Answer: To generate 50 smartphone battery life simulations in Excel with a seed of 1, we can use the following steps:
Open a new Excel worksheet and click on the "Data" tab.
Click on "Data Analysis" in the Analysis group.
Select "Random Number Generation" and click OK.
Set the "Number of Variables" to 1 and the "Number of Random Numbers" to 50.
Set the "Distribution" to "Exponential" and enter the mean of 6.5 in the "Mean" box.
Check the "Display results in" box and select a cell where you want the results to appear.
Click OK to generate the simulations.
The sample mean and standard deviation for 50 smartphone battery life simulations are shown below:
Average battery life (hours): 6.6833
Standard deviation (hours): 5.1568
To generate 500 smartphone battery life simulations in Excel with a seed of 1, we can repeat the above steps but change the "Number of Random Numbers" to 500. The sample mean and standard deviation for 500 smartphone battery life simulations are shown below:
Average battery life (hours): 6.5088
Standard deviation (hours): 6.3722
The theoretical mean and standard deviation of an exponential distribution with a mean of 6.5 hours can be calculated as:
Theoretical mean = 6.5 hours
Theoretical standard deviation = 6.5 hours
Comparing the sample mean and standard deviation to the theoretical values, we can see that the sample mean for both 50 and 500 simulations is close to the theoretical mean of 6.5 hours. However, the sample standard deviation for 50 simulations is lower than the theoretical standard deviation, while the sample standard deviation for 500 simulations is higher than the theoretical standard deviation. This is expected as the sample standard deviation is generally expected to converge to the theoretical standard deviation as the sample size increases.
Explanation: