The Bureau of Labor Statistics’ American Time Use Survey showed that the amount of time spent using a computer for leisure varied greatly by age. Individuals age 75 and over averaged 0.30 hour (18 minutes) per day using a computer for leisure. Individuals ages 15 to 19 spend 0.9 hour per day using a computer for leisure. If these times follow an exponential distribution, find the proportion of each group that spends:
4.
value:
10.00 points
a.
Less than 17 minutes per day using a computer for leisure. (Round your answers to 4 decimal places.)
b. More than two hours. (Round your answers to 4 decimal places.)
Proportion
Age 75 and over
Ages 15 to 19
c.
Between 34 minutes and 102 minutes using a computer for leisure. (Round your answers to 4 decimal places.)
d.
Find the 30th percentile. Seventy percent spend more than what amount of time? (Round your answers to 2 decimal places.)
Amount of time for individuals ages 75 and over minutes
Amount of time for individuals ages 15 to 19
Proportion
Age 75 and over
Ages
876
To solve these questions, we need to use the exponential distribution formula. The formula for an exponential distribution is:
f(x) = λe^(-λx)
where f(x) is the probability density function (pdf), λ is the rate parameter, and x is the time variable in this case.
Let's solve each part step by step:
a. To find the proportion of individuals age 75 and over who spend less than 17 minutes per day using a computer for leisure, we need to integrate the exponential distribution function from 0 to 0.3 (18 minutes in hours):
Proportion = ∫[0 to 0.3] λe^(-λx) dx
Since we don't have the rate parameter, λ, we need to find it first. We can do this by using the average time spent on the computer per day for this age group (0.3 hours):
λ = 1 / Average time = 1 / 0.3 = 3.3333
Now we can calculate the proportion:
Proportion = ∫[0 to 0.3] 3.3333e^(-3.3333x) dx
Calculating this integral will give us the answer for part a.
b. To find the proportion of individuals age 15 to 19 who spend more than two hours (120 minutes) using a computer for leisure, we need to integrate the exponential distribution function from 0.9 to infinity:
Proportion = ∫[0.9 to infinity] λe^(-λx) dx
Using the same rate parameter calculated earlier (λ = 3.3333), we can calculate the proportion by integrating the exponential function from 0.9 to infinity.
c. To find the proportion of individuals in both age groups who spend between 34 minutes and 102 minutes using a computer for leisure, we need to calculate the cumulative distribution function (CDF) and subtract the CDF values for these two time values:
Proportion = CDF(102 minutes) - CDF(34 minutes)
d. To find the 30th percentile, we need to find the time value for which the cumulative distribution function (CDF) is equal to 0.3. We can use the formula:
CDF(x) = 1 - e^(-λx)
Set this equation equal to 0.3 and solve for x.
To find the time for which 70% of individuals spend more time, we need to find the inverse of the CDF for the given proportion (1 - 0.7). In other words, we are looking for the x value for which CDF(x) = 0.3.
Once we have the rate parameter λ and the integrals or inverse CDFs, we can calculate the proportions and time values for each part of the question.
Note: Since the actual values for λ and the integrals are missing in the question, you will need to calculate them using the information provided.