Normal Distribution
Trusha's phone keeps track of her screen time. Over the past year, her screen time has followed an approximately normal distribution with a mean of 255 minutes and a standard deviation of 29 minutes. Suppose a randomly selected day from the past year is selected and Trusha's screen time from that day is recorded.
Identify and classify the random variable of interest.
Screen Time (quantitative)
Every hour, Trusha's phone randomly selects one of the photos from her "Favorites" album for her lock screen wallpaper with replacement. In Trusha's "Favorites" album, 3.8% of the photos are of her dog, Mance. For one 24-hour period, Trusha records the number of lock screen photos that are of Mance.
Define the random variable of interest and state how it is distributed.
x = # of lock screen photos that are of Trusha's dog Mance in one 24-hour period.
Binomial Distribution with n = 24 and p = 0.038
Every hour, Trusha's phone randomly selects one of the photos from her "Favorites" album for her lock screen wallpaper with replacement. In Trusha's "Favorites" album, 3.8% of the photos are of her dog, Mance. For two straight weeks (336 hours), Trusha records the number of lock screen photos that are of Mance.
Is the 10% condition met? Explain.
The 10% condition is not needed since Trusha's lock screen chooses photos from her favorites album with replacement.
Calculate the probability that Trusha's screen time on a randomly selected day is less than 4 hours.
0.302
Calculate the probability that Trusha's lock screen will display Mance as the wallpaper exactly 2 times in a 24-hour period.
0.167
Is the large counts condition met? Explain.
Yes, 336(0.038)=12.768 and 336(0.962)=323.232 are greater than or equal to 10.
Suppose five days are randomly selected. Calculate the probability that Trusha's screen time is less than 4 hours for none of the five days.
0.1657
Calculate the probability that Trusha's lock screen will display Mance as the wallpaper at most 2 times in a 24-hour period.
0.939
Why is checking the 10% condition and the large counts condition important?
We check 10% when sampling w/o replacement since one of the binomial criteria is independence (for calculating probabilities). We check large counts so that the normal approximation can be used.
Trusha’s friend Mariana has also been tracking her screen time on her phone over the past year. Her screen time is approximately normally distributed with a mean of 245 minutes and a standard deviation of 37 minutes. Suppose a day from the past year is randomly selected for Trusha and a separate day from the past year is randomly selected for Mariana. Describe the distribution of the difference in screen time for Trusha and Mariana.
Mean = 10 minutes
Standard Deviation = 47.011 minutes
Calculate the probability that Trusha's lock screen will display Mance as the wallpaper at least 2 times in a 24-hour period.
0.231
Calculate the mean and standard deviation of the distribution of number of Mance photos in 336 lock screen photos.
Mean = 12.768
SD = 3.505
Calculate the probability that the screen time for Trusha's randomly selected day is less than the screen time for Mariana's randomly selected day.
0.416
Calculate the probability that Trusha's lock screen will not display Mance as the wallpaper at all in a 24-hour period.
0.395
Calculate the probability that the number of Mance photos on Trusha's lock screen in 336 hours is more than 15.
0.262