Marlene has noticed that she spends a lot of time in front of a screen. She spends all day working in front of a computer screen, she spends time relaxing and watching her favourite shows in front of a TV screen and before she goes to sleep she spends time looking at things on her phone screen. Marlene thinks she should probably cut down on the time she spends on her phone before going to sleep. On any given night, Marlene believes that the amount of time she spends on her phone is uniformly distributed between 30.9 and 71.8 minutes. Assume that the amount of time Marlene spends on her phone is independent from night to night. Calculate the variance of the total amount of time that Marlene spends on her phone at night this weekend (Friday night, Saturday night and Sunday night). Do not include any units in your answer.
Question
Marlene has noticed that she spends a lot of time in front of a screen. She spends all day working in front of a computer screen, she spends time relaxing and watching her favourite shows in front of a TV screen and before she goes to sleep she spends time looking at things on her phone screen. Marlene thinks she should probably cut down on the time she spends on her phone before going to sleep. On any given night, Marlene believes that the amount of time she spends on her phone is uniformly distributed between 30.9 and 71.8 minutes. Assume that the amount of time Marlene spends on her phone is independent from night to night.
Calculate the variance of the total amount of time that Marlene spends on her phone at night this weekend (Friday night, Saturday night and Sunday night). Do not include any units in your answer.
Solution
To calculate the variance of the total amount of time that Marlene spends on her phone at night this weekend, we need to first find the mean and variance of the amount of time she spends on her phone each night.
The mean of the amount of time Marlene spends on her phone each night is the average of the minimum and maximum values of the uniform distribution, which is:
(mean) = (30.9 + 71.8) / 2 = 51.35
The variance of the amount of time Marlene spends on her phone each night is calculated using the formula for the variance of a uniform distribution, which is:
(variance) = ((max - min)^2) / 12
where max is the maximum value of the distribution and min is the minimum value of the distribution. Substituting the values we have:
(variance) = ((71.8 - 30.9)^2) / 12 = 325.04
Now, to find the variance of the total amount of time that Marlene spends on her phone at night this weekend, we need to use the fact that the variance of the sum of independent random variables is equal to the sum of their variances. Since Marlene's phone usage on each night is independent, we can add their variances to get the variance of the total amount of time she spends on her phone this weekend:
(variance of total) = (variance of Friday) + (variance of Saturday) + (variance of Sunday)
(variance of total) = 3 * 325.04 = 975.12
Therefore, the variance of the total amount of time that Marlene spends on her phone at night this weekend is 975.12.
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