Answer:
1a
![P(39 < X < 48 ) = 0.8767](https://tex.z-dn.net/?f=P%2839%20%3C%20%20X%20%3C%2048%20%20%29%20%3D%200.8767)
1b
95% of all sample means will fall between ![40.1 < \mu < 49.9](https://tex.z-dn.net/?f=%2040.1%20%20%3C%20%20%5Cmu%20%3C%2049.9%20)
1c
![\= x = 41. 795](https://tex.z-dn.net/?f=%5C%3D%20x%20%3D%2041.%20795)
2
![n = 25](https://tex.z-dn.net/?f=n%20%3D%20%2025)
Step-by-step explanation:
From the question we are told that
The mean is ![n = 45](https://tex.z-dn.net/?f=n%20%20%20%3D%20%2045)
The population standard deviation is ![\sigma = 10](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%2010)
The sample size is n = 16
Generally the standard error of the mean is mathematically represented as
![\sigma_{x} = \frac{ \sigma}{\sqrt{n} }](https://tex.z-dn.net/?f=%5Csigma_%7Bx%7D%20%3D%20%20%5Cfrac%7B%20%5Csigma%7D%7B%5Csqrt%7Bn%7D%20%7D)
=> ![\sigma_{x} = \frac{ 10 }{\sqrt{16 } }](https://tex.z-dn.net/?f=%5Csigma_%7Bx%7D%20%3D%20%20%5Cfrac%7B%2010%20%7D%7B%5Csqrt%7B16%20%7D%20%7D)
=> ![\sigma_{x} = 2.5](https://tex.z-dn.net/?f=%5Csigma_%7Bx%7D%20%3D%202.5)
Generally the probability that the sample mean will be between 39 and 48 minutes is
=> ![P(39 < X < 48 ) = P(-2.4 < Z< 1.2 )](https://tex.z-dn.net/?f=P%2839%20%3C%20%20X%20%3C%2048%20%20%29%20%3D%20%20P%28-2.4%20%3C%20Z%3C%201.2%20%29)
=> ![P(39 < X < 48 ) = P( Z< 1.2 ) - P(Z < -2.4)](https://tex.z-dn.net/?f=P%2839%20%3C%20%20X%20%3C%2048%20%20%29%20%3D%20%20P%28%20Z%3C%201.2%20%29%20-%20P%28Z%20%3C%20%20-2.4%29)
From the z table the area under the normal curve to the left corresponding to 1.2 and -2.4 is
=> ![P( Z< 1.2 ) = 0.88493](https://tex.z-dn.net/?f=P%28%20Z%3C%201.2%20%29%20%3D%200.88493)
and
![P( Z< - 2.4 ) = 0.0081975](https://tex.z-dn.net/?f=P%28%20Z%3C%20-%202.4%20%29%20%3D%200.0081975)
So
![P(39 < X < 48 ) = 0.88493 -0.0081975](https://tex.z-dn.net/?f=P%2839%20%3C%20%20X%20%3C%2048%20%20%29%20%3D%200.88493%20-0.0081975)
=> ![P(39 < X < 48 ) = 0.8767](https://tex.z-dn.net/?f=P%2839%20%3C%20%20X%20%3C%2048%20%20%29%20%3D%200.8767)
From the question we are told the confidence level is 95% , hence the level of significance is
![\alpha = (100 - 95 ) \%](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%28100%20-%2095%20%29%20%5C%25)
=> ![\alpha = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.05)
Generally from the normal distribution table the critical value of is
![Z_{\frac{\alpha }{2} } = 1.96](https://tex.z-dn.net/?f=Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%3D%20%201.96)
Generally the margin of error is mathematically represented as
![E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }](https://tex.z-dn.net/?f=E%20%3D%20Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%2A%20%20%5Cfrac%7B%5Csigma%20%7D%7B%5Csqrt%7Bn%7D%20%7D)
=>
=>
Generally the 95% of all sample means will fall between
![\mu -E < and \mu +E](https://tex.z-dn.net/?f=%5Cmu%20%20-E%20%3C%20%20and%20%20%20%5Cmu%20%20%20%2BE)
=> ![45 -4.9\ and \ 45 + 4.9](https://tex.z-dn.net/?f=45%20%20-4.9%5C%20%20%20and%20%5C%20%2045%20%20%2B%204.9)
Generally the value which 90% of sample means is greater than is mathematically represented
![P( \= X > \= x ) = 0.90](https://tex.z-dn.net/?f=P%28%20%5C%3D%20X%20%3E%20%20%5C%3D%20x%20%20%29%20%3D%200.90)
=> ![P( \= X > \= x ) = P( \frac{\= X - \mu }{ \sigma_x} > \frac{\= x -45 }{ 2.5} ) = 0.90](https://tex.z-dn.net/?f=P%28%20%5C%3D%20X%20%3E%20%20%5C%3D%20x%20%20%29%20%3D%20%20P%28%20%5Cfrac%7B%5C%3D%20X%20%20-%20%5Cmu%20%7D%7B%20%5Csigma_x%7D%20%3E%20%20%5Cfrac%7B%5C%3D%20x%20%20-45%20%7D%7B%202.5%7D%20%20%29%20%3D%200.90)
=> ![P( \= X > \= x ) = P( Z > z ) = 0.90](https://tex.z-dn.net/?f=P%28%20%5C%3D%20X%20%3E%20%20%5C%3D%20x%20%20%29%20%3D%20%20P%28%20Z%20%3E%20%20z%20%20%29%20%3D%200.90)
Generally from the z-table the critical value of 0.90 is
![z = -1.282](https://tex.z-dn.net/?f=z%20%3D%20-1.282)
![\frac{\= x -45 }{ 2.5} = -1.282](https://tex.z-dn.net/?f=%5Cfrac%7B%5C%3D%20x%20%20-45%20%7D%7B%202.5%7D%20%20%3D%20-1.282)
=> ![\= x = 41. 795](https://tex.z-dn.net/?f=%5C%3D%20x%20%3D%2041.%20795)
Considering question 2
Generally we are told that the standard deviation of the mean to be one fifth of the population standard deviation, this is mathematically represented as
![s = \frac{1}{5} \sigma](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B1%7D%7B5%7D%20%5Csigma)
Generally the standard deviation of the sample mean is mathematically represented as
![s = \frac{\sigma }{ \sqrt{n} }](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%20%7D%7B%20%5Csqrt%7Bn%7D%20%7D)
=> ![\frac{1}{5} \sigma = \frac{\sigma }{ \sqrt{n} }](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B5%7D%20%5Csigma%20%3D%20%5Cfrac%7B%5Csigma%20%7D%7B%20%5Csqrt%7Bn%7D%20%7D)
=> ![n = 5^2](https://tex.z-dn.net/?f=n%20%3D%20%205%5E2)
=> ![n = 25](https://tex.z-dn.net/?f=n%20%3D%20%2025)