Carl
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’s records the number of lock screen photos that are of Mance.
Define the random variable of interest and state how the variable is distributed.
RVI: X=# of lock screen photos of Mance in one 24-hour period
B(24, 0.038)
B - Mance/Not
I - sampling with replacement, could be anything
N - 24 hours & once per hour, so n=24
S - probability always 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’s records the number of lock screen photos that are of Mance.
Is the 10% condition needed? If so, is it met? Explain.
The 10% condition is not needed since Trusha's lock screen chooses photos from her favorites album with replacement.
The 10% condition is only needed for establishing independence when sampling without replacement.
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.
Calculate the probability that Trusha’s screen time on that randomly selected day is less than 4 hours.
N(255, 29); Region: left; Area: P(X<240)=0.302
OR
N(4.25, 0.483), Region: left, Area: P(X<4)=0.302
MUST INCLUDE A CURVE DRAWN, LABELED, AND SHADED
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’s records the number of lock screen photos that are of Mance.
Calculate the probability that Trusha’s lock screen will display Mance as the wallpaper exactly 2 times in a 24-hour period.
B(24, 0.038)
P(X=2)=0.167
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’s records the number of lock screen photos that are of Mance.
Is the large counts condition met? Explain.
n=336 p=0.038
336(0.038) = 12.768
336(1-0.038)=323.232
Since both 12.768 and 323.232 are greater than or equal to 10, we can use a 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.
Suppose five days from the past year are randomly selected. Calculate the probability Trusha’s screen time is less than 4 hours for none of the randomly selected days.
B(5, 0.302)
P(X=0)=0.166
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’s records the number of lock screen photos that are of Mance.
Calculate the probability that Trusha’s lock screen will display Mance as the wallpaper at most 2 times in a 24-hour period.
B(24, 0.038)
P(X<=2)=0.939
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’s records the number of lock screen photos that are of Mance.
Why is checking the conditions important?
We have to check the 10% condition when sampling without replacement to verify independence.
We have to check the large counts condition to verify the use of a 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.
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.
What is the expected difference in screen time?
Bonus: what is the standard deviation of the difference?
mean difference: 10 min
stdev: 47.011 min
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’s records the number of lock screen photos that are of Mance.
Calculate the probability that Trusha’s lock screen will display Mance as the wallpaper at least 2 times in a 24-hour period.
B(24, 0.038)
P(X>=2)=0.231
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’s records the number of lock screen photos that are of Mance.
Calculate the mean and standard deviation of the distribution of number of photos of Mance in 336 lock screen photos.
mean=np=336(0.038)=12.768 photos
stdev=sqrt(np(1-p))=sqrt(336(0.038)(0.962))=3.505 photos
BONUS
Trusha: mean 255, stdev 29
Marianna: mean 245, stdev 37
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
N(10, 47.011)
P(D <0)=0.416
MUST INCLUDE A CURVE DRAWN, LABELED, AND SHADED
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’s records the number of lock screen photos that are of Mance.
Calculate the probability that Trusha’s lock screen will NOT display Mance as the wallpaper in a 24-hour period.
B(24, 0.038)
P(X=0)=0.395
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’s records the number of lock screen photos that are of Mance.
Calculate the probability that the number of Mance photos on Trusha’s lock screen in 336 hours is more than 15.
N(12.768, 3.505)
P(X>15)=0.262
MUST INCLUDE A CURVE DRAWN, LABELED, AND SHADED