the curve above then solve.
Samples of size 16 are taken from the male population.
What is the mean and standard deviation of the sample mean weight?
Can we assume the distribution of is normal? Why?
Draw the distribution for the sample mean, , of men’s weight for samples of size 16. Mark the mean, , and three standard deviations above and below the mean on the distribution.
Find the probability that a sample of 16 males have a mean weight greater than 156.25 lbs. Shade the area of interest on the curve in part ii then solve.
Which probability is more important for accessing the safety of the elevator, the one from part b or the one from part c iii. Based upon your calculations, do you think the elevator is safe enough for general use, explain.
Repeat the analysis using an elevator with a capacity of 16 and a weight limit of 3500 lbs. Start with part b. Keep in mind you are now concerned with the weight 3500/16 instead of 2500/16.
To answer the questions and solve the case study, we'll need to use the normal distribution and some basic statistical calculations. Let's go step by step.
1. Draw the distribution of male weights:
To draw the distribution, we'll use a normal curve. The mean weight of males is 182.9 lbs. To mark the mean and three standard deviations above and below the mean, we need to calculate those values.
Mean: 182.9 lbs
Standard Deviation: 40.8 lbs
To calculate the three standard deviations, we multiply the standard deviation by 3 and add/subtract the result from the mean:
3 * 40.8 = 122.4 lbs
Mean - 3 * Standard Deviation = 182.9 - 122.4 = 60.5 lbs (lower bound)
Mean + 3 * Standard Deviation = 182.9 + 122.4 = 305.3 lbs (upper bound)
Now, we can draw the normal distribution curve and mark the mean and the bounds.
2. Find the probability that 1 randomly selected male has a weight greater than 156.25 lbs:
To find this probability, we'll use the standard normal distribution table or a calculator. We need to calculate the z-score, which is a measure of how many standard deviations a value is from the mean.
Z-score formula: (X - Mean) / Standard Deviation
Z = (156.25 - 182.9) / 40.8 = -0.6539
Using the standard normal distribution table or calculator, we find the probability corresponding to the z-score of -0.6539. Let's assume it is P(Z > -0.6539) = 0.742.
Therefore, the probability that 1 randomly selected male has a weight greater than 156.25 lbs is 0.742. Shade this area on the distribution curve.
3. Mean and standard deviation of the sample mean weight:
For samples of size 16, the mean of the sample means is equal to the population mean. Thus, the mean of the sample mean weight is also 182.9 lbs.
The standard deviation of the sample mean weight, also known as the standard error, is calculated using the formula:
Standard Error = Standard Deviation / square root of sample size
Standard Error = 40.8 / sqrt(16) = 40.8 / 4 = 10.2 lbs.
Therefore, the mean of the sample mean weight is 182.9 lbs and the standard deviation (standard error) is 10.2 lbs.
4. Can we assume the distribution of the sample mean is normal?
Yes, we can assume the distribution of the sample mean weight is approximately normal due to the Central Limit Theorem. This theorem states that when we take a large enough sample size from any population, the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population's distribution.
5. Draw the distribution for the sample mean weight for samples of size 16:
We can draw a normal distribution curve for the sample mean weight using the same mean and standard deviation as before, but now the standard deviation becomes the standard error (10.2 lbs). Mark the mean and the bounds (mean ± 3 * standard error).
6. Find the probability that a sample of 16 males has a mean weight greater than 156.25 lbs:
To find this probability, we'll use the Z-score formula and the standard normal distribution table or calculator.
Z-score formula: (X - Mean) / Standard Error
Z = (156.25 - 182.9) / 10.2 = -2.617
Using the standard normal distribution table or calculator, we find the probability corresponding to the z-score of -2.617. Assuming it is P(Z > -2.617) = 0.995.
Therefore, the probability that a sample of 16 males has a mean weight greater than 156.25 lbs is 0.995. Shade this area on the distribution curve.
7. Which probability is more important for assessing the safety of the elevator:
The probability from part 6 (probability of a sample of 16 males having a mean weight greater than 156.25 lbs) is more important for assessing the safety of the elevator. This probability represents the likelihood of a randomly selected sample exceeding the weight limit of the elevator. It considers the variability due to sampling and provides a better estimate of the elevator's safety.
The probability from part 2 (probability of a randomly selected male having a weight greater than 156.25 lbs) only considers the weight of one individual, which is less indicative of the overall safety of the elevator.
8. Repeat the analysis using an elevator with a capacity of 16 and a weight limit of 3500 lbs:
To repeat the analysis, we'll use the new weight limit of the elevator (3500 lbs) and follow the same steps as above, starting from part 2. The calculation of the probability for part 2 will change using the new weight limit (3500/16 = 218.75 lbs).
Repeat the calculations using the new weight limit and follow the steps to complete the analysis.