Answer:
7. μ=204.9 and σ=5.4968
8. μ=75.9 and σ=0.7136
9. p=0.9452
Step-by-step explanation:
7. - Given that the population mean =204.9 and the standard deviation is 81.90 and the sample size n=222.
-The sample mean,
is calculated as:
![\mu_x=\mu=204.9, \mu_x=sample \ mean](https://tex.z-dn.net/?f=%5Cmu_x%3D%5Cmu%3D204.9%2C%20%5Cmu_x%3Dsample%20%5C%20mean)
-The standard deviation,
is calculated as:
![\sigma_x=\frac{\sigma}{\sqrt{n}}\\\\=\frac{81.9}{\sqrt{222}}\\\\=5.4968](https://tex.z-dn.net/?f=%5Csigma_x%3D%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B81.9%7D%7B%5Csqrt%7B222%7D%7D%5C%5C%5C%5C%3D5.4968)
8. For a random variable X.
-Given a X's population mean is 75.9, standard deviation is 9.6 and a sample size of 181
-#The sample mean,
is calculated as:
![\mu_x=\mu\\\\=75.9](https://tex.z-dn.net/?f=%5Cmu_x%3D%5Cmu%5C%5C%5C%5C%3D75.9)
#The sample standard deviation is calculated as follows:
![\sigma_x=\frac{\sigma}{\sqrt{n}}\\\\=\frac{9.6}{\sqrt{181}}\\\\=0.7136](https://tex.z-dn.net/?f=%5Csigma_x%3D%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B9.6%7D%7B%5Csqrt%7B181%7D%7D%5C%5C%5C%5C%3D0.7136)
9. Given the population mean, μ=135.7 and σ=88 and n=59
#We calculate the sample mean;
![\mu_x=\mu=135.7](https://tex.z-dn.net/?f=%5Cmu_x%3D%5Cmu%3D135.7)
#Sample standard deviation:
![\sigma_x=\frac{\sigma}{\sqrt{n}}\\\\=\frac{88}{\sqrt{59}}\\\\=11.4566](https://tex.z-dn.net/?f=%5Csigma_x%3D%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B88%7D%7B%5Csqrt%7B59%7D%7D%5C%5C%5C%5C%3D11.4566)
#The sample size, n=59 is at least 30, so we apply Central Limit Theorem:
![P(\bar X>117.4)=P(Z>\frac{117.4-\mu_{\bar x}}{\sigma_x})\\\\=P(Z>\frac{117.4-135.7}{11.4566})\\\\=P(Z>-1.5973)\\\\=1-0.05480 \\\\=0.9452](https://tex.z-dn.net/?f=P%28%5Cbar%20X%3E117.4%29%3DP%28Z%3E%5Cfrac%7B117.4-%5Cmu_%7B%5Cbar%20x%7D%7D%7B%5Csigma_x%7D%29%5C%5C%5C%5C%3DP%28Z%3E%5Cfrac%7B117.4-135.7%7D%7B11.4566%7D%29%5C%5C%5C%5C%3DP%28Z%3E-1.5973%29%5C%5C%5C%5C%3D1-0.05480%20%5C%5C%5C%5C%3D0.9452)
Hence, the probability of a random sample's mean being greater than 117.4 is 0.9452