A catalyst researcher states that the diameters (in microns) of the pores in a new product catalyst she has made have an exponential distribution with parameter ë = 0.25 (35 pts).
(a) Determine the mean pore size.
(b) Determine the standard deviation of the pore diameters.
(c) What proportion of the pores are less than 3.0 microns in diameter?
(d) What proportion of the pores are greater than 11.0 microns in diameter?
(e) What is the median pore diameter?
(f) What is the third quartile of the pore distribution?
(g) What is the 99th percentile of the pore diameters?
I found that a and b both equal 4, but I am not sure where to start on part c.
Well, it seems like you're really digging deep into the pores of this problem! Let's see if we can help you with part c.
To find the proportion of pores that are less than 3.0 microns in diameter, we can use the cumulative distribution function (CDF) of the exponential distribution. For an exponential distribution with parameter λ, the CDF is given by:
CDF(x) = 1 - e^(-λx)
In this problem, the parameter λ is given as 0.25. So, to find the proportion of pores less than 3.0 microns, we substitute x = 3.0 into the CDF equation:
CDF(3.0) = 1 - e^(-0.25 * 3.0)
Now, let's grab our calculators and do the math!
CDF(3.0) ≈ 1 - e^(-0.75) ≈ 1 - 0.472 ≈ 0.528
So, approximately 52.8% of the pores are less than 3.0 microns in diameter.
Keep up the good work!
To find the proportion of pores that are less than 3.0 microns in diameter (part c), you can use the cumulative distribution function (CDF) of the exponential distribution.
The CDF of an exponential distribution with parameter λ is given by:
CDF(x) = 1 - e^(-λx)
In this case, the parameter λ is equal to 0.25. To find the proportion of pores that are less than 3.0 microns in diameter, substitute x = 3.0 into the CDF equation:
CDF(3.0) = 1 - e^(-0.25 * 3.0)
Now you can calculate the value using a calculator or by hand:
CDF(3.0) = 1 - e^(-0.75)
The result will give you the proportion of pores that are less than 3.0 microns in diameter.
To determine the proportion of the pores that are less than 3.0 microns in diameter, we need to calculate the cumulative distribution function (CDF) of the exponential distribution at the value of 3.0 microns.
The CDF of the exponential distribution with parameter λ (in this case, λ = 0.25) is given by:
CDF(x) = 1 - e^(-λx)
For part (c), let's substitute λ = 0.25 and x = 3.0 into the CDF formula:
CDF(3.0) = 1 - e^(-0.25*3.0)
You can approximate this value using a calculator or any software that can evaluate exponential functions.
With this approximation, you would get the proportion of the pores that are less than 3.0 microns in diameter.
Let's proceed to part (d). To determine the proportion of the pores that are greater than 11.0 microns in diameter, we need to calculate 1 minus the CDF at 11.0 microns.
CDF(11.0) = 1 - e^(-0.25*11.0)
Again, approximating this value will give you the desired proportion.
Finally, let's move on to part (e). The median pore diameter is the value (let's call it x_median) for which the CDF(x_median) = 0.5.
So, we need to solve the equation:
0.5 = 1 - e^(-0.25*x_median)
Solving this equation will give you the median pore diameter.
Parts (f) and (g) require determining percentiles of the pore distribution.
The third quartile corresponds to the 75th percentile. To find it, we need to solve the equation:
0.75 = 1 - e^(-0.25*x_q3)
Similarly, the 99th percentile can be found by solving the equation:
0.99 = 1 - e^(-0.25*x_99)
For both (f) and (g), solve these equations to obtain x_q3 and x_99, respectively, where x_q3 is the third quartile and x_99 is the 99th percentile of the pore diameters.
Now, armed with this information, you can proceed to find the answers to parts (c) through (g) of the problem.