Answer:
a
i So the approximate distribution of
is
and ![\sigma_{\= X} = 0.783](https://tex.z-dn.net/?f=%5Csigma_%7B%5C%3D%20X%7D%20%3D%200.783)
ii So the approximate distribution of
is
and ![\sigma_{\= Y} = 0.645](https://tex.z-dn.net/?f=%5Csigma_%7B%5C%3D%20Y%7D%20%3D%200.645)
b
the approximate distribution of
is
and ![\sigma_{\= X - \=Y}=1.029](https://tex.z-dn.net/?f=%5Csigma_%7B%5C%3D%20X%20%20-%20%5C%3DY%7D%3D1.029)
Here we can see that the mean of the approximate distribution is negative which tell us that this negative value of the data for
sample are more and their frequency occurrence is higher than the positive values
c
the value of
is
Step-by-step explanation:
From the question we are given that
The expected tensile strength of the type A steel is ![\mu_A = 103 ksi](https://tex.z-dn.net/?f=%5Cmu_A%20%3D%20103%20ksi)
The standard deviation of type A steel is ![\sigma_A = 7ksi](https://tex.z-dn.net/?f=%5Csigma_A%20%3D%207ksi)
The expected tensile strength of the type B steel is
The standard deviation of type B steel is ![\sigma_B = 5 \ ksi](https://tex.z-dn.net/?f=%5Csigma_B%20%3D%205%20%5C%20ksi)
Also the assumptions are
Let
be the sample average tensile strength of a random sample of 80 type-A specimens
Here ![n_a =80](https://tex.z-dn.net/?f=n_a%20%3D80)
Let
be the sample average tensile strength of a random sample of 60 type-B specimens.
Here ![n_b = 60](https://tex.z-dn.net/?f=n_b%20%3D%2060)
Let the sampling distribution of the mean be
![\mu _ {\= X} = \mu](https://tex.z-dn.net/?f=%5Cmu%20_%20%7B%5C%3D%20X%7D%20%3D%20%5Cmu)
![=103](https://tex.z-dn.net/?f=%3D103)
Let the sampling distribution of the standard deviation be
![\sigma _{\= X} = \frac{\sigma }{\sqrt{n_a} }](https://tex.z-dn.net/?f=%5Csigma%20_%7B%5C%3D%20X%7D%20%3D%20%5Cfrac%7B%5Csigma%20%7D%7B%5Csqrt%7Bn_a%7D%20%7D)
![= \frac{7}{\sqrt{80} }](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B7%7D%7B%5Csqrt%7B80%7D%20%7D)
![=0.783](https://tex.z-dn.net/?f=%3D0.783)
So What this mean is that the approximate distribution of
is
and ![\sigma_{\= X} = 0.783](https://tex.z-dn.net/?f=%5Csigma_%7B%5C%3D%20X%7D%20%3D%200.783)
For ![\= Y](https://tex.z-dn.net/?f=%5C%3D%20Y)
The sampling distribution of the sample mean is
![\mu_{\= Y} = \mu](https://tex.z-dn.net/?f=%5Cmu_%7B%5C%3D%20Y%7D%20%3D%20%5Cmu)
![= 105](https://tex.z-dn.net/?f=%3D%20105)
The sampling distribution of the standard deviation is
![\sigma _{\= Y} = \frac{\sigma }{\sqrt{n_b} }](https://tex.z-dn.net/?f=%5Csigma%20_%7B%5C%3D%20Y%7D%20%3D%20%5Cfrac%7B%5Csigma%20%7D%7B%5Csqrt%7Bn_b%7D%20%7D)
![= \frac{5}{\sqrt{60} }](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B5%7D%7B%5Csqrt%7B60%7D%20%7D)
![= 0.645](https://tex.z-dn.net/?f=%3D%200.645)
So What this mean is that the approximate distribution of
is
and
Now to obtain the approximate distribution for ![\=X - \= Y](https://tex.z-dn.net/?f=%5C%3DX%20%20-%20%5C%3D%20Y)
![E (\= X - \= Y) = E (\= X) - E(\= Y)](https://tex.z-dn.net/?f=E%20%28%5C%3D%20X%20-%20%5C%3D%20Y%29%20%3D%20E%20%28%5C%3D%20X%29%20-%20E%28%5C%3D%20Y%29)
![= \mu_{\= X} - \mu_{\= Y}](https://tex.z-dn.net/?f=%3D%20%20%5Cmu_%7B%5C%3D%20X%7D%20-%20%5Cmu_%7B%5C%3D%20Y%7D)
![= 103 -105](https://tex.z-dn.net/?f=%3D%20103%20-105)
![= -2](https://tex.z-dn.net/?f=%3D%20-2)
The standard deviation of
is
![\sigma_{\= X - \=Y} = \sqrt{\sigma_{\= X}^2 - \sigma_{\= Y}^2}](https://tex.z-dn.net/?f=%5Csigma_%7B%5C%3D%20X%20%20-%20%5C%3DY%7D%20%3D%20%5Csqrt%7B%5Csigma_%7B%5C%3D%20X%7D%5E2%20-%20%5Csigma_%7B%5C%3D%20Y%7D%5E2%7D)
![= \sqrt{(0.783)^2 + (0.645)^2}](https://tex.z-dn.net/?f=%3D%20%5Csqrt%7B%280.783%29%5E2%20%2B%20%280.645%29%5E2%7D)
![=1.029](https://tex.z-dn.net/?f=%3D1.029)
Now to find the value of ![P(-1 \le \=X - \= Y \le 1)](https://tex.z-dn.net/?f=P%28-1%20%5Cle%20%5C%3DX%20-%20%5C%3D%20Y%20%20%5Cle%201%29)
Let us assume that ![F = \= X - \= Y](https://tex.z-dn.net/?f=F%20%3D%20%5C%3D%20X%20-%20%5C%3D%20Y)
![= P [\frac{-1 -E (F)}{\sigma_F} \le Z \le \frac{1-E(F)}{\sigma_F} ]](https://tex.z-dn.net/?f=%3D%20P%20%5B%5Cfrac%7B-1%20-E%20%28F%29%7D%7B%5Csigma_F%7D%20%5Cle%20Z%20%5Cle%20%20%5Cfrac%7B1-E%28F%29%7D%7B%5Csigma_F%7D%20%5D)
![= P[\frac{-1-(-2)}{1.029} \le Z \le \frac{1-(-2)}{1.029} ]](https://tex.z-dn.net/?f=%3D%20P%5B%5Cfrac%7B-1-%28-2%29%7D%7B1.029%7D%20%20%5Cle%20%20Z%20%5Cle%20%20%5Cfrac%7B1-%28-2%29%7D%7B1.029%7D%20%5D)
![= P[0.972 \le Z \le 2.95]](https://tex.z-dn.net/?f=%3D%20%20P%5B0.972%20%5Cle%20Z%20%5Cle%202.95%5D)
![= P(Z \le 0.972) - P(Z \le 2.95)](https://tex.z-dn.net/?f=%3D%20P%28Z%20%5Cle%200.972%29%20-%20P%28Z%20%5Cle%202.95%29)
Using the z-table to obtain their z-score
![= 0.8345 - 0.9984](https://tex.z-dn.net/?f=%3D%200.8345%20-%200.9984)
![= -0.1639](https://tex.z-dn.net/?f=%3D%20-0.1639)