1. A student organization held a contest where groups of students could sign up to make a launching device to launch small pumpkins. After each launch, the horizontal distance would be measured from where the pumpkin was launched to where it initially makes impact with the ground. Records are kept after the event each year, and a small sample of measured distances from random groups and trials is shown below:
Pumpkin Horizontal Distances: \( X=\{89,93,91,97,86,95,102,94,104\} \)
a. What is the mean of this sample?
b. What is the median of this sample?
c. What is the sample standard deviation of this sample?
d. What is the population standard deviation of this sample?
2. Answer the following questions using the data from Problem 1. and the standard normal distribution tables (located at the end of this assignment document). The population mean and population standard deviation are as follows: \( \mu=94.56 \quad \sigma=5.48 \)
a. What is \( \phi(-1.65) \) ?
b. Using the given population mean and standard deviation, what distance would the normalized number of \( z=-1.65 \) equate to?
c. What is \( \phi(2.47) \) ?
d. Using the given population mean and standard deviation, what distance would the normalized number of \( z=2.47 \) equate to?
e. What is \( \mathrm{P}(\mathrm{z}>2.47) \) ?
3. The following data is real data and has not yet been normalized. Answer the questions by first converting to a standard normal z value.
a. If \( \mu=130 \) and \( \sigma=20 \), what is \( \mathrm{P}(\mathrm{X}<180) \) ?
b. If \( \mu=-25 \) and \( \sigma=4 \), what is \( \mathrm{P}(\mathrm{X}<-28) \) ?
c. If \( \mu=12 \) and \( \sigma=1 \), what is \( \mathrm{P}(\mathrm{X}>14.5) \) ?
d. If \( \mu=72 \) and \( \sigma=10 \), what is \( P(X>90) \) ?
4. The breaking resistance (strength) for a concrete column can be modeled as a normal variable R with \( \mu_{R}=140,000 \mathrm{lb} \) and \( \sigma_{\mathrm{m}}=18,000 \mathrm{lb} \). The load applied to the column can be modeled as a normal variable L with \( \mu_{\mathrm{L}}=70,000 \mathrm{lb} \) and \( \mathrm{\sigma}_{\mathrm{L}}=17,000 \mathrm{lb} \).
- What is the probability that a particular load will cause a particular column to fail?
