due 11/21/2024 before midnight via Learning Suite 25 possible points
Problems 2-4 are adapted from Principles of Statistics for Engineers and Scientists by William Navidi.
Your company manufacturers widgets. The table below lists the probability mass function associated with the number of widgets a random customer orders.
| x | 1 | 2 | 3 | 4 | 5 |
| p(x) | 0.4 | 0.25 | 0.2 | 0.1 | 0.05 |
a) Find the mean number of widgets a customer ordered
b) Find the variance of the number of widgets ordered
Resistors labeled 100 \(\Omega\) have true resistances that are between 80 \(\Omega\) and 120 \(\Omega\).Let X be the resistance of a randomly chosen resistor. The probability density function of X is given by
\[f(x) = \begin{cases} \frac{x - 80}{800} & 80 < x < 120\\ 0 & \text{ otherwise} \end{cases}\]a) What proportion of resistors have resistances less than 90 \(\Omega\)?
b) Find the mean resistance.
c) Find the standard deviation of the resistances.
The article “Application of Surgical Navigation to Total Hip Arthroplasty” (T. Ecker and S. Murphy, Journal of Engineering in Medicine, 2007: 699–712) reports that in a sample of 123 hip surgeries of a certain type, the average surgery time was 136.9 minutes with a standard deviation of 22.6 minutes.
a. Find a 95% confidence interval for the mean surgery time for this procedure.
b. A surgeon claims that the mean surgery time is between 133.9 and 139.9 minutes. With what level of confidence can this statement be made?
c. Approximately how many surgeries must be sampled so that a 95% confidence interval will specify the mean to within \(\pm\)3 minutes?
d. Find a 98% lower confidence bound for the mean time.
Five measurements are taken of the octane rating for a particular type of gasoline. The results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3. Find a 99% confidence interval for the mean octane rating for this type of gasoline.
Let’s look again at the ANSUR Data on knee heights from the last lab. The knee height dataset for combined males and females is here for your convenience. Import the data and plot a histogram, probability distribution, and cumulative distribution. You can create all three plots using different inputs to the hist function in matplotlib.pyplot. Plot the cumulative distribution with density=True so that the last bin ends at 1. Make sure to try different numbers of bin sizes to settle on something that looks good.
The probability distribution looks close to normal, so let’s fit a normal distribution using the norm.pdf function in scipy.stats. The pdf function requires an x vector, which should span the range of the data, and it requires a mean and standard deviation, which you should calculate from the data. Plot the probability distribution function on the same plot as the probability distribution created above using hist. You might need to revisit the number of bins you used in your plot if the fit doesn’t look very good.